Mechanical Properties of Cementitious Materials at Microscale 9811968829, 9789811968822

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Ya Wei Siming Liang Weikang Kong

Mechanical Properties of Cementitious Materials at Microscale

Mechanical Properties of Cementitious Materials at Microscale

Ya Wei · Siming Liang · Weikang Kong

Mechanical Properties of Cementitious Materials at Microscale

Ya Wei Department of Civil Engineering Tsinghua University Beijing, China

Siming Liang School of Civil Engineering Sun Yat-sen University Guangzhou, China

Weikang Kong Department of Civil Engineering Tsinghua University Beijing, China

ISBN 978-981-19-6882-2 ISBN 978-981-19-6883-9 (eBook) https://doi.org/10.1007/978-981-19-6883-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To Jiren, Jiyuan, Liangchun, Yulan, Dewei, and Xiaohong.

Preface

The measurement of micro scale mechanical properties of cementitious materials has gained interest in the last one to two decades. This book grows out of the ten years of research work by the authors in this area, with the intention to (1) provide the methods of preparing the samples for the micro scale mechanical testing, (2) address the techniques for measuring and analyzing the elastic modulus, the stiffness, and the fracture toughness of cementitious materials at micro scale by instrumented indentation, (3) describe a method for measuring and interpreting creep behavior of cementitious materials at micro scale, and (4) demonstrate the homogenization method for obtaining the mechanical properties of cementitious materials across scales. This book is designed primarily for use at the undergraduate level, but it can also serve as a guide for professionals working with cementitious materials and other composite materials. The information in this book is illustrated by large amounts of figures, tables, text, and the references, which is helpful to a wide readership in the various fields of civil engineering and materials science. Although we have cited and compiled a list of references that we believe are most relevant to the text, listing such a limited number of references does not represent the breadth and depth of this field. We sincerely apologize to those authors whose works were not cited. Finally, we would like to express our gratitude to our students and colleagues at Tsinghua University for their valuable input and collaboration. This book would not have been completed without the supports from our families; this book is dedicated to them. Beijing, China Beijing, China Guangzhou, China

Ya Wei Weikang Kong Siming Liang

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objective of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4

2

Samples Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mixing, Casting, and Curing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Casting and Curing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hydration Stoppage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Direct Drying Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Organic Solvent Exchange Method . . . . . . . . . . . . . . . . . . 2.4 Epoxy Impregnation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Grinding and Polishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Polishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Images of Polished Sample . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Surface Roughness Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Influence of Epoxy Impregnation on Characterizing Microstructure and Small-Scale Mechanical Properties by Different Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 On Quality of Optical Microscopic Images . . . . . . . . . . . 2.8.2 On Quality of BSE Images . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 On Surface Elevation Quantified by SPM . . . . . . . . . . . . . 2.8.4 On Characterizing ITZ Between Aggregate and the Paste Matrix (ITZagg-paste ) . . . . . . . . . . . . . . . . . . . 2.8.5 On Characterizing ITZ Between Residual C3 S Clinker and Hydration Product . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6 8 9 10 11 12 13 14 16 18 19 23

24 25 26 27 28 33

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3

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2.8.6 On the Small-Scale Mechanical Properties . . . . . . . . . . . 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 36 37

Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Instrumented Indentation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Development of Indentation Technique . . . . . . . . . . . . . . 3.2.2 Geometry of Indenter Probes . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Method of Indentation Test . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Indentation Scale in Multiphase Materials . . . . . . . . . . . . 3.2.5 Calculating Microscale Mechanical Properties . . . . . . . . 3.2.6 Method of Grid Indentation Test . . . . . . . . . . . . . . . . . . . . 3.2.7 Microindentation (MI) Test . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Scanning Probe Microscopy (SPM) Technique . . . . . . . . . . . . . . . 3.3.1 Development of SPM Technique . . . . . . . . . . . . . . . . . . . . 3.3.2 Test Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Continuous Stiffness Measurement Test . . . . . . . . . . . . . . . . . . . . . 3.5 Nanoscratch Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Development of Nanoscratch Technique . . . . . . . . . . . . . 3.5.2 Testing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Parameter p(d)A(d) for Different Indenter . . . . . . . . . . . . 3.5.4 Redefining Vertical Loading Rate . . . . . . . . . . . . . . . . . . . 3.5.5 Calibration of Scratching Probe . . . . . . . . . . . . . . . . . . . . . 3.5.6 Start Point of Scratching . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Scanning Electron Microscope (SEM) Test and Associated Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Development of SEM Technique . . . . . . . . . . . . . . . . . . . . 3.6.2 Principle of SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Principle of BSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Principle of EDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 X-Ray Computed Tomography (CT) Technique . . . . . . . . . . . . . . . 3.7.1 Development of X-Ray CT . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Principle of X-Ray CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Commonly Used CT Scanners . . . . . . . . . . . . . . . . . . . . . . 3.8 Mercury Intrusion Porosimetry (MIP) Technique . . . . . . . . . . . . . . 3.8.1 Development of MIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Principle of MIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Calculation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 42 42 43 45 47 53 54 56 56 57 60 63 63 63 65 69 70 70 71 71 72 73 74 75 75 76 76 79 79 80 82 83 84

Contents

4

Phase Quantification by Different Techniques . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Phase Quantification by Backscattered Electron (BSE) Mapping Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Identifying Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Phase Segmentation from BSE Images . . . . . . . . . . . . . . . 4.2.3 Quantifying Degree of Hydration . . . . . . . . . . . . . . . . . . . 4.3 Phase Quantification by Energy Dispersive Spectrometer (EDS) Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Variation of Chemical Composition by EDS Line Scanning Across the Featured Phases . . . . . . . . . . . . . . . . 4.3.2 Chemical Composition of IP and OP in OPC and Slag-Blended Systems . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Phase Quantification by Nanoindentation Technique . . . . . . . . . . . 4.4.1 Categorization and Mechanical Property of Phase by Discrete NI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Deconvolution of Grid Indentation Data . . . . . . . . . . . . . . 4.4.3 Fraction of Single Phase with Different w/cm Ratios and Slag Contents . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Phase Distribution and Mechanical Properties of CSH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Nanoindentation (NI)-Based Degree of Hydration . . . . . 4.4.6 Comparison of Different Techniques in Estimation of DOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Methods of Data Extraction from Mapping . . . . . . . . . . . 4.5.2 Mapping Images Based on Different Measuring Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Resolution of Modulus Mapping . . . . . . . . . . . . . . . . . . . . 4.5.4 Quantifying Thickness of CSH Layer . . . . . . . . . . . . . . . . 4.5.5 Quantifying Thickness of ITZ Between C3 S and Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Phase Quantification by Nanoscratch Technique . . . . . . . . . . . . . . 4.6.1 Scratching Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Phase Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Probability Distribution of Fracture Toughness . . . . . . . . 4.6.4 Fracture Toughness of Individual Phases . . . . . . . . . . . . . 4.6.5 Thickness of Clinker and Hydrates Assessed from Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.6 Thickness of ITZ Assessed from Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Phase Quantification by X-Ray Computed Tomography . . . . . . . . 4.7.1 Reducing Background Noise of Original CT Image by Image Filtering . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

91 91 92 92 94 94 96 96 98 99 99 101 101 102 105 105 106 107 109 111 111 113 114 115 116 120 122 124 125 128 128

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4.7.2

Determining Gray Level Range of Phases for Threshold Segmentation . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Determining Phase Boundary Using Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 3D Reconstruction of Microstructure . . . . . . . . . . . . . . . . 4.7.5 Calculation of Degree of Hydration Based on CT Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Porosity Characterization and Permeability Prediction of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Pore Characteristics Quantified from Different Techniques . . . . . 5.2.1 Pore Characteristics from MIP Technique . . . . . . . . . . . . 5.2.2 Pore Characteristics from BSE Technique . . . . . . . . . . . . 5.2.3 Pore Characteristics from X-Ray CT Technique . . . . . . . 5.2.4 Comparison of Pore Size Distribution Quantified by Different Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Calculated Permeability Based on Katz-Thompson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Katz-Thompson Equation Theory for Permeability . . . . 5.3.2 Assessment of Katz-Thompson Equation . . . . . . . . . . . . . 5.3.3 Determination of L c and L max . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Results of Permeability Prediction from the Katz-Thompson Equation . . . . . . . . . . . . . . . . . . 5.4 Calculated Permeability Based on GEM Method . . . . . . . . . . . . . . 5.4.1 General Effective Media (GEM) Theory for Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Calculation for k h and k l . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Results of Permeability Prediction from the GEM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Computed Permeability Based on Navier–Stokes Method . . . . . . 5.5.1 Navier–Stokes Theory for Permeability . . . . . . . . . . . . . . 5.5.2 Six Conditions of Space Step Direction . . . . . . . . . . . . . . 5.5.3 Process of Calculating Permeability Based on Navier–Stokes Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Brief Introduction of Stokes Permeability Solver . . . . . . 5.5.5 Results of Permeability Prediction from the Navier–Stokes Method . . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130 133 136 138 140 141 145 145 146 146 151 156 158 160 160 162 162 163 167 167 168 170 172 172 174 179 179 180 183 184

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6

7

xiii

Testing and Analysis of Micro Elastic Properties . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Cyclic Loading Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Wave Propagation Method . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Dielectric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Resonance-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Small-Scale Mechanical Properties from Indentation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Least-Square Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . 6.3.3 Clustering Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Discrete Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Dynamic Method Based on Indentation Technique . . . . . . . . . . . . 6.4.1 Pixel Spacing Effect on Storage Modulus . . . . . . . . . . . . 6.4.2 Effect of Surface Roughness on Storage Modulus . . . . . 6.4.3 Effect of Magnitude of Applied Load: Quasi-Static and Dynamic Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Continuous Stiffness Measurement (CSM) of Homogeneous Mechanical Properties at Small-Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Analysis of Small-Scale Mechanical Properties . . . . . . . . . . . . . . . 6.5.1 Influence of Indentation Load . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Influence of Loading Rate/Loading Time . . . . . . . . . . . . . 6.5.3 Influence of Holding Time . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Influence of Unloading Time . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Influence of Mixture Proportion of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.6 Influence of Curing Age of Cementitious Materials . . . . 6.5.7 Influence of Loading Strain Rate . . . . . . . . . . . . . . . . . . . . 6.5.8 Comparison of Modulus Measured by Quasi-Static and Dynamic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.9 Comparison of Elastic Modulus Measured by NI and Macroscopic Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189

Testing and Analysis of Micro Fracture Properties . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Lawn–Evans–Marshall (LEM) Method by Indentation . . . . . . . . . 7.2.1 Typical Crack Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 K C for Different Crack Patterns . . . . . . . . . . . . . . . . . . . . 7.2.3 Typical Studies of LEM Method . . . . . . . . . . . . . . . . . . . . 7.3 Energy-Based Method by Indentation . . . . . . . . . . . . . . . . . . . . . . .

235 235 236 237 239 241 243

190 190 193 195 196 198 199 202 204 206 208 208 208 211

213 216 216 217 219 220 221 222 223 225 226 229 229

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Contents

7.3.1

Energy-Based Method with an Excursion in Load–Displacement Curve . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Energy-Based Method with No Excursion in Load–Displacement Curve . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Improved Energy-Based Method with No Excursion in Load–Displacement Curve . . . . . . . . . . . . . . 7.3.4 Typical Studies of Energy-Based Method . . . . . . . . . . . . . 7.4 Scratch Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Fracture Properties Across Scale for Cementitious Materials . . . . 7.5.1 Development of Macro Fracture Toughness and Fracture Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Development of Micro Fracture Properties of Unreacted Clinker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Development of Micro Fracture Properties of Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Development of Micro Fracture Properties of ITZ . . . . . 7.5.5 Recovery of Fracture Toughness During Later Curing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.6 Effect of Frost Attack on Fracture Properties of CSH and CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Effect of Nano Materials on Macro Fracture Properties of Paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Effect of Nanomaterials on Micro Fracture Properties of Clinker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Effect of Nanomaterials on Micro Fracture Properties of Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Effect of Nanomaterials on Micro Fracture Properties of ITZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 Effect of Nanomaterials on ITZ Thickness . . . . . . . . . . . . 7.6.6 Strengthening Mechanism of Fracture Properties by Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.7 Overall Strengthening Effectiveness by Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Comparison of Fracture Properties Between Micro and Macro Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Testing and Analysis of Micro Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Creep Test of Cementitious Materials at Macro Scale . . . . . . . . . . 8.2.1 Uniaxial Compressive Creep Test . . . . . . . . . . . . . . . . . . . 8.2.2 Uniaxial Tensile Creep Test . . . . . . . . . . . . . . . . . . . . . . . .

245 246 248 251 257 259 260 262 265 267 269 271 272 272 275 276 278 280 282 284 284 287 288 293 293 294 295 296

Contents

8.2.3 Flexural Creep Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Restrained Creep Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Multiaxial Creep Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Quantifying Micro Creep and Creep Recovery of Cementitious Materials Based on Indentation Technique . . . . . 8.4 Creep Property of Calcium Silicate Hydrate by NI . . . . . . . . . . . . 8.5 Influence of Experimental Protocol . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Influence of Indentation Load . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Influence of Loading Rate/Loading Time . . . . . . . . . . . . . 8.5.3 Influence of Holding Time . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Influence of the Experimental Protocols of MI Test on the Long-Term Creep Rate of Cement Paste . . . . 8.5.5 Influence of Loading Strain Rate . . . . . . . . . . . . . . . . . . . . 8.6 Creep Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Ratio of Recoverable Creep to the Total Creep of Cement Paste at Micro Scale and Its Comparison to the Macroscopic One . . . . . . . . . . . . . . . . 8.6.2 Modeling of Recovery of Indentation Depth of Cement Paste at Micro Scale . . . . . . . . . . . . . . . . . . . . . 8.6.3 Creep Recovery Rate of Cement Paste at Micro Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Influence of Mixture and Environment . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Influence of Mixture Proportion of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Influence of Microstructures of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Influence of Curing Age of Cementitious Materials . . . . 8.7.4 Influence of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.5 Influence of Relative Humidity . . . . . . . . . . . . . . . . . . . . . 8.8 Implications of Logarithmic-Type Creep Development of Cement Pastes at Micro Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Multiscale Prediction of Elastic Modulus of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Multiscale Model of Cementitious Materials . . . . . . . . . . . . . . . . . 9.3 Determination of Volume Fraction of Individual Phase at Different Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Volume Fraction at CSH Scale . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Volume Fraction at Cement Paste Scale . . . . . . . . . . . . . . 9.3.3 Volume Fraction at Mortar Scale . . . . . . . . . . . . . . . . . . . . 9.3.4 Volume Fraction at Concrete Scale . . . . . . . . . . . . . . . . . . 9.4 Perfect Versus Imperfect Interface Conditions . . . . . . . . . . . . . . . .

xv

297 298 300 301 304 306 306 310 313 314 316 319

319 324 324 327 327 333 338 341 342 345 349 350 355 355 356 361 364 366 369 370 371

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Contents

9.5

Micromechanics-Based Homogenization Method . . . . . . . . . . . . . 9.5.1 Mori-Tanaka Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Self-consistent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Generalized Self-consistent Method . . . . . . . . . . . . . . . . . 9.5.4 Differential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Micromechanics-Based Homogenization Method Considering Interface Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Prediction of Elastic Modulus at Cement Paste Scale . . . . . . . . . . 9.7.1 Homogenized Results by Different Methods . . . . . . . . . . 9.7.2 Effect of the Clinker Modulus . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Effect of the Fractions of LD CSH and HD CSH . . . . . . 9.7.4 Effect of the Effective Modulus of Pore . . . . . . . . . . . . . . 9.8 Prediction of Elastic Modulus at Mortar Scale . . . . . . . . . . . . . . . . 9.8.1 Effect of Volume Fraction of Sand . . . . . . . . . . . . . . . . . . . 9.8.2 Effect of Water on the Development of Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Prediction of Elastic Modulus at Concrete Scale . . . . . . . . . . . . . . 9.9.1 Effect of Water-To-Cement Ratio . . . . . . . . . . . . . . . . . . . 9.9.2 Effect of Curing Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.3 Effect of Coarse Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Effect of Imperfect Interface on Homogenized Elastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.1 Effect of Imperfect Interface on Elastic Modulus at Cement Paste Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.2 Effect of Imperfect Interface on Elastic Modulus at Mortar Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.3 Effect of Imperfect Interface on Elastic Modulus at Concrete Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.4 Sensitivity of Interface Property to Elastic Modulus . . . . 9.10.5 Applicability of Homogenization Method Considering Perfect and Imperfect Interface . . . . . . . . . . 9.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Multiscale Prediction of Creep Property of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Numerical Methods for Creep Prediction . . . . . . . . . . . . . . . . . . . . 10.2.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Fast Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . 10.3 Micromechanics-Based Homogenization Methods for Creep Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Laplace-Carson Transformation . . . . . . . . . . . . . . . . . . . . . 10.3.2 Elastic–Viscoelastic Correspondence Principle . . . . . . . . 10.3.3 Mori–Tanaka Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 377 379 379 381 382 385 385 388 388 390 391 391 391 392 393 394 394 396 396 398 399 401 404 405 406 411 411 412 412 422 426 426 428 429

Contents

10.3.4 Self-consistent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Generalized Self-consistent Method . . . . . . . . . . . . . . . . . 10.3.6 Representation of Inclusion Geometry in Micromechanics-Based Homogenization . . . . . . . . . . . 10.3.7 Micromechanics-Based Homogenization Method Considering Interface Property . . . . . . . . . . . . . . . . . . . . . 10.3.8 Micromechanics-Based Homogenization of Coupled Creep-Damage of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.9 Micromechanics-Based Homogenization Method Considering Aging Creep Property . . . . . . . . . . . . . . . . . . 10.4 Prediction of Creep at Cement Paste Scale . . . . . . . . . . . . . . . . . . . 10.4.1 Homogenized Creep Property by Using Different Homogenization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Homogenized Creep Property by Considering Different Phase Geometries . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Homogenized Creep Property by Considering Imperfect Interface Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Sensitivity of Interface Property to Creep Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Homogenized Creep Property of Cement Paste Considering Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.6 Influence of Porosity on Creep Development of Cement Paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.7 Influence of Applied Stress on Creep Development of Cement Paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Prediction of Creep at Mortar and Concrete Scale . . . . . . . . . . . . . 10.5.1 Prediction of Creep Property of Mortar at Different Curing Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Prediction of Creep Property of Early-Age Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Prediction of Creep Property of Concrete Subject to Different Dryings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Prediction of Non-aging Creep Property of Fiber-Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . 10.5.5 Influence of ITZ on the Creep of Concrete . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

430 430 431 432

433 436 440 440 442 444 446 448 449 450 452 452 452 453 455 457 458 459

Abbreviations

a/c ratio AFM AFm Aft AGG BSE C2 S C3 A C3 S C4 AF CASH CDF CH CL CNTs COF CSH CSM CTOD DOH DT EDS EM EMI FEA FEM FFT FIB FT GEM GSC

Aggregate-to-cement ratio Atomic force microscope Monosulfate Ettringite Aggregate Backscattered electron Dicalcium silicate Tricalcium aluminate Tricalcium silicate Tetracalcium aluminoferrite Calcium aluminum silicate hydrate Cumulative distribution function Calcium hydroxide (or portlandite) Clinker Carbon nanotubes Coefficient of friction Calcium silicate hydrate Continuous stiffness measurement Crack tip opening displacement Degree of hydration Differential Energy-dispersive X-ray spectroscopy Expectation–Maximization Effective mixed interphase Finite element analysis Finite element method Fast Fourier transform Focused ion beam Freeze–thaw cycles General effective media Generalized self-consistent xix

xx

HD CSH HP IDD IP ITZ K-T LD CSH LEM LVDT MI MIP MLE MT N2 -BET NI NMR NS OP OPC PDF RAC RH RVE SC SE SEM SPM TGA TXM UHD CSH UHPC w/b ratio w/c ratio w/cm ratio X-ray CT

Abbreviations

High-density calcium silicate hydrate Hydration product Interaction direct derivative Inner product Interfacial transition zone Katz–Thompson equation Low-density calcium silicate hydrate Lawn–Evans–Marshall method Linear voltage displacement transducer Microindentation Mercury intrusion porosimetry Maximum likelihood estimation Mori–Tanaka N2 Brunauer–Emmett–Teller nitrogen gas adsorption Nanoindentation Nuclear magnetic resonance Nanosilica Outer product Ordinary Portland cement Probability density function Recycled aggregate concrete Relative humidity Representative volume element Self-consistent Secondary electron Scanning electron microscope Scanning probe microscopy Thermal gravimetric analyzer Transmission X-ray microscope Ultrahigh-density calcium silicate hydrate Ultrahigh-performance concrete Water-to-binder ratio Water-to-cement ratio Water-to-cementitious ratio X-ray computed tomography

Chapter 1

Introduction

Abstract Currently, cementitious materials have become the most widely-used civil engineering materials around the world, which have been used to construct concrete structures such as buildings, dams, tunnels, bridges, and roads, etc. Keywords Cementitious materials · Interfacial phase · Microstructures · Micromechanical properties · Multiple scales

1.1 Background Currently, cementitious materials have become the most widely-used civil engineering materials around the world, which have been used to construct concrete structures such as buildings, dams, tunnels, bridges, and roads, etc. Due to the increasing urbanization of developing countries and the growing world’s population, cementitious materials will continue to be the most produced and consumed civil engineering material, and there will be an increasing demand for cement, as shown in Fig. 1.1. After the concrete structures are put into use, they may be degraded by various factors such as mechanical loads, environmental actions, and chemical attacks, which would weaken the safety and durability of concrete structures. It is widely accepted that the macroscopic deterioration of concrete depends greatly on its microstructures and the mechanical properties of the constituted phases (Mehta & Monteiro, 2014). As typical composite materials, cementitious materials possess multiphase at different scales, the mechanical properties and the morphologies may vary significantly with the constituted phases. Moreover, the interfacial phase is normally generated between different phases, which forms the weak region and has a negative effect on the macroscopic properties of cementitious materials. To enhance the macroscopic performance of concrete materials and to conduct optimization of the mixture proportions, it is of importance to investigate the microstructures and mechanical properties of cementitious materials at different scales. The complex microstructures and the large property contrast in different phases bring difficulties to investigating the property of different phases at micro scale. For example, due to the high property contrast between different phases, it is difficult to obtain the sample with a flat and smooth surface by grinding and polishing, which is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_1

1

Fig. 1.1 Predicted cement demand to produce cementitious materials in Organization for Economic Co-operation and Development Nations (Zhang et al., 2020)

1 Introduction

Cement demand (Billion tons)

2

6 5 Other developing countries

4 3

India

2 China

1 0 1990

OECD other industrial nations 2000

2010

2020 Year

2030

2040

2050

a prerequisite for the nanoindentation test. Due to the mixing of the hydrate products, it is not easy to assess directly the morphology and the mechanical properties of the individual phase (e.g. calcium hydroxide) in the cementitious materials. Moreover, the cementitious materials will exhibit time-dependent microstructures and mechanical properties due to the hydration reaction of cement, which will also bring great challenges to the multiscale investigation of microstructures and mechanical properties. Over the past several decades, many attempts have been made to study the cementitious materials at different scales, which include the experimental and the theoretical methods. However, the multiscale features of cementitious materials are far from being well understood. The research on microstructures and properties of cementitious materials at micro scale is still a hot and tough task.

1.2 Objective of the Book The authors of this book have conducted systematic researches on the multiscale characterization of the microstructures and the properties of cementitious materials for more than 10 years. Several successful experiences concerning the testing and modeling of the microstructures and the properties of cementitious materials at different scales have been obtained, which include the sample preparation before the experiments to measure the microstructures and the properties of cementitious materials, and the important experimental details for the nano/microindentation technique, the dynamic modulus mapping technique, the nanoscratch technique, the scanning electron microscope technique, and the X-ray computed tomography technique. Although valuable findings have also been reported by other researchers, there exist limited books to summarize the related techniques and methods for the multiscale measurement and prediction of the microstructures and the properties of cementitious materials. Therefore, the authors made a comprehensive review and summarize the key points of the relevant multiscale technique and methods in this book, which

1.3 Organization of the Book

3

are expected to provide guides for the readers who are interested in these researches and promote the research development in relevant fields.

1.3 Organization of the Book Chapter 2 introduces the key points during the preparing process for mixing, casting, curing, hydration stoppage, epoxy impregnation, grinding, polishing, and storage, as well as the effects of the preparing process on the measured results at micro scale. Chapter 3 reviews the history, test principle, calculation method and test parameter setting of the techniques for measuring the microstructure and the small-scale mechanical properties of cementitious materials, which include NI/MI technique, SPM technique, nanoscratch technique, SEM technique, X-ray CT technique, and MIP technique. Chapter 4 illustrates the existing methods of identifying individual phases and quantifying their size and mechanical properties by using the advanced techniques introduced in Chapter 3. Chapter 5 summarizes the existing techniques and methods to measure the pore parameters and predict the intrinsic permeability of cementitious materials, including the MIP, BSE, X-ray CT scanning pore measuring techniques, the Katz-Thompson method, the general effective media (GEM) method, and the Navier–Stokes method for permeability prediction. Chapter 6 reviews the existing methods to measure the elastic modulus of cementitious materials at both macro and micro scales, and the measured elastic modulus of cementitious materials at different scales by the indentation technique are presented and discussed. Chapter 7 introduces the existing methods for the fracture toughness measurement of the film materials, the rock materials, and the cementitious materials at the micro scales, which include the Lawn-Evans-Marshall method, the energy method, and the nanoscratch method. The development of fracture toughness of the pure cement paste and the nano materials-modified cement pastes by the nanoscratch method is also presented. Chapter 8 aims to summarize the current research efforts concerning the testing and analysis of creep and creep recovery of cementitious materials at both macro and micro scales, and the measured creep and creep recovery of cementitious materials at different scales by the indentation technique are presented and discussed. Chapter 9 discusses the existing multiscale models to predict the elastic modulus of cementitious materials by the micromechanics-based homogenization method, including the Mori–Tanaka method, the self-consistent method, the generalized selfconsistent method, and the differential method, and the predicted elastic modulus of cementitious materials at different scales are presented and discussed. Chapter 10 illustrates the existing multiscale models to predict the creep property of cementitious materials by micromechanics-based homogenization methods and

4

1 Introduction

the predicted creep of cementitious materials at different scales are presented and discussed.

References Mehta, P.K., Monteiro, P.J.M. (2014). Concrete: microstructure, properties, and materials. McGrawHill Education. Zhang, W., Zheng, Q., Ashour, A., & Han, B. (2020). Self-healing cement concrete composites for resilient infrastructures: A review. Composites Part b: Engineering, 189, 107892.

Chapter 2

Samples Preparation

Abstract Sample preparation is the first step and the prerequisite for successfully conducting the micro scale tests. This chapter summarizes the key points during the preparing process of cementitious materials samples in terms of the procedures of mixing, casting, curing, hydration stoppage, epoxy impregnation, grinding, polishing, and storage. The major conclusions include but not limited to which the epoxy impregnation is encouraged to be used in preparing the samples for NI test, but not for MI test and for characterizing the ITZagg-paste properties by using SPM test, especially for high w/cm ratio cementitious materials. The fine grinding process should adopt less vertical force, lower grinding rate, and shorter grinding time as possible until a slight reflection phenomenon is observed, etc. The content of this chapter can serve as a detailed guidance for preparing qualified cementitious materials samples for the microstructure and the mechanical tests conducted at micro scale. Keywords Dispersion of nano materials · Epoxy impregnation · Grinding · Polishing · Surface roughness

2.1 Introduction Sample preparation is the first step for conducting the microscale test, and it is also a prerequisite for the successful testing. This chapter introduces the entire process of the cement-based sample preparation for measuring its microstructure and the smallscale mechanical properties. Considering the difficulties in the sample preparation process, the authors review and summarize the key points for mixing, casting, curing, hydration stoppage, epoxy impregnation, grinding, polishing, and storage, as well as the effects of the preparing process on the measured results at microscale. Some modifications on epoxy impregnation process for cementitious materials are made in this chapter to improve the quality of epoxy impregnation. Moreover, through the authors’ extensive attempts and the accumulated experiences, the universal methods of grinding and polishing for cementitious materials are discussed and summarized in this chapter.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_2

5

6

2 Samples Preparation

In the following chapters, the sample preparation follows the process described in this chapter if there is no specific illustration. For some special tests, if specific treatments are conducted during the sample preparation process, they will be explained in detail in the relevant chapters.

2.2 Mixing, Casting, and Curing 2.2.1 Mixing The suitable mixing machine should be selected according to the characteristics of materials to be mixed, and the selection of mixing time and mixing rate should also be varied with the materials. If the very fine particles, such as the nano-SiO2 (NS) and the carbon nanotubes (CTNs), are added to concrete, they should be first evenly dispersed in the cement paste to achieve good mechanical and durability properties. Due to the strong van der Waals force between nano particles, nanomaterials often exist in the form of aggregated bundles, which are difficult to be uniformly dispersed in the cementitious materials. And thus, defects can be formed in the cement paste and have negative impacts on the mechanical properties of concrete. Dispersion method of the nanoparticles can be categorized as the physical and the chemical methods (Krishnamoorti, 2007). The physical methods utilize the mechanical mixing and the ultrasonic dispersion to destroy the van der Waals forces between nanoparticles (Li et al., 2019). For example, Zhao et al. (2018) used ultrasonic to evenly disperse TiO2 nanoparticles in polyacrylate latex. While chemical dispersion methods utilize surfactants to graft functional groups on the surface of nanoparticles. For example, She et al. (2018) enhanced the dispersion of SiO2 nanoparticles in suspension by using polymethyl-hydrogen siloxane surfactants. Currently, surfactants such as the polyethylene glycol and the polycarboxylate are commonly used for the dispersion of nano-carbon materials (Xu et al., 2015b; Zhang et al., 2019b). In this chapter, the methods for dispersing nanomaterials followed by mixing with cementitious materials are proposed. For mixing the nano-strengthening cementitious materials, it is suggested that these nanomaterials should be first dispersed in water before mixed with the cement paste. For dispersing the carbon nanotubes or graphene, polycarboxylate surfactant should be used at a content of 20% of the mass of the carbon nanotubes or graphene (Xu et al., 2015b). The steps for dispersing and mixing carbon nanotubes, for example, are as follows: (1) Weigh the required polycarboxylate surfactant according to the mixture proportion, and dissolve it in the deionized water and stir evenly. The carbon nanotubes (CNTs) are then added to the above prepared surfactant solution and stirred fully. (2) Conduct the ultrasonic dispersion of carbon nanotubes by using ScientzII D ultrasonic cell grinder (as shown in Fig. 2.1 a) under the power of 360 W. Each dispersion time is set as 5 min, and 6 dispersions are required to achieve a uniform state. (3) Defoamer the solution by using the tributyl phosphate (2‰ of the

2.2 Mixing, Casting, and Curing

7

cement weight) after each ultrasonic dispersion process during which the bubbles can be generated in the solution. The carbon nanotubes powder and the well dispersed carbon nanotubes solution are illustrated in Fig. 2.1b and c, respectively. For the dispersion of nano-SiO2 , the weight of the nano-SiO2 particles should be determined first according to the mixture proportion. Then, the nano-SiO2 particles are added to the deionized water, and fully stirred by the mixer. The following dispersion method is the same as the steps (2) and (3) of dispersing carbon nanotubes. The nano-SiO2 powder and the well dispersed nano-SiO2 solution are illustrated in Fig. 2.2. It is worth noting that the polycarboxylate surfactant can only maintain the effective dispersion of the carbon nanotubes for several hours, therefore, the appropriate

(a)

(b)

(c)

Fig. 2.1 Photos of a Scientz-II D ultrasonic cell grinder, b carbon nanotubes powder, and c the well-dispersed carbon nanotubes solution

(a)

(b)

Fig. 2.2 Photos of a nano-SiO2 powder, and b the well-dispersed nano-SiO2 solution

8

2 Samples Preparation

(a)

(b)

Fig. 2.3 State of cement paste with w/c ratio of 0.3 containing nano-SiO2 under the condition of a low-speed mixing, and b high-speed mixing for 5 min

amount of solution should be prepared according to the actual usage and be used immediately. Due to the high water absorption of nano-SiO2 , the workability of the cement paste will be significantly reduced after being mixed with nano-SiO2 . Therefore, the cement paste and the dispersed nano-SiO2 solution should be mixed under the high-speed mixing mode. Figure 2.3 shows the nano-SiO2 cement paste with w/c ratio of 0.3 mixed at low speed (rotation speed of 140 r/min and revolution speed of 125 r/min) and high speed (rotation speed of 285 r/min and revolution speed of 62 r/min) for 5 min, respectively. It is found that the nano-SiO2 cement paste mixed at the low speed is agglomerated and looks very dry and hard, and it is difficult to be cast and compacted. However, after high-speed mixing for 5 min, the cement paste with good consistency and fluidity is obtained. This is consistent with the findings of Geiker et al. (2006). Therefore, we suggest using high-speed mixing mode to mix cement paste containing nano-SiO2 . For mixing the carbon nanotubes solution with the cement, low speed mixing should be adopted to prevent excessive bubble generated in cement paste.

2.2.2 Casting and Curing The microscale mechanical properties measurements, such as the nanoindentation (NI) and the nanoscratch tests, are usually conducted on the small-sized sample. The size of sample can be as small as with a diameter of F 25 mm, which is obtained by casting materials into a plastic tube with size of F 25 × 100 mm. For X-ray computed tomography (CT), the samples are sometimes poured in a plastic tube with a smaller size of F 5 × 10 mm. The size of the sample depends highly on the testing device, and the researchers may adjust the tube size according to their own measuring needs. It is observed that a large number of bubbles are trapped in the paste during the process of casting into tubes. After many try and error, the authors recommend laying the tube horizontally and sending the paste to the bottom of the tube by a medicine

2.3 Hydration Stoppage

9

(b)

(a) Demolding

Plastic tube

25mm

100mm

Hardened paste

(c)

(d) 20mm

Fig. 2.4 Photos of a sample before demolding, b sample after demolding, c cutting process, and d sample after cutting

spoon to reduce the bubbles generated in the paste. In addition, the paste should be poured and vibrated layer by layer to a full vibration condition. However, it should be noted that the diameter of the test tube should be carefully selected for paste to be appropriately cast and vibrated. Attention should be paid to the bleeding phenomenon, especially for the paste with higher w/c ratio (such as the paste with w/c ratio of 0.5 or higher). Although slight bleeding cannot be avoided and is considered acceptable, serious bleeding should be mitigated by continuously rotating the tube until the initial set of the cement paste. After the initial set, the sample is placed under 20 °C environment for sealed curing. The samples are then demolded upon reaching the target age of testing by gently tapping the plastic tube (Fig. 2.4). The demolded sample is then sawed into short discs with size of ϕ 25 × 20 mm from the middle part of the sample (Fig. 2.4c and d). The two ends of each sawed cylinders are grinded by using the silicon carbide paper to produce relatively parallel ends prepared for the next step epoxy impregnation.

2.3 Hydration Stoppage For some tests, such as MIP and TGA, the free water in the sample should be removed artificially before testing. For other tests, such as the NI and the nanoscratch test, hydration stoppage is not strictly necessary but is generally done, because the low relative humidity after stoppage helps to minimize carbonation.

10

2 Samples Preparation

Two methods are most commonly used to remove the free water from the pores in cementitious materials (Zhang & Scherer, 2011). One is the direct drying method, such as the oven drying and the freeze-drying method, the other one is the organic solvent exchange method, which uses ethanol, isopropanol or acetone to dissolve water in the organic solvent, and then removes them together by evaporation.

2.3.1 Direct Drying Method Oven drying Oven drying is probably the most widely used drying technique conducted under the temperature typically ranging from 60 to 105 °C at the atmospheric pressure (101 kPa). When the specimen reaches a constant mass (typically less than 0.1% of mass change per day), the drying is considered to be completed. The oven drying is easy to be conducted, but it has a strong destructive effect on the samples, including the damage to the microstructure, the loss of the chemically bound water, and the accuracy of the testing age (Scrivener et al., 2016). Table 2.1 lists the previously used drying temperature and drying time of the oven drying method. When the drying temperature is low (60 °C), the chemically bound water in the AFt and AFm can be well preserved, but it may take longer time to dry, and the hydration cannot be stopped at the desired age. When the drying temperature is high, the free water can be removed at 105 °C quickly, which can ensure the accurate control of the testing age. However, the chemically bound water in AFt and AFm may be decomposed after 24 h’ hightemperature drying (Zhang & Glasser, 2000), which will lead to the underestimation of the ratio of hydration product in TGA test and the overestimation of porosity in MIP test. Freeze-drying Freeze-drying method is to place the samples directly into the liquid nitrogen, or place the vessel that contains samples into the liquid nitrogen. Some literatures introduced the steps of freeze-drying method including the immersion time in liquid nitrogen, and the temperature and pressure of freeze drying, as shown in Table 2.2. When the samples are immersed in the liquid nitrogen, freezing of pore water is almost instantaneous, and the cement hydration is arrested immediately. After Table 2.1 Summary of oven drying method used in literatures Literature

Method

Moukwa and Aitcin (1988) Samples are dried in an oven at 105 ± 1 °C for 24 h Gallé (2001)

Samples are dried in ventilated ovens with temperature of 105 ± 1 °C at the atmospheric pressure for 7 days

Korpa and Trettin (2006)

Samples are dried in an oven at 105 ± 1 °C for 24 h

Collier et al. (2008)

Samples are dried in an oven at 60 °C for 7 days

2.3 Hydration Stoppage

11

Table 2.2 Summary of freeze-drying method used in literatures Literature

Method

Collier et al. (2008)

Samples are frozen in liquid nitrogen (−196 °C) for 2 h, and dried at a temperature of −48 °C and a vacuum of 5 Pa for 4 days

Zeng et al. (2013)

Samples are frozen in liquid nitrogen (−196 °C) for 5 min, and dried at a temperature of −56 °C and a vacuum of 8 ~ 13 Pa for 24 h

Konecny and Naqvi (1993) Samples are frozen in liquid nitrogen (−196 °C) for 15 min, and dried at a temperature of −90 °C and a vacuum of 3.3 Pa for 16 h Gallé (2001)

Samples are frozen in liquid nitrogen (−196 °C) for 5 min, and dried at a temperature of −20 °C and a vacuum of 6 Pa

Zhang and Scherer (2011)

Samples are frozen in liquid nitrogen (−196 °C), and dried at a temperature of −77.7 °C and a vacuum of 4.8 Pa for 2 days

freezing, the samples are vacuumed in a freeze dryer at a low temperature and pressure (as shown in Table 2.2). Under the vacuum environment, the water molecules sublime directly from solid ice crystals to gas without passing through the liquid state, so the capillary stresses induced by normal drying can be eliminated. After removing the vacuum environment, the samples return to its initial conditions of room temperature and atmospheric pressure, and the freeze-drying is completed. All the above direct drying methods remove only the water (whether it is free water or bound water), but not the ions in pore solution. For example, in the study of Rößler et al. (2008), when there is high concentration of K+ and SO4 2− ions in the pore solution, the precipitate (CaK2 (SO4 2− )2 · H2 O) will be formed during the drying process. With the sublimation of the free water, the precipitates are retained in the pores and critically damage the pore structure.

2.3.2 Organic Solvent Exchange Method The principle of organic solvent exchange method is to use organic solvent to dilute the water in the pores, or make the pore water dissolved in the solvent, and then the water can be evaporated together with the solvent. The organic solvent exchange method is easy to perform and has minor effect on the microstructure of cement paste. Ethanol and isopropanol are proved to be the commonly used solvents, because they almost have no effects on the hydrates, and will not cause dehydration of the hydrates (Scrivener et al., 2016). Methanol and acetone are also common organic solvents, but considered not suitable for the remove of the free water from the pores in cement paste because they can react with the hydrates and cause a complete decomposition of AFt (Khoshnazar et al., 2013). On the other hand, in the alkaline environment of cement paste, the condensation reaction of acetone may occur, and the products can hardly be removed at a lower temperature.

12

2 Samples Preparation

In summary, the direct drying method and the solvent exchange method have their own advantages and disadvantages, and are suitable for different samples. Although the direct drying method is easy to operate, it may cost much time for removing all the water, which is not conducive to ensure the precise age of early-age samples. Organic solvents seem to be more suitable for the termination of hydration. Although some solvents such as methanol and acetone can change the microstructure and composition of hydration products, ethanol and isopropanol are proved to be reliable exchange solvents to remove the free water.

2.4 Epoxy Impregnation The process of impregnating the pores of cementitious materials with epoxy has been a prevalent sample preparation method for microstructure characterization. After solidification, the impregnated epoxy can protect the pore structure in cementitious materials from damage during the grinding and polishing process. Consequently, the artificial cracks, pits, and patches generated during the grinding and polishing process can be reduced or even avoided. Based on the vacuum impregnation process proposed in previous studies (Bisschop & Van Mier, 2002; Kjellsen et al., 2003; Struble & Stutzman, 1989; Stutzman & Clifton, 1999; Wong & Buenfeld, 2006), some modifications are made in this chapter. The changes include a grinding step before impregnation (Fig. 2.5a), the face-up placement of the sample during the impregnation (Fig. 2.5b and c), and the maintenance of a 5-min period under the vacuum state. The impregnation process used in this chapter can be divided into four steps. Step 1 The surface of the disc samples to be investigated is grinded by a P120 grade silica carbide paper to increase the roughness of sample surface and to increase the amount of the epoxy to be impregnated. Prior to impregnation,

(a)

(b)

Pressure indicator Mold

Fig. 2.5 a Grinding and polishing, b the schematic of vacuum chamber and the epoxy impregnation process (Liang et al., 2020)

2.5 Grinding and Polishing Fig. 2.6 Epoxy impregnated mortar sample

13

Hardened epoxy

Mortar sample

the samples should be dried to remove the free water, because the water can interfere with the polymerization of the epoxy. Step 2 The grinded disc sample with size of F 25 × 20 mm is placed face up at the bottom of a 30 mm diameter mold, then the mold is placed in the chamber which is pumped down to 30 mbar and maintained for 5 min to evacuate the entrapped air in the open pores of the sample, as shown in Fig. 2.5b. The reason for the face-up placement is that the epoxy can easily impregnate the sample from top to bottom rather than from bottom to top. Step 3 The epoxy is fed to the mold by a plastic tube. The quantity of epoxy fed is regulated by a valve. After the top surface of the disc samples is covered with epoxy, nitrogen is let gradually into the vacuum chamber until the pressure of the chamber reaches 660 mbar, which is lasted for 45 min to let the epoxy enter the pore of concrete. Step 4 The impregnated samples are cured at the atmospheric pressure for 24 h, and the epoxy impregnated mortar sample is shown in Fig. 2.6.

2.5 Grinding and Polishing The roughness of sample surface is one of the main factors affecting the results of the small-scale mechanical tests, since the Oliver-Pharr method for micro mechanical property calculation assumes that the sample surface is ideally flat. Otherwise, the large surface roughness might destroy the self-similarity feature used in the analytical derivations for modeling the indented materials. Theoretically, the smoother the sample surface, the more accurate the microscopic mechanical test results. Therefore, the impregnated disc samples should be processed by grinding and polishing (as shown in Fig. 2.7) before the microscale mechanical test. The grinding and polishing process can be divided into three steps: rough grinding, fine grinding, and polishing. The polishing scheme might be different for different samples and the polishing instruments. In this section, a polishing scheme suitable for the ordinary cement paste and cement mortar is introduced.

14

2 Samples Preparation

Console Groove Head plate

Base plate + silicon carbide papers Fig. 2.7 Buehler automatic grinding—polishing machine

2.5.1 Grinding Figure 2.8 shows the photo of grinding. The first process is rough grinding which aims to make the height of the sample meet the requirements of the test instrument as well as to quickly remove the excessive epoxy covered on the surface of the samples. For the rough grinding process, it is recommended to use P180 grade silicon carbide paper, and the vertical force can be set as 20 N or more, the rotation rate of the head plate and the base plate can be set as 80 rpm and 120 rpm, respectively. The rotation direction of the head plate is opposite to that of the base plate, that is, the rotation rate of the sample relative to the silicon carbide paper is 200 rpm. Since rough grinding is mainly to remove the covered epoxy, the rough grinding rate of 200 rpm is applicable to all kinds of cement-based materials samples. Larger vertical force and higher relative rotation rate can accelerate the progress of rough grinding, which saves time but might cause minor damage to the sample. Due to the rapid grinding rate, more attention should be paid to ensure that the top and bottom surface of the sample are parallel. The rough grinding should be stopped to avoid wearing off the impregnated epoxy when the hardened paste surface is about to be exposed. The control of the rough grinding time relies very much on the experiences of the operators. Fig. 2.8 Photo of grinding

Loading bar Head plate Base plate + silicon carbide papers

Sample

2.5 Grinding and Polishing

15

The second process is fine grinding, which aims to ensure the sample surface a preliminary flatness. P400 and P1200 grade silicon carbide papers are recommended successively for fine grinding. To prevent the vibration generated in the fine grinding process from damaging the microstructure of paste, the fine grinding process should adopt as less vertical force, lower grinding rate, and shorter grinding time as possible, because the impregnated epoxy can be completely removed by the long-time fine grinding. The vertical load should be set to be less than 10 N. The rotation rate of the head plate and the base plate can be set to be 80 rpm and 160 rpm, respectively. The fine grinding rate is reduced by setting the rotation direction of the head plate to be the same as that of the base plate, that is, the rotation rate of the sample relative to the silicon carbide paper is 80 rpm. The fine grinding time can be set to be 30– 60 s. After completing the fine grinding process, a slight reflection phenomenon should be observed on the surface of the sample. Otherwise, the sample should be abandoned and replaced by a new one for regrinding, because the depth of the epoxy impregnation may be not sufficient for a new grinding process. The fine grinding step is essential, as the polishing will not be successful if the fine grinding step is skipped. Moreover, the grade of the silicon carbide paper used in fine grinding should be increased gradually. The complete grinding process is recommended in Table 2.3. Experiences show that if the P1200 grade silicon carbide paper is directly used for fine grinding while skipping the P400 grade, the sample surface cannot show reflection after polishing, indicating the failure of grinding and polishing. Table 2.3 The steps and methods of grinding process Grinding process

Type of silicon carbide paper

Vertical load

Rotation rate

Time

Rough grinding

P180 grade

≥ 20 N

80 rpm for head plate and 160 rpm for base plate (opposite rotation direction)

Depend on the thickness of epoxy covered on the paste surface

Fine grinding

P400 grade

≤ 10 N

80 rpm for head plate and 160 rpm for base plate (same rotation direction)

30 ~ 60 s

P1200 grade

5 N

80 rpm for head plate and 160 rpm for base plate (same rotation direction)

Less than 30 s

16

2 Samples Preparation

2.5.2 Polishing Figure 2.9 shows the photo of the polishing process, which aims to remove the scratches left by the grinding process to achieve a smoother sample surface required by the micro experimental tests. The polishing pastes containing diamond with particle size of 9, 3, 1 and 0.25 µm are suggested to be used successively. Special polishing plate should be used for each polishing paste to prevent the blending of different particle sized paste. The optimum protocol of polishing varies from sample to sample and from machine to machine, so only general guidelines can be given. The rotation rate of the head plate and the base plate in polishing process should be greater than that in grinding process, so as to prevent diamond grain from generating new scratches on the sample surface. The rotation rate of the head plate and the base plate can be set as 120 rpm and 240 rpm, respectively. The rotation direction of the head plate is opposite to that of the base plate, that is, the rotation rate of the sample relative to the polishing plate is 360 rpm. No matter which polishing paste is used, the rotation rate of the head and the plate need not to be changed during the entire polishing process, because the rotation rate is not the main factor affecting the polishing once it is determined appropriately. During the polishing process, the polishing paste should be sprayed continuously to keep the polishing plate wet. If the polishing plate is dry, the increase of friction on the sample surface will result in the failure of polishing. The type of samples and the size of the diamond grain in polishing paste will affect the polishing time and the vertical load applied during polishing. The load applied is typically 15 ~ 25 N for pastes but can be 40 N for mortars and concretes. When decreasing the diamond grain size, the load is usually slightly increased. Meanwhile, it is best to go through the progressively smaller diamond grains with the increasing polishing times. The polishing time by using 9 µm particle sized polishing paste should not be too long and the polishing vertical load should not be too large (15 N is enough), because the 9 µm particle sized polishing paste can also wear the sample surface. When using the polishing paste with smaller particle size, the polishing time should last longer and the vertical load can be slightly larger. The appropriate polishing method is recommended in Table 2.4. However, for most

Sample + polishing pastes Head plate Base plate + polishing plate

Fig. 2.9 Photo of polishing

2.5 Grinding and Polishing

17

Table 2.4 The steps and methods of polishing process Type of polishing paste (µm)

Rotation rate

9

120 rpm for head plate and 15 240 rpm for base plate 30 ~ 60 (opposite rotation 90 ~ 120 direction) 120 ~ 180

3 1 0.25

Polishing time (min)

Vertical load (N) 15 20 25 25

conditions, 0.25 µm particle sized polishing paste will not improve the smoothness of the sample surface, so normally it is not used during the polishing process. After polishing, ethanol should be used to clean the sample in a ultrasonic cleaning machine to remove the residuals left on the sample surface. Most importantly that the water should not be used to clean the sample at any time, because it will cause further hydration on the surface of the sample. The polishing plate can be cleaned by the pure water, but it should be fully dried before the next use. Due to the multiphase feature of the cement-based materials and the difference of the physical properties between each phase, the probability of successful polishing is very low (generally less than 30% by the authors’ experience). Therefore, it is necessary to be patient during the polishing process. Meanwhile, due to the different mixture proportions of the cement-based materials, the rotation rate, the polishing time, the magnitude of the vertical load, and other parameters used in the polishing process should be adjusted accordingly, which requires a lot of operations to accumulate experiences. Failed and successfully polished surfaces are shown in Fig. 2.10, respectively. For the well-polished sample, the surface can show good reflection. Experienced researchers can even judge whether the roughness meets the microscale test requirement by the degree of reflection. At present, there is no standardized method for surface preparation of cementitious materials. The methods adopted by different researchers are quite different, which are summarized in Table 2.5. (a)

(b)

Fig. 2.10 Sample with a failed polished surface, and b well-polished surface judged by the reflection of sample surface

18

2 Samples Preparation

Table 2.5 Summary of the surface preparation of cementitious materials for mechanical test at micro scale Authors

Material

Grinding

Polishing

Experimental parameter

Mouret et al. (2001)

Paste/mortar

silicon carbide with grit from 30 to 4 µm

1 µm particle sized polishing paste

cleaned in an ultrasonic bath filled with alcohol

Kjellsen et al. (2003)

Paste

diamond discs with grit of 90, 30, and 15 µm

6, 3, 1, and 0.25 µm particle sized polishing paste

moderate force and a low rotation speed applied for polishing

Wong & Buenfeld (2006)

Mortar

P120, P220, P500, P1000, and P1200 silicon carbide papers

9, 6, 3, 1 and 0.25 µm particle sized polishing paste

70 rpm and 7 N applied force, polishing time was about 5 min, and cleaned ultrasonically in acetone

Mondal et al. (2007)

Paste

silicon carbide papers with grit of 6.5 µm

0.1 µm particle sized polishing paste

–

Miller et al. (2008)

Paste

P120 abrasive paper

1 µm particle sized polishing paste

60 rpm, polishing time was 8 h

Sakulich & Li (2011)

PVA reinforced mortar

P1200 grinding paper

15, 9, 6, 3, 1, and 0.5 µm particle sized polishing paste

15 N force, 150 rpm complimentary rotation

Wei et al. (2017a) Paste

P180, P400, and P1200 silicon carbide papers

9, 3, and 1 µm particle sized polishing paste

cleaned in an ultrasonic bath filled with alcohol

Zhang et al. (2019a)

P180, P240, P400, P600, P800, and P1200 silicon carbide paper

6, 3, 1, and 0.25 µm particle sized polishing paste

soaked into an ultrasonic bath between each polishing step

Paste

2.5.3 Images of Polished Sample Figure 2.11 shows the optical microscope images of the impregnated concrete sample before and after grinding and polishing. It is found that the grinding process helps to grind off the epoxy covered on the sample surface (Fig. 2.11b), as the blurring caused by epoxy (Fig. 2.11a) is removed. The polishing process helps to obtain the smooth surface (Fig. 2.11c) in which the hydration products and the unreacted clinker can be observed very clearly under the microscope.

2.6 Surface Roughness Examination (a

Coarse aggregate

19 (c

(b Unreacted clinker

Fine aggregate

Hydration product

Coarse aggregate

500μm

500μm

500μm

Fig. 2.11 Optical images of concrete sample after a epoxy impregnation, b grinding, and c polishing process (Liang et al., 2020)

(a)

Top surface

(c)

(b)

Bottom surface

5mm

100μm

Fig. 2.12 a The schematic of the vertical section of concrete sample, b the optical image of the vertical section, and c the BSE image of the vertical section near the top surface of the impregnated concrete sample (Liang et al., 2020)

The depth of epoxy impregnation was quantified after all the property characterizing is completed. The impregnated sample was cut open to examine the impregnation depth, as shown in Fig. 2.12. Unfortunately, the impregnated portion or the impregnated depth is not distinguishable from either the optical microscope image or the SEM image. Nevertheless, the epoxy is believed to be impregnated into the samples, which is proved from the remarkable differences in the morphology, microstructure, and mechanical properties between the impregnated and the non-impregnated samples discussed in Sect. 2.8.

2.6 Surface Roughness Examination Miller et al. (2008) conducted a systematic experimental study to develop a criterion for surface roughness of cement paste for NI test. Figure 2.13 shows the AFM images of the polished surface during different stages of polishing and a photograph of the sample showing the final polished surface. It can be seen that the surface roughness will be improved as the polishing time increases. To quantify the surface roughness of the cementitious material, Miller et al. (2008) introduces the root-mean-squared average (RMS) of the topography of the surface, which can be expressed as Eq. (2.1). Figure 2.14 shows the variation of surface

20

2 Samples Preparation

(a) rough ground sample surface

(d) after 4 h of polishing

(b) after 1 h of polishing

(c) after 2 h of polishing

(e) after 8 h of polishing

(f) sample after 8 h of polishing

Fig. 2.13 AFM images of different stages of polishing process and a photograph of the sample showing the final polished surface (Miller et al., 2008)

roughness of the sample with the polishing time for the three different AFM scanned sizes. It can be seen that after 2–4 h of polishing, there is little difference in roughness values. Moreover, the roughness values at different sampling sizes are drastically different. Therefore, it is suggested that the RMS roughness value must be reported along with the scanned size in order to compare different roughness conditions. [ | | Rq = ]

M N 1 Σ Σ 2 z N · M i=1 j=1 i j

(2.1)

where, N and M are the number of pixels on the two edges of the scanned rectangular area, and zij is the height at position (i, j). The previous studies (Bobji & Biswas, 1999; Kim et al., 2006) on roughness and NI experiments show that the increase of roughness will result in the lower estimation of test results and the increased dispersion of the results. Figure 2.15 displays the relationship between the NI deconvolution results (indentation modulus, contact hardness, and volume fraction of each phase) and the surface roughness of the three 400 Rq (nm)

Fig. 2.14 Variation of surface roughness of sample with polishing time for three different AFM scanned sizes (Miller et al., 2008)

10μm×10μm 50μm×50μm 2μm×2μm

300 200 100 0 0

2

4 6 Polishing time (h)

8

10

21 0.8

200 Volume fraction

Indentation modulus (GPa)

2.6 Surface Roughness Examination

150 100 50

0.4 0.2 0

0 0

Hardness (GPa)

0.6

50

100 Rq (nm)

150

0

200

12 10 8 6 4 2 0

50

100 Rq (nm)

150

200

High density CSH Ultra-high density CSH Clinker 0

50

100 Rq (nm)

150

200

Fig. 2.15 Average indentation modulus, hardness, and volume fraction vs. surface roughness of high density CSH, ultra-high density CSH and clinker (Miller et al., 2008)

phases (high density CSH, ultra-high density CSH, and clinker) in cement paste with different surface roughness. It can be seen that the measured indentation modulus, contact hardness, and volume fraction of each phase will not change significantly when the surface roughness is below about 100 nm which was achieved after 2 h of polishing. Based on the experimental results, Miller et al. (2008) recommended that the root-mean-squared roughness (Rq ) of the sample surface should be less than 0.2 hmax in an area with the AFM image edge size of 200 hmax , (where hmax is the maximum indentation depth), which have been adopted by many researchers (Wei et al., 2017a; Xiao et al., 2013). Normally, when testing the mechanical properties of single phase in cement-based materials, a vertical load of less than 4 mN would be applied on the surface of the sample in the NI tests (the selection of the load will be described in detail in Chap. 3), and the indentation depth is generally within the range of 80–400 nm which can be used to estimate the surface roughness required for small-scale indentation test of each individual phase. A large number of existing literatures have provided the surface roughness that can be adopted, as summarized in Table 2.6. The requirements for the surface roughness of the samples in the nanoscratch test can refer to that in the NI test. The maximum vertical load of less than 14 mN was adopted in the nanoscratch test (Wei et al., 2021a), and the maximum scratch depth (more than 500 nm) is much greater than the maximum indentation depth (about 200 nm) in NI test. Therefore, the roughness requirements for NI test fully meet the requirements for nanoscratch test. The surface roughness of samples available for nanoscratch test in the literatures is summarized in Table 2.6, and Fig. 2.16a shows a sample with surface roughness meets the nanoscratch requirement.

22

2 Samples Preparation

Table 2.6 Summary of the surface roughness of cementitious materials for different testing techniques Literature

Technique

Materials

Properties

Rq (nm)

Area (µm)

Miller et al. (2008)

NI

Cement paste

Hardness and indentation modulus

20

50 × 50

Long et al. (2018)

Indentation modulus, hardness, and elastic modulus

< 100

100 × 100

Xu et al. (2015a)

Indentation modulus and 59.4 hardness

40 × 40

Wei et al. (2016)

Hardness and indentation modulus

80

50 × 50

Slag-blended cement paste

Hardness and indentation modulus

40

35 × 35

Cement paste + Nano-SiO2

Friction force, scratch depth, and coefficient of friction

23.5

20 × 20

Wei et al. (2021a)

Cement paste

Friction force and fracture toughness

47

40 × 40

Kong et al. (2021)

Cement paste

Friction force, coefficient of friction, and fracture toughness

39

40 × 40

Wei et al. (2021b)

Cement paste + nano-SiO2 /carbon nanotube

Friction force and fracture toughness

57/47/44

40 × 40

Cement paste

Storage modulus and loss modulus

Wei et al. (2018a) Xu et al. (2017)

Xu et al. (2015a)

Nanoscratch

SPM

Li et al. (2015)

59.4

40 × 40

93.9

50 × 50

Wei et al. (2017b)

38.2

35 × 35

Wei et al. (2018b)

13

35 × 35

(a)

(b)

Rq=13nm

Fig. 2.16 Surface roughness of paste sample that meets the requirement of a nanoscratch test (Wei et al., 2021a), and b SPM test (Wei et al., 2018b)

2.7 Storage

23

The in-situ continuous measurements of the surface roughness of samples by scanning probe microscopy (SPM) are more reliable than that measured in a randomly selected area. The SPM technique can measure the surface elevation as well as the storage modulus simultaneously of the target area by applying a very small highfrequency dynamic load (for example 4 ± 3.5 µN in Wei et al., 2017b and 2018a). The test principle will be described in detail in Chap. 3. For the well-polished surface, the surface elevation varies in the range of −300 ~ 100 nm along the scanning line, and the displacement amplitude of the indenter tip is only at the scale of nanometers. This implies that the displacement amplitude is two orders of magnitude less than the differences of the elevation of the sample surface. Therefore, Gao et al. (2018) and Kim et al. (2006) believed that the surface roughness of 10–200 nm has a minor effect on the stability of the measured storage modulus. The surface roughness of samples available for SPM experiment in the literature is summarized in Table 2.6, and Fig. 2.16b shows a cement paste sample with surface roughness meets the SPM test requirement.

2.7 Storage The storage of the polished sample before microscale testing is very crucial. The sample should be stored in a vacuum environment, because the water and carbon dioxide in the air will cause carbonation on the polished sample surface. Carbonation occurs mainly at a relative humidity of between 50 and 80% (Mmusi et al. 2009). However, Yang et al. (2003) observed carbonation even under the 30% relative humidity. The comparison between the carbonized and the non-carbonized surfaces of the polished samples is shown in Fig. 2.17. The polished sample shown in Fig. 2.17a was stored at room temperature and exposed to the air for about 12 h (carbonation was caused by accident, and the temperature and relative humidity were not recorded), and serious carbonation occurred on the surface. The polished sample shown in Fig. 2.17b was stored in a vacuum storage box (as shown in Fig. 2.18) at room temperature for 3 days, and no carbonation occurred on its surface. Although one vacuum pumping can ensure the vacuum environment for 2–3 days, it is recommended to vacuum the storage box every day. If the samples need to be preserved for a long time, it is recommended to create a more professional vacuum environment. Meanwhile, the samples should be placed back to the storage box immediately after the microscale test to reduce the chances of carbonation. However, the carbonation during the tests is basically unavoidable. On the other hand, the sample surface cannot be touched directly by one’s bare hand, a pair of clean disposable medical gloves should always be worn when taking and placing the sample.

24

2 Samples Preparation

(a)

(b)

50 μm

50 μm

Fig. 2.17 SEM images of surface of a a carbonized cement paste sample, and b a non-carbonized cement paste sample

Fig. 2.18 Photo of the vacuum storage box and the stored samples

Vacuum storage box

Samples

2.8 Influence of Epoxy Impregnation on Characterizing Microstructure and Small-Scale Mechanical Properties by Different Techniques Vacuum impregnation process has been reported to be important and necessary to characterize the microstructures and the small-scale mechanical properties in a few studies (Bisschop & Van Mier, 2002; Kjellsen et al., 2003; Wong & Buenfeld, 2006). It has been proposed by Bisschop and Van Mier (2002) that the cracking induced by sample preparation for the SEM imaging can be reduced efficiently by the epoxy impregnation. Wong and Buenfeld (2006) concluded that the artefact generated in the sample preparation can be reduced by using the epoxy impregnation. It has also been reported by Kjellsen et al. (2003) that the microstructure can be stabilized by the epoxy and thus can withstand the external stresses from the grinding and polishing. Moreover, the gray level contrast between the solid phases and the pores in the BSE images can be enhanced by epoxy saturation (Struble & Stutzman, 1989). Up to now, the benefits of the epoxy impregnation on improving the quality of the microscopic images have been well studied. However, the influence of epoxy

2.8 Influence of Epoxy Impregnation …

25

impregnation on other quantified features including the surface elevation, the thickness of the interfacial transition zone (ITZ), and the mechanical properties at the micro and the sub-micro scales is still unclear for cementitious materials. Limited studies have conducted to measure the indentation modulus (Zhu et al., 2009) and the thickness of ITZ (Yio et al., 2014) of the samples with epoxy impregnation, but the influence of the epoxy impregnation on these properties was not analyzed. This section evaluates the effect of epoxy impregnation on the quality of the optical microscopic images and the BSE images, the surface elevation of the polished sample, the thickness of the ITZ around the aggregate or the residual clinker, and the mechanical properties of cement paste at the micro and the sub-micro scales. The results of this section are expected to promote the utilization of the epoxy impregnation in preparing the cementitious materials samples.

2.8.1 On Quality of Optical Microscopic Images The identification of the individual phases in the sample depends greatly on the quality of the optical microscope image. Figure 2.19 shows the similar areas covering both coarse aggregate and the paste matrix, which are selected from the non-impregnated and the impregnated polished concrete samples, respectively. The imaging quality is highly determined by the flatness of the polished sample surface. All individual phases including the aggregate, the residual clinker, and the hydration product can be exhibited clearly when the sample surface is flat enough. However, an unsatisfactory image may be obtained due to different elevations of different phases which prevents the optical microscope to focus on a proper plane. An optical microscope image of the non-impregnated 3C (concrete with w/c ratio of 0.3) sample is shown in Fig. 2.19a. It is seen that the aggregates raise from the paste matrix, and gaps will exist between the aggregates and the paste matrix. The reason lies in that the paste matrix is more porous than the aggregates, and thus the paste matrix is etched and the coarse aggregates are left after the polishing process. Besides, the paste matrix itself is blurry and the residual clinkers cannot be identified (a) Coarse Aggregate

(b)

Fine aggregate

Coarse Aggregate

Clinker Fine aggregate

100μm

Clinker

100μm

Fig. 2.19 Optical microscope images of a the non-impregnated, and b the impregnated concrete sample with w/c ratio of 0.3 (hereinafter referred to as 3C) after polishing (Liang et al., 2020)

26

2 Samples Preparation

from Fig. 2.19a. However, as shown in Fig. 2.19b, the impregnated epoxy can support the pore structure in the paste matrix of the impregnated sample, which contributes to resisting the etching during the polishing process, and thus a clear image was obtained.

2.8.2 On Quality of BSE Images It has been shown in the above section that the quality of the optical microscope image can be improved by the epoxy impregnation. However, the surface flatness does not affect the quality of BSE images significantly due to the fact that the different phases such as aggregate, clinker, hydration product, and the pore are identified according to their different gray levels in the BSE images. The local mean atomic number is the main factor that affects the local gray level in BSE images. The residual clinker will appear bright gray due to its greatest atomic number, while the hydration product appears dark gray and the pore appears black. As shown in the BSE image of the non-impregnated 3C sample (Fig. 2.20a), the details can be observed more clearly than that in the optical microscopic image (Fig. 2.19a). For instance, the crack within the cement paste and the gap between the aggregate and the paste matrix are clearly seen in the BSE image. However, no cracks and gaps are seen in the impregnated 3C sample as shown in Fig. 2.20b. Besides, the impregnated sample seems to contain more pores than that of the non-impregnated sample. Struble and Stutzman (1989) and Stutzman and Clifton (1999) also reported that the volume fraction of pores in concrete may be increased by the epoxy impregnation. The possible reason may be that the signal generated from the base and the side walls of the pores will make the pores invisible in the BSE image when the pores are not filled with epoxy. Therefore, the accuracy of the porosity obtained from the BSE image analysis by the gray level contrast can be improved by the epoxy impregnation. When the sample was not impregnated with epoxy, the porosity would be underestimated. (a)

Gap Clinker Fine aggregate

Crack Clinker

Coarse aggregate

(b) Coarse Aggregate

Clinker Hydration product

Hydration product 100μm

100μm

Fig. 2.20 Typical BSE images of a the non-impregnated and b the impregnated 3C sample after polishing (Liang et al., 2020)

2.8 Influence of Epoxy Impregnation …

27

2.8.3 On Surface Elevation Quantified by SPM It can be seen from the optical microscope images (Fig. 2.19) that the Δhagg-paste defined as the elevation difference between aggregate and the paste matrix is smaller in the impregnated sample than that of the non-impregnated sample. This is because the paste matrix is softer than the aggregate, the paste matrix is easier to be polished off than the aggregate, and the large Δhagg-paste is always induced in the non-impregnated sample by the polishing process. The Δhagg-paste measured by SPM is shown in Fig. 2.21. Figures 2.21a and a’ show the extracted elevation mapping within the area of 35× 35 µm for the nonimpregnated and the impregnated 3C samples, respectively. Figures 2.21b and b’ display the topographies obtained from the SPM experiments where the part with higher elevation will be denoted by the light yellow, while the dark yellow represents the part with lower elevation. Figures 2.21c and c’ display the elevation variation along the lines selected across the aggregate and the matrix in the non-impregnated and the impregnated samples. In the non-impregnated sample (Fig. 2.21c), Δhagg-paste can be up to nearly 1 µm. While Δhagg-paste is only about 0.05 µm if the epoxy is impregnated (Fig. 2.21c’). Considering that the main purposes of the sample preparation is to obtain the flat surface and thus to provide clear SPM or optical microscopic images, it is recommended to use the epoxy impregnation for sample preparation to study the microstructure and micromorphology of the concrete. Non-impregnated sample

Impregnated sample

(a)

(a')

Coarse Aggregate

(c) 20

Scanning line in (c)

Elevation (103 nm)

Crack

(b)

Rq=410nm

Coarse Aggregate

Hydration product

Hydration product

10μm (b')

Agg.

15

Clinker

(c')100

0.9μm

10

Scanning line in (c')

5 0

Paste

Elevation (nm)

Clinker

Agg.

50

0.05μm

0 -50

Paste

-100

-5 0

10 20 30 Distance (μm)

40

Rq=46nm

0

10 20 30 Distance (μm)

40

Fig. 2.21 Surface elevation of a–c the non-impregnated and (a’–c’) the impregnated 3C sample. a and a’ are BSE images; b and b’ are the surface topography of the selected area from a and a’; c and c’ are the variation of surface elevation measured by SPM (Liang et al., 2020)

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2 Samples Preparation

2.8.4 On Characterizing ITZ Between Aggregate and the Paste Matrix (ITZagg-paste ) Quantifying ITZagg-paste Thickness Based on BSE Image Analysis The microstructures of the ITZ between the aggregate and the paste matrix (ITZagg-paste ) has been reported to be different from that of the paste matrix (Diamond & Huang, 2001; Liao et al., 2004; Scrivener et al., 2004). The microstructure of ITZagg-paste varies along the distance from the aggregate until it approaches to be the same as the paste matrix (Scrivener et al., 2004), and the distance of the gradient can be taken as the ITZagg-paste thickness (Diamond & Huang, 2001; Liao et al., 2004). Scrivener et al. (2004) has proposed a method to quantify the ITZagg-paste thickness by extracting strips in BSE image starting from the boarder of aggregate towards the paste matrix. Based on the gray level analysis, the fraction of different phases can be quantified, and it is possible to determine the ITZagg-paste thickness based on the variation of clinker fraction and porosity in different strips. However, it is noted that the samples were not impregnated with epoxy for quantifying ITZagg-paste thickness in the literatures (Diamond & Huang, 2001; Liao et al., 2004). As shown in Sect. 2.8.2, the remarkable gaps and the excess artificial cracks in paste matrix can be seen from the BSE images of the non-impregnated samples, which may affect the fraction of clinkers or pores quantified based on the gray level analysis of BSE images. As a result, the ITZagg-paste thickness may be misestimated. To assess the influence of epoxy impregnation on quantifying the ITZagg-paste thickness, several strips with width of 10 µm are trimed from the BSE image starting from the aggregate boarder until 100 µm away from the aggregate boarder, which are shown in Figs. 2.22a and b. Since it has been reported that the ITZagg-paste thickness was less than 50 µm (Diamond & Huang, 2001; Liao et al., 2004; Scrivener et al., 2004), the strips with thickness of up to 100 µm from the aggregate boarder to cover greater area was analyzed for meaningful extraction of ITZagg-paste information. Image-Pro software was employed to analyze the BSE image of each strip to determine the fractions of clinker and pore phases. The gray level threshold segmentation method is used for phases identification, which will be detailed in Chapter 4. As illustrated in Fig. 2.22c, the valley bottom in the gray level histogram can be used to identify the gray level threshold of clinker, the “overflow” method proposed by Wong et al. (2006) (detailed in Sect. 4.2.2) is used to determine the gray-level threshold of pore, while the gray level range of the hydration product lies in between the thresholds of the clinker and the pore. According to the different gray level of different phases, it is possible to calculate the fraction of these phases. The ITZagg-paste thickness is taken as the range over which the fractions of clinkers and pores change gradually. Figure 2.22d shows the variation of the clinker fraction and the porosity with the distance from the aggregate boarder. The data point is calculated according to the average values of fifteen BSE images of the non-impregnated and the impregnated concrete samples with w/c ratios of 0.3 (3C) and 0.5 (5C), respectively. It is seen from

2.8 Influence of Epoxy Impregnation …

29

(c) 2.5

(a)

Pore

HP

2

Clinker

0 10 20 30 40 50 60 70 80 90 100 μm

(b)

Frequency (%)

1.5 1 Critical point

Valley bottom

0.5 0 0

0 10 20 3040 50 60 70 80 90100 μm

(d) 0.25

ITZ

0.6

Paste matrix

0.4

0.15

Porosity

Clinker fraction

3C-non-impregnated 3C-impregnated 5C-non-impregnated 5C-impregnated

0.5

0.20

0.10 0.05

50 100 150 200 250 Gray level

0.3 0.2 0.1 0.0

0.00 0

20 40 60 80 100 Distance from aggregate (μm)

0

20 40 60 80 100 Distance from aggregate (μm)

Fig. 2.22 The schematic of the extracted strips from the BSE images for quantifying phase fraction of a impregnated and b non-impregnated concrete samples with w/c ratio of 0.3 (3C), c the schematic of the segment of pore, hydration product (HP), and clinker from gray level analysis, and d the variation of clinker fraction and porosity with the distance from the aggregate border for 3C and 5C (concrete with w/c ratio of 0.5) (Liang et al., 2020)

Fig. 2.22d that the ITZagg-paste thickness is about 40 µm, the clinker fraction increases gradually while the porosity decreases gradually along the direction towards the surrounding paste from the aggregate boarder. This phenomenon in ITZagg-paste can be accounted for by the wall effect, which suggests that ITZagg-paste thickness is similar to the clinker size, and the residual clinker can hardly exist in ITZagg-paste (Scrivener et al., 2004). Besides, ITZagg-paste around the aggregate possesses a higher w/c ratio and contains less clinkers and more pores due to the local bleeding around the aggregate (Elsharief et al., 2003). The clinker fraction of the paste matrix outside ITZagg-paste is in the range of 10–15 and 5% for 3C and 5C concretes, respectively, which is in good agreement with the clinker fraction in the hardened paste sample with w/c ratio of 0.3 and 0.5 cured for 1 year (Yio et al., 2014).

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2 Samples Preparation

It can be seen from Fig. 2.22d that the quantified ITZagg-paste thickness by BSE images is about 40 µm for both 3C and 5C samples. It seems that the ITZagg-paste thickness is not influenced by the epoxy impregnation. However, the clinker fraction is lower and the porosity is higher within the ITZagg-paste in the non-impregnated sample as compared with the impregnated one, which can be explained by the significant gap between the aggregate and the surrounding paste matrix in the non-impregnated sample. The gap in the BSE images appears black and thus is treated as pore phase, which will lead to different estimations of the fractions of the individual phases in the non-impregnated and the impregnated samples for the ITZ phase. Quantifying ITZagg-paste thickness and mechanical property based on SPM modulus mapping SPM modulus mapping can be used to quantify the ITZagg-paste thickness through inspecting the variation of the storage modulus of the aggregate and the surrounding paste along the lines across a distance (95 µm in this case). The BSE image and the storage modulus mapping of the same area of the non-impregnated sample are shown in Figs. 2.23a and b, respectively. The field size of the BSE image and the storage modulus mapping is 95 × 35 µm. The variations of the storage modulus and the gray level along the lines shown in the BSE image are displayed in Figs. 2.23c, d and e, respectively. The range near the aggregate possessing smaller storage modulus is taken as ITZagg-paste . The thickness of ITZ in the non-impregnated sample shown in Figs. 2.23c, d and e is about 25 µm and the corresponding modulus is about 5–15 GPa. While the ITZagg-paste in the impregnated sample is narrower and has a higher storage modulus as compared with that of the non-impregnated one. As displayed in Figs. 2.23c’, d’ and e’, the ITZagg-paste in the impregnated sample is about 15 µm in width, and the corresponding modulus is about 20 GPa. However, Fig. 2.23 shows that the features of ITZagg-paste in terms of the gray level of different phases are less distinguishable than the storage modulus, which implies that the storage modulus may be act as a better indicator for identifying ITZagg-paste . Figure 2.24 summarizes the differences between the non-impregnated and the impregnated 3C sample in terms of the ITZagg-paste thickness, the gray level, and the storage modulus of the individual phases. The average modulus along the lines from the aggregate boarder to the paste matrix is plotted in Fig. 2.24a. The starting point of x-axis in Fig. 2.24a denotes the aggregate border. It is found that the thickness of ITZagg-paste in the non-impregnated sample identified from the SPM modulus mapping is greater than that in the impregnated sample. The reason lies in that the impregnated epoxy fills into the pores in the ITZagg-paste and supports the microstructure, which may result in the similar storage modulus of the ITZ to that of the paste matrix. The modulus images of the impregnated samples may ignore the existence of ITZagg-paste when it is compared to the non-impregnated samples. As a result, it is not encouraged to use the epoxy impregnation to characterize the ITZagg-paste thickness when the SPM modulus mapping technique is used.

2.8 Influence of Epoxy Impregnation …

31

Non-impregnated sample (a)

(c)150

Scanning line (d)

100

50

50 0

0

(a’)

Scanning line (c')

ITZ

HP CL

100

150 100

50

50 0

0 30 60 Distance (μm)

HP

90

Gray Level

90

0

SM GL

SM GL

CL 250

150 100 50

0

200

50

SM GL

CL 250 HP HP 200

50

(e’)

100

50

30 60 Distance (μm)

AGG

90

200

250

150

0

150

0

Storage modulus (GPa)

Storage modulus (GPa)

CL HP

100

0

30 60 Distance (μm)

100

(b’)

HP

0 0

Impregnated sample (c’) AGG ITZ CL 150

Scanning line (e')

AGG ITZ CL

50

0

SM GL

90

Scanning line (d')

(d’) 150

100

50

Gray Level

30 60 Distance (μm)

150

(e)

150

250

Gray Level

CL 250 HP 200

100

0

HP

0

30 60 Distance (μm)

90

AGG ITZ HP CL HP CL HP 150

0 SM GL 250 200 Gray Level

HP

Gray Level

ITZ CL

Storage modulus (GPa)

Storage modulus (GPa)

AGG

CL

200

Storage modulus (GPa)

(b)

150

ITZ HP

100

Scanning line (e)

(d)

AGG

Gray Level

Storage modulus (GPa)

Scanning line (c)

100

150 100

50

50 0

0

30 60 Distance (μm)

90

0 SM GL

Fig. 2.23 Quantification of the ITZagg-paste thickness of the non-impregnated a–e and the impregnated 3C sample a’–e’ by SPM and BSE. a, b and a’, b’ the BSE image and the SPM modulus mapping image of the target area. c–e and c’–e’ the variation of storage modulus (SM) and gray level (GL) along the scanning lines (Liang et al., 2020)

32

2 Samples Preparation

Impregnated

Storage modulus (GPa)

(a) 50 40

ITZ≈15μm

30 20 10

ITZ≈25μm

0 0

240

10

(c)

Impregnated Non-impregnated

180 120 60 0 ITZ

HP

CL

20 30 Distance (μm)

Storage modulus (GPa)

Gray level

(b)

Non-impregnated

AGG

120

40 Impregnated Non-impregnated

90 60 30 0 ITZ

HP

CL

AGG

Fig. 2.24 a Variation of the storage modulus with the distance to the aggregate, the comparison between b the gray level and c the storage modulus of individual phases in the non-impregnated and the impregnated 3C sample (Liang et al., 2020)

The average values of the gray level and the storage modulus of different phases in the impregnated and the non-impregnated samples are shown in Figs. 2.24b and c. It is seen that the BSE gray levels of the same phase change dramatically between the non-impregnated and the impregnated samples. The reason lies in that the gray level is influenced by the preset parameters such as the lightness, the contrast and the working distance of the BSE measurement. These parameters may always vary when different samples are observed. Therefore, the gray level cannot be taken as constant to identify each individual phase. On the contrary, the storage modulus of the same phase does not vary significantly with the tested samples. The storage moduli of the aggregate, the clinker, and the hydration product in both the non-impregnated and the impregnated samples are 85 GPa, 65 GPa, and 25 GPa, respectively. Therefore, the storage modulus seems a more objective indicator to identify different phases in concrete. The measured storage modulus of the ITZagg-paste in the non-impregnated sample is about 12 GPa, which is notably smaller than that measured in the impregnated sample (22 GPa), this may be related to the fact that the porous ITZagg-paste is weakened by the artificial cracks during the polishing process in the non-impregnated sample, meanwhile the ITZagg-paste can be improved by the epoxy in the impregnated sample. Since the modulus of the hydration products in the non-impregnated sample

2.8 Influence of Epoxy Impregnation …

33

is similar to that of the impregnated samples, the epoxy may only strengthen the materials with high porosity more significantly than that with low porosity, which will be further validated by the measured modulus of the paste with high w/c ratio discussed in Sect. 2.8.6.

2.8.5 On Characterizing ITZ Between Residual C3 S Clinker and Hydration Product Figure 2.25 reveals that an ITZ can be identified between the residual C3 S clinker and the hydration product (ITZC3S-HP ) for both non-impregnated and impregnated paste samples through SPM modulus mapping. As show in Fig. 2.25, the thickness of ITZC3S-HP ranges roughly between 1 and 2 µm, which has also been observed by Xu et al. (2015a) by using NI and modulus mapping. Due to the low storage modulus, the ITZC3S-HP might lead to a weak bond between C3 S and the surrounding hydration product, which might contribute to the initiation of the mechanical failure of cementitious materials under the external loads (Liang et al., 2017b). It has been suspected that the ITZC3S-HP is produced by the artificial ‘pull-out’ of the C3 S clinker during the polishing process (Kjellsen & Justnes, 2004), and the large elevation difference between the C3 S clinker and the hydration product (ΔhC3S-HP ) can account for the measured weak mechanical properties of ITZC3S-HP . However, ΔhC3S-HP in both the impregnated and the non-impregnated samples is no greater than 200 nm (Fig. 2.25), which is less than Δhagg-paste of the non-impregnated sample (about 900 nm) as shown in Fig. 2.21. Since the Δhagg-paste of the non-impregnated sample does not result in the extremely low modulus, the low modulus within the area between the C3 S clinker and the hydration product is not mainly affected by the ΔhC3S-HP . The ITZC3S-HP was also observed in the high-resolution BSE images especially in the early-age cement paste (Kjellsen & Justnes, 2004). More images within the area around the C3 S clinkers in both 3C and 5C samples are shown in Fig. 2.26. It can be seen that the epoxy impregnation doesn’t affect the thickness and storage modulus of ITZC3S-HP . The ITZC3S-HP in both the impregnated and the non-impregnated samples is clearly exhibited due to its low storage modulus colored by white, however, the ITZC3S-HP is difficult to be entified in the BSE images of the corresponding areas.

2.8.6 On the Small-Scale Mechanical Properties The above sections have revealed the merits of epoxy impregnation on characterizing microstructure and morphology of the concrete, which include avoiding the artificial cracks and maintaining the flat surface during the polishing process. The mechanical properties at the micro and sub-micro scales, including the indentation modulus by

2 Samples Preparation

IP ITZ

Scanning line (c')

Impregnated sample

(b')

80

OP

ITZ

40

90

0 60 -40 30

-80

0

-120 0

C3S

(a')

IP

Elevation (nm)

OP

C3S

5

10 15 Distance (μm)

(c')80

C3S OP

60

IP

ITZ

SM ELE

20

IP ITZ

40

100 50 0 -50 -100

20

Elevation (nm)

(b)

Storage modulus (GPa)

Non-impregnated sample

(c) 120

Storage modulus (GPa)

(a)

Scanning line (c)

34

-150

0 0

10 20 30 Distance (μm)

-200 SM ELE

Fig. 2.25 Comparison between the surface elevation and the storage modulus of a–c the nonimpregnated and a’–c’ the impregnated paste sample with w/c ratio of 0.3 (3P), a, b, a’ and b’ the topography and modulus mapping images of target area in the samples. c and c’ the variation of storage modulus (SM) and surface elevation (ELE) of different phases along the scanning line (Liang et al., 2020)

NI and MI, are often measured on the polished paste samples. How the mechanical properties will be affected by the epoxy impregnation is of significance. The moduli at micro level and sub-micro level of both the impregnated and the non-impregnated paste samples were measured by instrumented indentation techniques. The NI with a maximum load of 2 mN was conducted to measure the indentation modulus of the outer hydration product (OP) at CSH scale since the OP is more porous than IP, and thus the modulus of OP is strengthened more by the impregnated epoxy. The corresponding maximum indentation depth is less than 400 nm. MI with a maximum indentation depth of 30 µm was used to measure the indentation modulus of cement paste, which is expected to reflect the homogeneous property of cement pastes (Liang et al., 2017a). The indentation modulus of the non-impregnated and the impregnated paste samples measured by NI and MI are shown in Fig. 2.27. It is seen that the epoxy impregnation has a minor effect at the CSH scale because the differences between

2.8 Influence of Epoxy Impregnation …

35

Non-impregnated sample 3P

3P

5P

5P ITZ

C3S

ITZ

ITZ C3S

C3S ITZ

C3S

C3S C3S

C3S C3S 5μm

5μm

10μm

10μm

Impregnated sample 3P

5P

3P

5P

ITZ

ITZ

C3S

ITZ C3S

ITZ C3S

C3S

C3S C3S C3S

C3S 10μm

5μm

5μm

5μm

Fig. 2.26 ITZ between the residual C3 S clinker and the hydration product identified by SPM modulus mapping in the non-impregnated and the impregnated paste samples with w/c ratio of 0.3 (3P) and 0.5 (5P) and the BSE images of the corresponding areas (Liang et al., 2020)

the measured moduli of the non-impregnated and the impregnated samples can be neglected, which is consistent with the finding by Zhu et al. (2009). For the sample with the same mixture proportions, the differences on the average modulus of hydration product between the non-impregnated and the impregnated samples is only about 1%. However, the measured modulus at the paste scale is affected significantly by the epoxy. The indentation modulus in the 5P sample is improved by the impregnated epoxy, which can increase from 16.9 to 21.1 GPa. The reason lies in that the abundant voids and capillary pores existing at the paste scale in the 5P sample are easily filled

36

2 Samples Preparation

NI

Indentation modulus (GPa)

30

MI

25 20 15 10

Hydration product

5 0

Clinker 5μm

3C impregnated 3C non-impregnated 5C impregnated 5C non-impregnated

100μm

Fig. 2.27 Comparison between the indentation modulus of the non-impregnated and the impregnated paste samples measured by NI and MI (Liang et al., 2020)

with epoxy, and thus enhance the microstructure. On the contrary, the epoxy impregnation doesn’t affect the indentation modulus of 3P at the paste scale significantly, because the denser structure of the low w/c ratio cement paste are hardly influenced by the epoxy impregnation. Based on the measured storage modulus of ITZagg-paste in Sect. 2.8.4 and the indentation modulus of paste at different scales in this section, the conclusion that the phases in the materials with high porosity (e.g. the ITZagg-paste in both 3C and 5C samples) are susceptible to the impregnated epoxy can be drawn. In order to characterize the mechanical property, it is encouraged to use the epoxy impregnation to prepare sample for NI test to improve the flatness of surface during the polishing process. However, the epoxy impregnation is not recommended to characterize the ITZagg-paste properties and the MI modulus at the paste scale.

2.9 Summary The appropriate dispersion and mixing methods should be selected according to the characteristics of cementitious materials. The dispersed nanomaterials should be used immediately due to its short period to maintain the effective dispersion. During the casting of the small-size sample, the paste should be poured and vibrated layer by layer to a full vibration, and the severe bleeding of the paste with high w/c ratio should be strictly controlled and avoided. The direct drying method and the organic solvent exchange method are the two commonly used methods to remove the free water from cementitious materials. Although the rapid drying process is easy to conduct, it may change the microstructure of cementitious materials and remove only the free water without removing the ions in the pore solution. Organic solvent method seems more suitable for hydration stoppage, particularly the ethanol and isopropanol are recommended as solvents.

References

37

The fine grinding step is essential, as the polishing will not be successful if the fine grinding step is skipped. The fine grinding process should adopt as less vertical force, lower grinding rate, and shorter grinding time as possible until a slight reflection phenomenon observed on the surface of the sample. The polished sample should be stored in a vacuum environment to prevent carbonation. The epoxy impregnation should be used to prepare the samples for microscale test, as the impregnated epoxy can provide the pore structure with the resistance to reduce the artificial cracks, pits, and patches created during the grinding and polishing process to improve the flatness of the sample surface. In the mechanical property characterization, the epoxy impregnation is encouraged to be used in preparing the samples for nanoindentation test without worrying about its influence on the measured modulus of hydration product at the CSH scale. However, it is not recommended to be used in preparing the samples for MI test at the paste scale and for characterizing the properties of ITZagg-paste by using SPM technique, because the epoxy would fill into the pores in the ITZagg-paste and densify its microstructure which may lead to an improved small-scale mechanical property.

References Bisschop, J., & Van Mier, J. G. M. (2002). How to study drying shrinkage microcracking in cementbased materials using optical and scanning electron microscopy. Cement and Concrete Research, 32(2), 279–287. Bobji, M. S., & Biswas, S. K. (1999). Deconvolution of hardness from data obtained from nanoindentation of rough surfaces. Journal of Materials Research, 14(6), 2259–2268. Collier, N. C., Sharp, J. H., Milestone, N. B., Hill, J., & Godfrey, I. H. (2008). The influence of water removal techniques on the composition and microstructure of hardened cement pastes. Cement and Concrete Research, 38(6), 737–744. Diamond, S., & Huang, J. (2001). The ITZ in concrete–a different view based on image analysis and SEM observations. Cement and Concrete Composites, 23(2), 179–188. Elsharief, A., Cohen, M. D., & Olek, J. (2003). Influence of aggregate size, water cement ratio and age on the microstructure of the interfacial transition zone. Cement and Concrete Research, 33(11), 1837–1849. Gallé, C. (2001). Effect of drying on cement-based materials pore structure as identified by mercury intrusion porosimetry: A comparative study between oven-, vacuum-, and freeze-drying. Cement and Concrete Research, 31(10), 1467–1477. Gao, X., Wei, Y., & Huang, W. (2018). Critical aspects of scanning probe microscopy mapping when applied to cement pastes. Advances in Cement Research, 30(7), 293–304. Geiker, M. R., Bøhm, A., Kjeldsen, A. M. (2006). On the effect of mixing on property development of cement pastes. In Proceedings pro052: International RILEM Conference, Volume Changes of Hardening Concrete: Testing and Mitigation, 20–23 Aug 2006. Lyngby, Denmark, RILEM Publications S.A.R.L., Bagneux, France, pp. 303–310. Khoshnazar, R., Raki, L., Beaudoin, J., & Alizadeh, R. (2013). Solvent exchange in sulphoaluminate phases; Part II: Monosulfate. Advances in Cement Research, 25(6), 322–331. Kim, J. U., Lee, J. J., Lee, Y. H., Jang, J., & Kown, D. (2006). Surface roughness effect in instrumented indentation: A simple contact depth model and its verification. Journal of Materials Research, 21(12), 2975–2978.

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Kjellsen, K. O., & Justnes, H. (2004). Revisiting the microstructure of hydrated tricalcium silicate-a comparison to Portland cement. Cement and Concrete Composites, 26(8), 947–956. Kjellsen, K. O., Monsoy, A., Isachsen, K., & Detwiler, R. J. (2003). Preparation of flat-polished specimens for SEM-backscattered electron imaging and X-ray microanalysis-importance of epoxy impregnation. Cement and Concrete Research, 33(4), 611–616. Konecny, L., & Naqvi, S. J. (1993). The effect of different drying techniques on the pore size distribution of blended cement mortars. Cement and Concrete Research, 23, 1223–1228. Kong, W. K., Wei, Y., Wang, Y. Q., & Sha, A. M. (2021). Development of micro and macro fracture properties of cementitious materials exposed to freeze-thaw environment at early ages. Construction and Building Materials, 271, 121502. Korpa, A., & Trettin, R. (2006). The influence of different drying methods on cement paste microstructures as reflected by gas adsorption: Comparison between freeze-drying (F-drying), D-drying, P-drying and oven-drying methods. Cement and Concrete Research, 36(4), 634–649. Krishnamoorti, R. (2007). Strategies for Dispersing Nanoparticles in Polymers. MRS Bulletin, 32(04), 341–347. Li, W., Kawashima, S., Xiao, J., Corr, D. J., Shi, C., & Shah, S. P. (2015). Comparative investigation on nanomechanical properties of hardened cement paste. Materials and Structures, 49(5), 1591– 1604. Li, G., Cui, H., Zhou, J., & Hu, W. (2019). Improvements of Nano-TiO2 on the Long-Term Chloride Resistance of Concrete with Polymer Coatings. Coatings, 9(5), 323. Liang, S. M., Wei, Y., & Gao, X. (2017a). Strain-rate sensitivity of cement paste by microindentation continuous stiffness measurement: Implication to isotache approach for creep modeling. Cement and Concrete Research, 100, 84–95. Liang, S., Wei, Y., & Wu, Z. (2017b). Multiscale modeling elastic properties of cement-based materials considering imperfect interface effect. Construction and Building Materials, 154, 567– 579. Liang, S. M., Wei, Y., Gao, X., & Qian, Z. D. (2020). Effect of epoxy impregnation on characterizing microstructure and micromechanical properties of concrete by different techniques. Journal of Materials Science, 55, 2389–2404. Liao, K. Y., Chang, P. K., Peng, Y. N., & Yang, C. C. (2004). A study on characteristics of interfacial transition zone in concrete. Cement and Concrete Research, 34(6), 977–989. Long, W. J., Gu, Y., Xiao, B. X., Zhang, Q., & Xing, F. (2018). Micro-mechanical properties and multi-scaled pore structure of graphene oxide cement paste: Synergistic application of nanoindentation, X-ray computed tomography, and SEM-EDS analysis. Construction and Building Materials, 179, 661–674. Miller, M., Bobko, C., Vandamme, M., & Ulm, F. J. (2008). Surface roughness criteria for cement paste nanoindentation. Cement and Concrete Research, 38(4), 467–476. Mmusi, M.O., Alexander, M.G., Beushausen, H.D. (2009). Determination of critical moisture content for carbonation of concrete. 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting, ICCRRR-2, 24–26 November 2008, Cape Town, South Africa: 359–364. Mondal, P., Shah, S. P., & Marks, L. (2007). A reliable technique to determine the local mechanical properties at the nanoscale for cementitiousmaterials. Cement and Concrete Research, 37(10), 1440–1444. Moukwa, M., & Aitcin, P. C. (1988). The effect of drying on cement pastes pore structure as determined by mercury porosimetry. Cement and Concrete Research, 18(5), 745–752. Mouret, M., Ringot, E., & Bascoul, A. (2001). Image analysis: A tool for the characterisation of hydration of cement in concrete–metrological aspects of magnification on measurement. Cement and Concrete Composites, 23(2), 201–206. Rößler, C., Eberhardt, A., Kuˇcerová, H., & Möser, B. (2008). Influence of hydration on the fluidity of normal Portland cement pastes. Cement and Concrete Research, 38(7), 897–906. Sakulich, A. R., & Li, V. C. (2011). Nanoscale characterization of engineered cementitious composites (ECC). Cement and Concrete Research, 41(2), 169–175.

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Scrivener, K. L., Crumbie, A. K., & Laugesen, P. (2004). The interfacial transition zone (ITZ) between cement paste and aggregate in concrete. Interf. Sci., 12(4), 411–421. Scrivener, K., Snellings, R., & Lothenbach, B. (2016). A Practical Guide to Microstructural Analysis of Cementitious Materials. CRC Press. She, W., Wang, X., Miao, C., Zhang, Q., Zhang, Y., Yang, J., & Hong, J. (2018). Biomimetic superhydrophobic surface of concrete: Topographic and chemical modification assembly by direct spray. Construction and Building Materials, 181, 347–357. Struble, L., & Stutzman, P. (1989). Epoxy impregnation of hardened cement for microstructural characterization. Journal of Material Science Letters, 8(6), 632–634. Stutzman, P. E., Clifton, J. R. (1999). Specimen preparation for scanning electron microscopy. In Proceedings of the international conference on cement microscopy. International cement microscopy association, vol. 21, pp. 10–22. Wei, Y., Gao, X., & Liang, S. (2016). Nanoindentation-based study of the micro-mechanical properties, structure, and hydration degree of slag-blended cementitious materials. Journal of Materials Science, 51(7), 3349–3361. Wei, Y., Liang, S., & Gao, X. (2017a). Indentation creep of cementitious materials: Experimental investigation from nano to micro length scales. Construction and Building Materials, 143, 222– 233. Wei, Y., Liang, S., & Gao, X. (2017b). Phase quantification in cementitious materials by dynamic modulus mapping. Materials Characterization, 127, 348–356. Wei, Y., Gao, X., & Liang, S. (2018a). A combined SPM/NI/EDS method to quantify properties of inner and outer C-S-H in OPC and slag-blended cement pastes. Cement and Concrete Composites, 85, 56–66. Wei, Y., Gao, X., & Liang, S. M. (2018b). Identification and characterization of hydration products in hardened cement paste based on modulus mapping and nanoindentation. Journal of the Chinese Ceramic Society, 46(8), 1043–1052. (In Chinese). Wei, Y., Kong, W. K., Wang, Y. Q., & Sha, A. M. (2021a). Multifunctional application of nanoscratch technique to characterize cementitious materials. Cement and Concrete Research, 140, 106318. Wei, Y., Kong, W. K., & Wang, Y. Q. (2021b). Strengthening mechanism of fracture properties by nano materials for cementitious materials subject to early-age frost attack. Cement and Concrete Composites, 119, 104025. Wong, H. S., & Buenfeld, N. R. (2006). Patch microstructure in cement-based materials: Fact or artefact. Cement and Concrete Research, 36(5), 990–997. Wong, H. S., Head, M. K., & Buenfeld, N. R. (2006). Pore segmentation of cement-based materials from backscattered electron images. Cement and Concrete Research, 36(6), 1083–1090. Xiao, J., Li, W., Sun, Z., Lange, D. A., & Shah, S. P. (2013). Properties of interfacial transition zones in recycled aggregate concrete tested by nanoindentation. Cement and Concrete Composites, 37, 276–292. Xu, J., Corr, D. J., & Shah, S. P. (2015a). Nanomechanical properties of C-S-H gel/cement grain interface by using nanoindentation and modulus mapping. Journal of Zhejiang University Science, 16(1), 38–46. Xu, S., Liu, J., & Li, Q. (2015b). Mechanical properties and microstructure of multi-walled carbon nanotube-reinforced cement paste. Construction and Building Materials, 76, 16–23. Xu, J., Corr, D. J., & Shah, S. P. (2017). Nanoscratch study of the modification effects of nanoSiO2 on C-S-H gel/cement grain interfaces. Journal of Materials in Civil Engineering, 29(9), 04017093. Yang, T., Keller, B., Magyari, E., Hametner, K., & Gunther, D. (2003). Direct observation of the carbonation process on the surface of calcium hydroxide crystals in hardened cement paste using an atomic force microscope. Journal of Materials Science, 38(9), 1909–1916. Yio, M. H. N., Phelan, J. C., Wong, H. S., & Buenfeld, N. R. (2014). Determining the slag fraction, water/binder ratio and degree of hydration in hardened cement pastes. Cement and Concrete Research, 56, 171–181. Zeng, Q., Li, K. F., Fen-Chong, T., & Dangla, P. (2013). Water removal by freeze drying of hardened cement paste. Drying Technology, 31(1), 67–71.

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Zhang, L., & Glasser, F. P. (2000). Critical examination of drying damage to cement pastes. Advances in Cement Research, 12(2), 79–88. Zhang, J., & Scherer, G. W. (2011). Comparison of methods for arresting hydration of cement. Cement and Concrete Research, 41(10), 1024–1036. Zhang, H., Šavija, B., Lukovi´c, M., & Schlangen, E. (2019a). Experimentally informed micromechanical modelling of cement paste: An approach coupling X-ray computed tomography and statistical nanoindentation. Composites Part b: Engineering, 157, 109–122. Zhang, Y., Li, S., Zhang, W., Chen, X., Hou, D., Zhao, T., & Li, X. (2019b). Preparation and mechanism of graphene oxide/isobutyltriethoxysilane composite emulsion and its effects on waterproof performance of concrete. Construction and Building Materials, 208, 343–349. Zhao, P., Wang, H., Wang, S., Du, P., Lu, L., & Cheng, X. (2018). Assessment of Nano-TiO2 enhanced performance for photocatalytic polymer-sulphoaluminate cement composite coating. Journal of Inorganic and Organometallic Polymers and Materials, 28, 2439–2446. Zhu, W., Fonteyn, M. T. J., Hughes, J., & Pearce, C. (2009). Nanoindentation study of resin impregnated sandstone and early-age cement paste specimens. Nanotechnology Construction, 3, 403–408.

Chapter 3

Experimental Techniques

Abstract In recent years, the advanced techniques have provided powerful tools for characterizing materials at micro- and nanoscales. This chapter reviews the developing history, test principle, parameter setting, and data analysis of a few techniques for measuring the microstructures and the small-scale mechanical properties of cementitious materials, including NI/MI, SPM, nanoscratch, SEM, X-ray CT, and MIP techniques. The instrumented indentation techniques can measure the mechanical properties at the micrometer and nanometer scales. The SPM technique can characterize the mechanical properties as well as the thickness of individual phases under the non-destructive high-resolution conditions. A new loading mode of the constant vertical loading rate is proposed for continuous fracture properties measurement by nanoscratch technique. The content of this chapter can deepen the understanding of current advanced techniques applied to cementitious materials for testing conducted at micro scale. Keywords Instrumented indentation · Indentation modulus · Contact creep compliance · Continuous stiffness measurement · Modulus mapping · Nanoscratch

3.1 Introduction Traditionally, the measurements on the properties of cement-based materials, such as deformation, strength, and durability, are conducted at the macroscale. In recent years, the advanced techniques have provided powerful tools for characterizing materials at micro- and nanoscales, and empowered many researchers to reveal how the composition, structure, and properties of materials at nanoscale influence their performance at macroscale. This chapter introduces the history, test principle, calculation method, and test parameter setting of the advanced techniques for measuring the microstructure and the small-scale mechanical properties of cementitious materials, including the NI/MI technique, the SPM technique, the CSM, the nanoscratch technique, the SEM technique, the X-ray CT technique, and the MIP technique.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_3

41

42

3 Experimental Techniques

3.2 Instrumented Indentation Test 3.2.1 Development of Indentation Technique Indentation hardness test might be the earliest method for measuring the mechanical property of materials due to its simplicity of operation, and the Mohs hardness table established in 1882 is the most famous application of the early indentation test. In this table, materials that can leave permanent scratches on another material are classified as harder materials. Based on this operation method, the indentation test can be conducted by pressing a material with known properties into another material with unknown properties, and the hardness of the indented material can be obtained by assessing the residual deformation after indentation. With the advance of technology, the indentation techniques developed with high spatial resolution have made it possible to characterize the mechanical properties of not only the bulk material, but also its constituent phases at micro and nanoscales. When the indentation depth is in nano- and micrometers scale, the corresponding experimental methods are called the NI test and the MI test, respectively. The NI and the MI test can be traced back to Brinell, in which a small ball was pushed into the surface of materials and the plastic properties of the materials were obtained (Bristowe et al., 1974; Tabor, 1970). Meanwhile, it is possible by using this method to determine Young’s modulus, hardness, and viscosity of material at the nanometer scale. It should be aware that this method was initially developed to characterize the macroscopically homogeneous materials, such as metals, glass, and ceramics. In recent years, the NI technique has been increasingly applied to characterize the heterogeneous and porous materials (such as cement-based composites) to reveal the mechanical properties of individual phase, such as the elastic modulus, hardness, and creep (Chen et al., 2010; Hu et al., 2014; Masoero et al., 2012; Vandamme et al., 2010). In addition, NI technique has been used to measure the mechanical properties of the interfacial transition zone (ITZ) of the aggregate and the fiber (Allison et al., 2012; Kang et al., 2013; Khedmati et al., 2018). These applications provide important insights on the understanding of the micromechanical properties and microstructure of cementitious materials.

3.2.2 Geometry of Indenter Probes The instrumented indentation technique was originally intended for application with sharp, geometrically self-similar indenters, such as the Berkovich triangular pyramid. A variety of axisymmetric indenter geometries including the spherical indenter were also proved to be useful (Oliver & Pharr, 1992). The shapes of the most commonly used indenter probes are summarized in Fig. 3.1, including the conical indenter, the spherical indenter, the flat indenter, the Berkovich indenter, the Cube-corner indenter, and the Vickers indenter. The Berkovich indenter, the Cube-corner indenter, and

3.2 Instrumented Indentation Test

Conical

43

Spherical

Flat Pyramidal indenter probes

90° Berkovich

Cube-corner

Vickers

Fig. 3.1 3D geometry of different indenter probes

the Vickers indenter are classified into the pyramidal indenter probes. Unlike the Berkovich indenter, the angle between the two edges of the Cube-corner indenter probe is 90°. Pyramidal indenters are often used to test glass and crystal materials. The spherical indenter probe is the most commonly used for measuring the soft materials, such as polymer materials. The flat indenter probe is rarely used in the actual indention test, but it is important from a theoretical point of view, because the contact surface between the flat probe and the indented surface is constant over the entire indentation process, and the analysis of the indentation contact problem can be greatly simplified. The pyramidal indenter probes (such as the Berkovich, Vickers, or Cube-corner indenter probe) are the commonly used indenter shapes due to their sharp geometry which allows for testing materials with smaller volumes compared to other probe geometries. It should be noted that the non-axisymmetric pyramidal probes are often approximated by an equivalent half-cone angel which ensures that the non-axisymmetric pyramidal probe and the axisymmetric probe have the same cross-sectional area at a given height for simplifying the analysis. The equivalent half-cone angle of the Berkovich, the Vickers, and the Cube-corner indenter probe is 70.32°, 70.32°, and 42.28°, respectively. Among all the pyramidal probes, the Berkovich probe is the most commonly used one.

3.2.3 Method of Indentation Test The indentation tests are classified into either NI or MI depending on the magnitude of the indentation depth, which is proved to be attractive ways to measure the small-scale mechanical behavior of materials because the load and the depth can be continuously recorded during the loading, the holding, and the unloading process at the nano and micro scales. The hardness (H) and the indentation modulus (M) of the indented materials can be calculated based on the measured load acting on the probe and the depth pressed into the surface of the sample.

44

3 Experimental Techniques

The typical loading–holding–unloading curve obtained with a Berkovich indenter is shown in Fig. 3.2a. The parameter P denotes the applied load, h is the indentation depth relative to the initial undeformed surface, Pmax and hmax are the maximum load and the maximum indentation depth, respectively. hf is the residual indentation depth after complete unloading, which represents the permanent depth of penetration after the indenter is fully unloaded. Another important parameter S is known as the contact stiffness, which is defined as the slope of the upper portion of the unloading curve during the initial stages of unloading: S = (dP/dh)|h=h max

(3.1)

The experiments have shown that the unloading curves can be well captured by the power-law relation: P = α(h − h f )m

(3.2)

where α and m are constants, and the value of m depends on the type of indenter, which is summarized in Table 3.1. The schematic diagram of the contact geometry of NI by a Berkovich indenter is shown in Fig. 3.2b. It is assumed that the behavior of the Berkovich indenter can be modeled by a conical indenter with an equivalent half-cone angel (70.32°) that gives the same depth-to-area relationship (Oliver & Pharr, 1992). On the other hand, the amount of sink-in (hs ) without considering the pile-up normally occurring in some elastic–plastic materials is given by:

Load (P)

(a)

Pmax

Holding

(b) ac

Loading S=dP/dh

Unloading

hs hmax

hf

Sample surface hf

hc

Unloaded Loaded

hmax

Indentation depth (h) Fig. 3.2 Schematic diagrams of a typical loading-holding-unloading curve, and b contact geometry of NI

Table 3.1 Value of m for different types of indenters

Indenter type

Value of m

Flat

1.0

Conical

2.0

Spherical

1.5

Pyramidal

1.5

3.2 Instrumented Indentation Test

45

h s = ε Pmax /S

(3.3)

where ε is a constant dependent on geometry of the indenter, and ε = 0.72 for a conical punch, ε = 1 for a flat punch. Normally, ε = 0.75 is recommended as the standard value for analysis. From Fig. 3.2, the contact depth can be calculated as: h c = h max − h s

(3.4)

A convenient estimation of contact depth hc is made by Eq. (3.5) based on the assumption of the elastic contact height-to-indentation depth relation: hc Pmax =1−ε h max S · h max

(3.5)

And thus, the projected area of contact AC that related to the indentation depth can be calculated as: AC = 24.5h C2

(3.6)

The geometry of the discrete indents made on the hydration product (HP) and the unreacted clinker in a paste sample are illustrated in Fig. 3.3a as an example. The NI test is conducted by using the indentation mode of the KEYSIGHT G200 nanoindenter equipped with a Berkovich probe. The maximum load is set as 4 mN with a loading speed of 0.4 mN/s. After holding for 5 s at the maximum load of 4 mN, it is unloaded to zero within 10 s. It can be seen that the size of the indent made on the unreacted clinker is smaller than that made on the hydration product, which is reasonable because of the less stiff feature of the hydration product compared to the unreacted clinker. This feature is also reflected in the load–indentation depth curve as shown in Fig. 3.3b, where greater indentation depth is observed at the less stiff phase.

3.2.4 Indentation Scale in Multiphase Materials It must be careful when measuring the micromechanical properties of cementitious materials by quasi-static NI technique since the heterogeneous characteristics of cementitious materials would affect the measured results. Generally, whether a material is heterogeneous or not depends on the length scale used in the observation. A separation of length scales L and D on which the composite properties and the heterogeneity properties are defined should be made for multiscale characterization, as shown in Fig. 3.4. L is the characteristic length of a composite, D represents the characteristic size of the heterogeneity in the composite, and d represents the characteristic size of the material volume that is activated by the indentation test in

46

3 Experimental Techniques

h of clinker and hydrates (nm)

(b)

0

200

400

600

800 5

1.25

Clinker

P for ITZ (mN)

Clinker

Hydrates

ITZ

1

4

0.75

3

0.5

2

0.25

1

Hydrate 0

0 0

50

100

150

P for clinker and hydrates (mN)

(a)

200

h of ITZ (nm) Fig. 3.3 a Indents made on polished surface by nanoindenter (Wei et al., 2018), and b measured NI load–depth curve of clinker, hydrates, and ITZ phase (Kong et al., 2021)

the multiphase material. The size of d can be roughly associated with a half-sphere projected underneath the indenter. The indentation response will depend mostly on the material within a distance d from the indenter tip. If d ≪ D, the indentation test may probe only one phase and therefore characterize its intrinsic phase properties. The parameter d can be determined by Eq. (3.7):

(a)

D L

Sample surface h Phase 1 d

Phase 2 Phase 1 (b) L

Phase 2 Sample surface h

Phase 2

Phase 1 Phase 1

d

D

Phase 2

Fig. 3.4 Schematic of the principle of indentation scale in multiphase materials. a at low indentation depths (h ≪ D) the individual phase can be identified, and b at large indentation depths (h ≫ D) the homogenized medium can be identified

3.2 Instrumented Indentation Test

47

) ⊓ ( M L(τ ≤ t) h d θ, , ν, α, = , d h H H D

(3.7)

where θ is the equivalent half-cone angle of the indenter, M is the indentation modulus, H is the hardness, ν is Poisson’s ratio, α is the friction angle, and L(τ ≤ t) is the contact creep compliance. The first five invariants (θ, M , ν, α, L(τH≤t) ) are the material properties, while the H last invariant, h/D, links the indentation depth to the characteristic size of the heterogeneity. Under the case of h/D ≪ 1, the indentation properties (M, H, L(t)) are representatives of the modulus, stiffness, and creep properties of the individual phase, respectively. In contrast, if h/D ≫ 1, the properties of composite materials can be measured. Therefore, by adopting suitable indentation force, one can measure not only the mechanical properties of the individual phase in cementitious materials but also those of the composites mixed by different phases. Generally speaking, small indentation depths of roughly h/D < 1/10 provide access to individual phase properties, while larger indentation depths h/D > 6 provide access to homogenized material properties of the composite (Constantinides & Ulm, 2006; Constantinides et al., 2006). Therefore, the selection of the indentation depth should be based on the characteristic size of the composite.

3.2.5 Calculating Microscale Mechanical Properties Indentation modulus, hardness, and elastic modulus The NI test on cement-based materials is complex, the measured results might include viscoelastic plasticity, heterogeneity, and anisotropy behavior of cement-based material. The effect of plasticity can be eliminated by analyzing the unloading curve to obtain the elastic properties of the material. By analyzing the initial part of the unloading curve as shown in Fig. 3.2a, the indentation modulus (M) and the indentation hardness (H) can be calculated as (Oliver & Pharr, 1992): √ M=

π S √ 2β AC

(3.8)

Pmax AC

(3.9)

H=

All quantities required to determine H and M are directly obtained from the load– depth (P − h) curves, with the exception of the projected area of contact Ac (Eq. (3.6)). S is contact stiffness; β is the geometrical correction factor, for the Berkovich probe used in this book, β = 1.034. Elastic contact mechanics provides a convenient framework for linking the measured indentation modulus, M, with the elastic properties of the indented material.

48

3 Experimental Techniques

The indentation modulus is directly related to the elastic modulus, E, and Poisson’s ratio, ν, of the indented material. Considering the flexibility of the indenter, the elastic modulus can be calculated as: 1 − νi2 1 1 − ν2 = + M E Ei

(3.10)

where E i = 1141 GPa and ν i = 0.07 (Oliver & Pharr, 1992) are the elastic modulus and Poisson’s ratio of the diamond indent tip. Creep Indentation technique has been increasingly used to characterize the elastic modulus (E) and hardness (H) of heterogeneous cementitious materials (Hu et al., 2014; Vandamme & Ulm, 2009; Velez et al., 2001). Meanwhile, it was introduced to measure the creep behavior of cementitious materials due to its superior advantage of reducing significantly the measuring time to observe the creep behavior (Nguyen et al., 2014; Vandamme & Ulm, 2009). NI with peak load of 2 mN was normally used to experimentally explore the creep phenomenon of CSH and its logarithmic creep behavior (Pichler & Lackner, 2009; Vandamme & Ulm, 2009, 2013). The typical indentation load (P)–depth (h) curve obtained from creep test containing loading, holding, and unloading processes is shown in Fig. 3.5. The loading and unloading rates should be set to ensure that each process is completed within 10 s. The holding period is 180 s, which is longer compared to the indentation modulus test. During the micro creep measurement, a rigid indentation probe is penetrated into the flat surface of a linear viscoelastic material. Based on the correspondence principle of viscoelasticity, the indentation force (P) and the indentation depth (h) satisfy the following equation (Zhang, 2014): (b) 0.6 180s

0.5

0.5

Pmax

0.4 P (N)

0.4 P (N)

(c) Δh

0.3 0.2

0.2

0.1

0.1

0

0 0

40

80 120 160 200 Time (s)

S=dP/dh

0.3

hf

Δh (μm)

(a) 0.6

hmax

0 1 2 3 4 5 6 7 8 h (μm)

0.1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

10 100 Time (s)

1000

Fig. 3.5 a loading configuration, b load–indentation depth curve, and c indentation depth increment during the 180 s’ holding period (Wei et al., 2017b)

3.2 Instrumented Indentation Test

49

⎧t P(t) = ϕ

M(t − τ )

d 1+1/ g h (τ )dτ dτ

(3.11)

0

where ϕ =

( ) 1/ g ⌈ g+1 2 ( ) √ 2 1 g g ( π g0 ) / 1+g ⌈ g+1 2

is parameter representing the effect of geometry of

(∞ the indentation probe, ⌈(x) = 0 t x−1 exp(−t)dt is the Euler Gamma function, g0 and g are the geometry coefficients of the indentation probe; M(t) is the indentation relaxation modulus. By applying the Laplace transform to the equation above, one can obtain the following equation: ⎡ ⎤ 𝓁[P(t)] = ϕs𝓁[M(t)]𝓁 h 1+1/ g (t)

(3.12)

where 𝓁 [·] is the Laplace transform operator; s is the Laplace parameter, and it can be a complex number. Here a parameter l(t) that is similar to compliance can be defined as: l(t) =

ϕ 1+1/ g h (t) Pmax

(3.13)

Performing the Laplace transform to the above equation yields: 𝓁[l(t)] =

ϕ ⎡ 1+1/ g ⎤ 𝓁 h (t) Pmax

(3.14)

When the indentation force is imposed in a Heaviside step, the indentation force can be expressed as: P(t) = Pmax H (t)

(3.15)

where H(t) is the Heaviside step function. Performing the Laplace transform to the above equation yields: / 𝓁[P(t)] = Pmax s

(3.16)

By combining Eqs. (3.12), (3.14), and (3.16), and defining: 𝓁[l(t)] = 𝓁[L(t)]

(3.17)

where L(t) is the contact creep compliance, which satisfies the following equation in Laplace domain: s𝓁[L(t)] =

1 s𝓁[M(t)]

(3.18)

50

3 Experimental Techniques

For a creep test with a Heaviside step load, rewriting Eq. (3.18) in time domain yields the time-dependent contact creep compliance expressed in the following equation (Vandamme et al., 2012): L(t) =

ϕ h(t)1+1/ g Pmax

(3.19)

Based on Eq. (3.19), the contact creep compliance can be calculated once the time–indentation depth, the maximum load, and the information on the geometry of the indentation probe are given. It should be noted that plasticity would occur under the sharp indentation probe even at very low indentation force due to the concentration of stress at the tip of indentation probe. Therefore, it is necessary to develop an alternative method to calculate the creep properties from indentation test. Vandamme et al. (2012) derived a method to calculate the creep property from a fictitious indentation test by introducing an instantaneous loading phase, an instantaneous unloading phase, and an instantaneous reloading and holding phase, which is shown in Fig. 3.6. Since most of the instantaneous plasticity occurs during the loading stage, the contact creep compliance L(t) is derived from the change in depth during the holding stage (➃) under the assumption that the time-independent plasticity occurs only during the loading phase but not during the reloading and holding phases. For any indentation probe with a given geometry, one can express the equivalent radius of the contact area (denoted as ac ) between the indenter and the test material as a function of indentation depth: ac = f (h)

(3.20)

It has been demonstrated that the expression of f (h) is not √ necessary to be found √2 M Ac with respect to the = β (Vandamme et al., 2012). By integrating dP dh π indentation depth, one can obtain the following equation:

(b) Load

Load

(a)

Time

Depth

Fig. 3.6 a Load versus time scheme; b load versus depth response of the thought experiment for ➀ instantaneous loading, ➁ instantaneous unloading, ➂ instantaneous reloading, and ➃ holding phase (Vandamme et al., 2012)

3.2 Instrumented Indentation Test

51

P = 2M F(h)

(3.21)

where F(h) is the primitive function of f (h) for which F(0) = 0. By rewriting Eq. (3.21) in the time domain for a linear viscoelastic material by applying the correspondence principle, one can obtain the following equation: P(t) = 2s𝓁[M(t)]𝓁[F(h(t))]

(3.22)

In the case of a step load, one can obtain the expression which links the contact creep compliance L(t) to indentation data in Laplace domain: 𝓁[M(t)] =

2 𝓁[F(h(t))] Pmax

(3.23)

After performing inverse Laplace transform on Eq. (3.23), one can obtain the timedependent contact creep compliance in the time domain as a function of indentation depth during the holding phase: L(t) =

2F(h(t)) Pmax

(3.24)

By differentiating Eq. (3.24) with respect to time, the following equation can be obtained: ˙ ˙ 2 f (h(t))h(t) 2ac (t)h(t) d 2F(h(t)) ˙ = = L(t) = dt Pmax Pmax Pmax

(3.25)

The contact radius ac (t) will evolve with time. However, it can be assumed that ac (t) is constant when the duration of holding stage is limited to several minutes. It has been shown by Vandamme (2008) that the contact radius ac (t) over a180s holding phase is estimated to vary about 5% for NI performed on cement paste by a Berkovich indentation probe; therefore, it is realistic to assume that the contact radius remains constant during the holding phase. By integrating Eq. (3.25) with respect to time from the beginning of holding stage, one can obtain the following equation: [L(t) − L(0)] = L(t) −

2ac ∆h(t) 1 = M Pmax

(3.26)

where ∆h(t) is the change of indentation depth during the holding phase. The difference L(t) -L(0) between the contact creep compliance L(t) and the reciprocal of indentation modulus L(0) = 1/M at the moment of loading is termed as the contact creep function.

52

3 Experimental Techniques

By using Eq. (3.26), the contact compliance can be calculated, which denotes the creep strain induced per unit stress at micro scale, similar to the concept “specific creep” at macro scale. The existing findings (Pichler & Lackner, 2009; Vandamme & Ulm, 2009, 2013; Wei et al., 2017b; Zhang et al., 2014) show that the contact creep function can be well-captured by a logarithmic function: ( ) t 1 [L(t) − L(0)] = ln 1 + C τ

(3.27)

where C is the creep modulus, which characterizes the long-term creep rate of cement paste (Vandamme & Ulm, 2009); τ is the characteristic time, which is related to the time at which creep starts exhibiting long-term logarithmic kinetics (Vandamme & Ulm, 2009). By making time derivation of Eq. (3.27), one can obtain the contact creep rate as: 1 d[L(t) − L(0)] = dt C × (t + τ )

(3.28)

Since the characteristic time (τ ) is usually within several seconds (Mallick et al., 2019; Pichler & Lackner, 2009; Vandamme & Ulm, 2013; Wei et al., 2017b; Zhang et al., 2014), the long-term creep rate at micro scale (i.e., t ≫ τ ) can be approximated as: 1 d[L(t) − L(0)] ≈ dt C ×t

(3.29)

A logarithmic function proposed by Vandamme (2008) was used to fit ∆h(t): ∆h(t) = x1 ln(x2 t + 1) + x3 t + x4

(3.30)

where x 1 relates to contact creep modulus (C) as in Eq. (3.27); x 2 is the reciprocal of characteristic time (Eq. (3.27)); x 3 is the rate of thermal drift during holding; x 4 is the correction for initialization of the creep phase. It should be noted that x 3 and x 4 have nothing to do with the material properties, and the curve fitting shows that these two parameters can be neglected. By combing Eqs. (3.26) and (3.30): L(t) − L(0) =

2ac [x1 ln(x2 t + 1)] Pmax

(3.31)

L(t) depends neither on the probe geometry nor on the load magnitude (Zhang et al., 2014). The typical contact creep compliance function [L(t) − L(0)] of the three individual phases in cement paste is shown in Fig. 3.7. By combining Eqs. (3.27) and (3.31), the contact creep modulus and the characteristic time can be written as:

3.2 Instrumented Indentation Test

53

35

OP IP Clinker

30

L(t) -1/M0 (×10-6/MPa)

Fig. 3.7 Typical contact creep function of outer product (OP), inner product (IP), and clinker in cement paste (Wei et al., 2017b)

25 20 15 10 5 0

0.1

1

10

100

Time (s)

C=

Pmax 2ac x1

(3.32)

1 x2

(3.33)

τ=

3.2.6 Method of Grid Indentation Test A large number of indentations can be performed at random locations on the surface of the multiphase material, as shown in Fig. 3.8a. Since cement paste is a multiphase composite, the NI test is often conducted as a grid test to cover large areas to fully take into account the property variations, as long as the spacing (δ) between each indent is large enough to ensure its statistical independence, as shown in Fig. 3.8b. To ensure statistical independence of the measured properties between two neighboring indentations, the spacing δ between indents must be greater than the characteristic size of the largest-sized heterogeneities. On the other hand, the indentation depth can roughly reflect which phases might be indented. The indent depth of about (a)

(c) Frequency

(b)

Phase B δ Phase A

Phase B

Phase A

Phase A Phase B

δ Properties

Fig. 3.8 Schematic diagram of a random indentation, b grid indentation, and c the histograms of the measured indentation properties

54

3 Experimental Techniques

100–200 nm is considered deep enough for capturing the mechanical behavior of the high-density (HD) CSH, while the low-density (LD) CSH may require deeper indent depth due to its greater size (Mondal et al., 2007; Richardson, 2004). A noticeable decrease of the obtained mechanical properties can be found when the indentation depth is greater than 500 nm (Davydov et al., 2011), which was suspected to be a reveal of different defects in the samples (Constantinides & Ulm, 2007). Statistical analysis or deconvolution on the large number of grid NI data is needed to extract the mechanical properties and the volume fraction of each individual phase by performing the maximum likelihood estimation for parameters of each phase. The deconvolution method will be described in detail in Sect. 6.3. It is generally believed that the randomly selected area for grid indentation can represent statically the whole sample, such that the phase distribution, the mechanical properties, and the volume fraction of each phase deconvoluted from the large amounts of indent data made on that area can be treated as the phase properties of that material. The test results can be displayed as histograms (or frequency plots) of the measured indentation properties (indentation modulus M, indentation hardness H, etc.), which in the case of the composite material displays several peaks (as shown in Fig. 3.8c). The mean value of each peak represents the mean property of each phase. The area below each curve of the histogram is a measure of the percentage of all indentations performed on the corresponding phase and is therefore a measure of the surface fraction of each phase. For a perfectly disordered material, the surface fractions and the volume fractions are identical, which is known as the Delesse principle (Delesse, 1847). Therefore, the volume fraction of each phase of the heterogeneous material can be obtained by analyzing the experimental frequency plot.

3.2.7 Microindentation (MI) Test The MI technique is based on the same working principle as the NI technique. The main difference is that the applied indentation load or indentation depth used in MI technique is much larger than that in NI. The MI technique is not used to test the properties of the individual phase, but to characterize the composite properties for hardened cement paste (Constantinides & Ulm, 2007; Randall et al., 2009). Nguyen et al (2014) measured the indentation creep of the LD CSH and HD CSH composite at micro scale by using the MI technique under the maximum load of 1 N and 5 N and investigated the effect of the porosity on the indentation mechanical properties. Zhang et al. (2014) measured the indentation creep of hardened cement paste by using MI technique under the maximum load of 20 N and concluded that micro creep behavior of hardened cement paste compares well to the macro creep of concrete. Wei et al. (2017b) also conducted MI test to measure the micromechanical properties of the composite phases of hardened pastes under the maximum loads of 0.1, 0.5, 1.5, and 5 N, as shown in Fig. 3.9. For test under the maximum load of 0.1 N, a smaller indent probe with a load capacity up to 0.5 N and a resolution of 0.05 mN was used, while

3.2 Instrumented Indentation Test

6

MI

NI 0.0025

5

0.002

P (N)

4

0

NI Pmax=2mN hmax=277nm

Δh MI Pmax=0.5N hmax=7um

0.001 P

1

MI Pmax=5N hmax=24um

0.0015

3 2

55

MI =0.1N

0.0005 hmax =3um max

MI Pmax=1.5N hmax=12um

0

NI

0

0.05

0.1

0.15

0.2

0.25

0.3

0

5

10

15 h (μm)

20

25

30

MI

Fig. 3.9 Typical indentation load–depth curves for NI and MI under different load magnitudes (Wei et al., 2017b)

for tests under the larger maximum load (0.5, 1.5, and 5 N), an indent probe with a load capacity up to 10 N and a resolution of 1 mN was used. These tests with the peak loads of 0.1, 0.5, 1.5, and 5 N are treated as MI, because the maximum indentation depth (hmax ) of the composite phases is greater than a few micrometers. The hmax ranges 3–24 µm for MI test under the maximum load ranging from 0.1 to 5 N (Wei et al., 2017b), while for NI that is only 277 nm under the maximum load of 2 mN. Similarities are revealed in the indentation load–depth curves under different magnitudes of peak loads. The differences of these curves lie in hmax under the maximum load, and the depth increment (Δh) during the holding period (as shown in Fig. 3.9) which is generally proportional to hmax . In general, fifty times increase in maximum indentation load may lead to eight times increase in the indentation depth, which clearly demonstrates the nonlinearity of mechanical properties of the hardened cement paste across different indented scales. Uniform properties of the cement paste can be measured by the MI test using a higher indentation load with a greater indented area. The BSE images of the indented areas under different peak loads are shown in Fig. 3.10, in which the projected area of the residual indent caused by the Berkovich indenter probe is identified as a triangle. The applied peak load increase from 0.5 to 5 N resulting in the corresponding edge length (a) of the triangle increasing from 47.5 to 128.8 µm. The composite phase, including the pore, the hydrate, and the unreacted phases, are all covered by the indent triangles, indicating that the composite phase rather than the individual phase is measured.

56

3 Experimental Techniques

(a)

(b)

(c)

(d)

Fig. 3.10 a Global view of indent grid under indentation load of 0.5 N, b individual indent under indentation load of 0.5 N, c 1.5 N, and d 5 N for w/c = 0.3 paste at the age of 5 months (Wei et al., 2017b)

3.3 Scanning Probe Microscopy (SPM) Technique 3.3.1 Development of SPM Technique Scanning probe microscopy (SPM) is a general term for various probes that are used for imaging and measuring surfaces on a fine scale. This technique is capable of mapping the local variation of the nanomechanical properties without causing plastic deformation to the material (Balooch et al., 2004). The principal ideas of SPM can be dated back to 1928, when Synge proposed a theoretical approach for overcoming diffraction limit in conventional optical microscopy by scanning a small subwavelength aperture in distance of 100 nm from the sample surface. Even if this pioneering research did not introduce most of the key parts of a SPM, its basic idea, a nanoscale probe in proximity to the surface, was already incorporated. The first instrument, which already consisted of the key SPM parts and concepts (scanner, probe, imaging by scanning), was built in 1971 by Young, R D. Ten years later, the first scanning tunneling microscope was built by Binning et al. (1982) who won the Nobel prize in 1986, followed by the first atomic force microscope (AFM) by Binning

3.3 Scanning Probe Microscopy (SPM) Technique

57

et al. (1986), and then followed by a large amount of different other microscopes and related analytical techniques in the next two decades. SPM mapping is a nanomechanical probe-based method but characterized as a non-destructive test with contact stiffness measurement at a small contact force of 2–10 µN (Asif et al., 2001; Balooch et al., 2004; Wilkinson et al., 2015). This technique relates each pixel in the image to the mechanical property values from each tip-sample response during sample surface scanning with an oscillation indenter tip. Therefore, images of mechanical property distribution can exhibit the spatial variation of different phases. SPM mapping excels in charactering the composite materials with sharp contrast of parameters of different components, such as carbon fiber in polymer matrix (Asif et al., 2001), the calcified tissues in human teeth (Balooch et al., 2004), and the organic-rich shales (Wilkinson et al., 2015). Recently, SPM mapping has been adopted to characterize the cementitious materials (Li et al., 2016; Wei et al., 2017a; Xu et al., 2015). Considering that it is conducted at a small contact force, SPM mapping technique enables the acquisition of modulus images with two orders of magnitude higher spatial resolution compared to the point-based methods such as the quasi-static NI (Li et al., 2016; Xu et al., 2015). The high spatial resolution helps to quantify the thickness of the inner product (IP) rims of both cement clinker and slag grain (Wei et al., 2017a). Moreover, the phase identification results by SPM mapping provide precise locations of indents when conducting the NI on individual phases. Li et al. (2016) compared the modulus of cementitious materials obtained by the SPM imaging and the NI and found that the elastic modulus obtained by the NI is greater than that obtained by the SPM mapping. However, the difference between them is not obvious. Xu et al. (2015) also used the SPM mapping technique to successfully characterize the interface transition zone (ITZ) between the unreacted clinker and the hydration product.

3.3.2 Test Method The SPM mapping technique utilizes the dynamic load to reflect the mechanical properties of the scanned region of the samples. By recording the displacement amplitude and the phase lag under the small dynamic load, various dynamic property distribution mappings of the scanned area can be obtained. Quantitative mapping in the form of SPM image is acquired using the direct force modulation operating mode of a Hysitron TriboScope nanoindentation. The mapping mode is accomplished by superimposing a small sinusoidal force on the larger quasistatic force during an indentation test. As illustrated in Fig. 3.11a, the data directly measured from a lock-in amplifier are the contact force, the displacement amplitude of the indenter tip, and the phase shift between the force and the displacement. From these quantitative parameters, the tip-sample contact stiffness and the damping coefficient of the contact can be calculated from the classical equation for a single freedom degree harmonic oscillator. If the geometry of the probe is known, the

58

3 Experimental Techniques

(a)

In-situ topography image

Scanning Probe Microscopy Mode

Diagram

Top plate

Storage Modulus

Center plate Bottom plate

20μm

Quasi-static & dynamic force Displacement

Indenter column

Modulus Mapping

Amplitude & Phase shift

Lock-in Amplifier Support spring

Loss Modulus

Specimen (b) Kf Ks

Ds

z(t)=z0sin(ωt+φ)

Ki

Di

F(t)=F0sinωt

Fig. 3.11 a Illustration of SPM mapping, and b physical model of measuring system (Gao et al., 2018)

storage modulus and the loss modulus can be calculated from the stiffness and the damping by applying the Hertzian contact mechanics. This testing system can be modeled as a physical system with force applied to a mass that is attached to the two fixed Voigt elements (Fig. 3.11b). The two elements represent the stiffness and damping of the transducer and the contact, respectively. The model yields a differential equation describing the relationship between the applied force and the motion: F(t) = F0 sin(ωt) = m(

dz d2 z ) + D( )+K z dt 2 dt

(3.34)

where F 0 is the amplitude of the applied sinusoidal force, m is the mass of the transducer, ω is the angular frequency, z is the sinusoidal displacement response, D = Di + Ds is the combined damping coefficient of system and the contact, K = K i + K s is the combined stiffness of the spring and the contact, K f is the stiffness of the system frame, which is much greater than that of the contact and thus can be neglected. The solution to Eq. (3.34) is: z = z 0 sin(ωt + ϕ)

(3.35)

3.3 Scanning Probe Microscopy (SPM) Technique

F0 z0 = ⎡ ⎤1 (K − mω2 )2 + (Dω)2 2

59

(3.36)

Dω where ϕ is the phase of the displacement, tan(ϕ) = K −mω 2. , ,, The storage modulus (E ) and the loss modulus (E ) of the contact can be calculated as (Herbert et al., 2008):

√ , E π F0 = cos ϕ √ 2 1−ν z0 2β A √ ,, π E F0 = sin ϕ √ 1 − ν2 z0 2β A

(3.37)

(3.38)

where ν is Poisson’s ratio of indenter, β is the geometrical correction factor, and β = 1.034 for Berkovich indenter, A is the projected contact area between the indenter and the sample. Storage modulus represents the capacity of the material to store energy, larger storage modulus means greater elastic property. Loss modulus represents the capacity of the material to dissipate energy as heat, and larger loss modulus represents more viscoelastic. They are generally used to characterize the viscoelastic response of a material. The SPM tests are designed to map the local variations of mechanical properties without causing plastic deformation to the materials. Therefore, the load is always chosen at the scale of 10–6 N to achieve the displacement amplitude at the scale of 10–9 m no matter it is applied on polymer or biomaterials. For the cement paste, different sets of loads, such as 2 ± 1.5, 4 ± 1.5, 4 ± 3.5 µN and 8 ± 3.5 µN (quasistatic ± dynamic), are applied to measure the mechanical properties of individual phases. In SPM mapping, the direct output parameters are the contact force on the sample surface, the displacement amplitude of the indenter tip at every pixel, and the phase shift between the contact force and the displacement. The modulus could be calculated based on these three parameters by Eqs. (3.37) and (3.38). The modulus mapping image distinguishes phases by their colors which reflect the local variation of storage modulus. For an area with slag grain, the BSE image and the gradient image are shown in Fig. 3.12a and b, respectively. The mapping color appears black, red, blue, and green for storage modulus varying from low to high (Fig. 3.12c–f). The black area represents the pores with the lowest storage modulus, the red area represents the outer product (OP) mixed with other hydration products, the blue area represents the inner product (IP) or other hydration products with higher storage modulus, and the unreacted grain (clinker or slag) appears green representing the highest modulus. The overall SPM image is shown in Fig. 3.12g, which is the overlapping of the four images shown in Fig. 3.12c–f.

60

3 Experimental Techniques

Void

Slag (a)

(c)

(d)

(g) (b)

(e)

(f)

Fig. 3.12 Modulus mapping-based morphology of slag-blended paste a BSE image, b gradient image, mapping image of c porosity, d outer product and other hydrates, e inner product, f unreacted grain, and g entire mapping image (Wei et al., 2017a)

3.4 Continuous Stiffness Measurement Test Continuous stiffness measurement (CSM) is another dynamic method based on indentation technique, capable of recording the force (P)–indentation depth (h) devel) opments under various strain rates. Based on the concept of true strain (ε = dh h dh 1 = . CSM is a (Nieh et al., 2002), the indentation strain rate is defined as ε˙ = dε dt h dt promising technique to illustrate strain rate effect on the time-dependent deformation of cement paste. Indeed, this technique has been successfully used for investigating the strain rate effect on the mechanical property of composites and multilayered materials (e.g., carbon-fiber composite and magnetic tapes) (Li & Bhushan, 2002). The schematic measuring system of CSM is shown in Fig. 3.13a and b. Similar to the SPM mapping technique (Sect. 3.3), a harmonic force F (t) = F 0 eiωt (F 0 is the amplitude of the applied sinusoidal force) is applied to the nominally increasing load (P) to the indenter during the measurement. The depth response of the indenter z (t) = z0 ei(ωt −ϕ ) (z0 is the amplitude of the sinusoidal depth) at the excitation frequency (ω) and the phase angle (ϕ) between the force and the depth are measured continuously as a function of depth. However, unlike the SPM mapping technique, the applied harmonic force is much smaller than the quasi-static force, which make the indentation force (P)–indentation depth (h) curve obtained from CSM look like that obtained from the quasi-static method based on indentation technique. A typical P–h curve obtained from the CSM measurement is shown in Fig. 3.13c. The second-order differential equation describing the relationship between the harmonic force and the response of indenter during the loading stage is (Li & Bhushan, 2002): M z¨ (t) + D z˙ (t) + K z(t) = F(t)

(3.39)

3.4 Continuous Stiffness Measurement Test

61 10 Kf

8

Dc

S

P (N)

Di

Ks

harmonic motion

dewell time=10s end of the harmonic motion

6 4 thermal drift time =100s

2 M

0 z(t)=zo e

(a)

i(wt-Φ)

0

F(t)=Fo e iwt

5

(b)

10 15 20 25 30 35 h (μm)

(c)

Fig. 3.13 Continuous stiffness measurement by instrumented indentation: a Agilent nanoindenter G200, b model of dynamic measuring system, c measured P–h curve (Liang et al., 2017)

where M is the moving mass of the transducer; D = Di + Dc is the effective damping coefficient, Di is the damping coefficient of the indenter, Dc is the damping coefficient of the contact; K = K s + S is the effective stiffness, K s is the stiffness of the transducer, S is the stiffness of the contact or contact stiffness of the indented material. By taking the first and second derivatives of z(t), one can obtain the following equation: ⎧

z˙ (t) = z0 i ωei(ωt−ϕ) z¨ (t) = −z0 ω2 ei(ωt−ϕ)

(3.40)

By substituting Eq. (3.40) into Eq. (3.39), one can obtain the following equation: −Mz0 ω2 ei(ωt−ϕ) + Dz0 iωei(ωt−ϕ) + Kz(t)ei(ωt−ϕ) = F0 eiωt

(3.41)

Multiplying every term in Eq. (3.41) by eiϕ /(z0 eiωt ) yields: −Mω2 + i Dω + K =

F0 iϕ e z0

(3.42)

Invoking Euler’s rule to substitute for eiϕ yields: −Mω2 + i Dω + K =

F0 (cos ϕ + i sin ϕ) z0

(3.43)

Based on Eq. (3.43), one can easily obtain the following equation: ⎧

−Mω2 + K = Fz00 cos ϕ Dω= Fz00 sin ϕ

(3.44)

62

3 Experimental Techniques

Therefore, the phase angle ϕ, by which the response lags the excitation depends on the components through the relationship: tan ϕ =

Dω K − Mω2

(3.45)

Then, applying this model to an indenter that is hanging free. When the indenter is not in contact, the model components are those of the indenter alone: K = K s and D = Di . The springs supporting the indenter shaft are the primary contributors to K s . If capacitive plates are used for sensing or actuation, they are the primary contributors to Di . Therefore, one can obtain the following equation: ⎧ ⎨ −Mω2 + K s = ⎩ Di ω =

F0 z0

| | sin ϕ |

F0 z0

| | cos ϕ |

free−hanging

(3.46)

free−hanging

Next, considering the situation when the indenter is in contact with the test material, as shown in Fig. 3.13b, K can be expressed as: 1 / + Ks K = / 1 Kf +1 S

(3.47)

where K f is the stiffness of the frame. Therefore, the contact stiffness can be expressed as: ⎤−1

⎡ ⎢ S=⎣

1

F0 z0

cos ϕ −

F0 z0

| | cos ϕ |

−

1 ⎥ ⎦ Kf

(3.48)

free−hanging

By substituting Eq. (3.46) into Eq. (3.48), one can obtain the following equation: ⎡ S=

1 1 − 2 (F0 /z 0 ) cos ϕ − (K s − Mω ) Kf

⎤−1 (3.49)

Once the contact stiffness is determined, the contact depth (hc ) and the contact area (Ac ) of the indented material can be easily calculated according to the type of indenter. Finally, the elastic modulus and contact hardness can be determined at any indentation depth (or indentation force) based on the Oliver–Pharr method (Oliver & Pharr, 1992), as shown in Sect. 3.2.5.

3.5 Nanoscratch Technique

63

3.5 Nanoscratch Technique 3.5.1 Development of Nanoscratch Technique The scratch test, first appeared in 1722, is a very old method to measure the hardness of materials. French scholars first proposed the concept of scratching hardness test, that is, to make a metal bar scratch on the surface of the material under a fixed force, and the hardness of the material is determined by the scratching characterization. In 1822, Friedrich Mohs first used the scratch method to compare the hardness of materials. In 1950, Heaven first carried out the scratch test on the film material and successfully measured the adhesion between the film material and the substrate (Bhushan et al., 1995). The friction properties of thin films were also successfully measured by using this technique (Charitidis et al., 2003; Huang et al., 2004). Subsequently, more researchers carried out research to promote the development of scratch technique and gradually advance it to nanoscratch technique, which can be used to conduct scratch test at nanoscale. The nanoscratch testing process is mainly composed of three steps. The first step is prescanning process with a normal force of 2 µN to prescan and measure the surface morphology and roughness of the sample. The second step is loading process with a target load (generally 10 mN) applied on the sample surface to complete the scratch process. Meanwhile, the scratching depth and the scratching length can be recorded by the instrument automatically. The third step is the post-scanning process with a normal force of 2 µN to post-scratch the tested surface to measure the residual scratch depth. Compared to the NI technique, nanoscratch can perform continuous mechanical properties measurement. Xu et al. (2017) investigated the properties of cementitious materials by nanoscratch test, and different phases were identified through the scratch depth and the coefficient of friction (COF). Hoover and Ulm (2015) and Akono and Ulm (2014, 2017) derived the formula based on the linear fracture mechanics theory for calculating the fracture toughness of materials from the scratched results, and applied it to various types of materials including cementitious materials, shales, and amorphous polymers. Wei et al. (2021) proposed a phases identification method based on the fracture toughness distribution by using the nanoscratch method.

3.5.2 Testing Method There are two loading modes in nanoscratch test, namely the slope loading mode and the constant loading mode. In the slope loading mode, the probe is indented to the sample at a constant vertical loading rate, which is usually used to study the critical failure load of films, and now can be used to characterize the fracture properties of cementitious materials, crystal material, and metallic materials. In the constant loading mode, the probe is indented to the sample at a constant vertical load, which

64

3 Experimental Techniques

(a)

Z

(b) FV (mN)

VV (mN/μm) Z

p(d)

A(d)

d VT (μm/s)

X

Y

Z

(c) FT (mN)

d

L

X

Fig. 3.14 Scratching mode from a side view, b front view, and c side view at the end of scratching (Wei et al., 2021)

is usually used to measure the coefficient of friction (COF) (Godara et al., 2007; Youn & Kang, 2006). The schematic illustration of the scratching with slope loading mode is shown in Fig. 3.14. X-axis represents the transverse scratching direction, Z-axis represents the vertical loading direction, V T is the transverse scratching speed, V V is the vertical loading rate, F V is the vertical force, F T = COF × F V is the transverse force and COF is the coefficient of friction, d is the scratching depth, L is the scratching length, A(d) is the projected area of the probed volume to the plane perpendicular to the transverse scratching direction, p(d) is the side length of the projected area A(d). The scratching depth (d) and the transverse scratching force (F T ) can be recorded during the scratching test, from which the fracture toughness (K C ), the fracture energy (ζ C ), and the scratching hardness (H) of the tested materials can be calculated by Eqs. (3.50)–(3.52), respectively (Akono & Ulm, 2017; Hoover & Ulm, 2015), and the definition of each parameter can be found in Fig. 3.14. KC = √

FT 2 p(d) A(d)

ζC = H=

(3.50)

K C2 M

(3.51)

FT A(d)

(3.52)

where M is the indentation modulus that can be measured by indentation test.

3.5 Nanoscratch Technique

65

3.5.3 Parameter p(d)A(d) for Different Indenter Previous studies have found that the properties of materials (including hardness, strength, friction and fracture properties, etc.) can be associated with the horizontal scratching force and the contact area between the probe and the scratched material projected in the scratch direction. The horizontal scratching force can be easily measured, but the geometry of different indenters is different, particularly, according to Eqs. (3.50) and (3.52), p(d)A(d) is an important parameter for calculating the fracture toughness. Therefore, it is necessary to quantify the geometric characteristics of different indenters. There are six commonly used indent probes, including conical indenter, spherical indenter, flat indenter, Berkovich indenter, Cube-corner indenter, and Vickers indenter, which have been displayed in Fig. 3.1. For the convenience of reference, the schematic diagrams of these six indenters are also shown in Fig. 3.15. (a) R

(c)

(b)

O

B

R

A A

R

O

O

α A (d)

C

(e)

B

O

B

C O

E E β

D α

Scratching direction

C’ A’ (O’, D’) B’ B

C A (O) θ D

B O

D

E

D α

βα

90°

A

A

(f)

F

A D’ A’ (O’, C’, E’) B’

(h) Scratching direction

β

(g)

C

C

D

B

A (O) θ E

Fig. 3.15 3D geometry of a conical indenter, b spherical indenter, c flat indenter, d Berkovich indenter, e Cube-corner indenter, and f Vickers indenter; top view of g triangular pyramid indenter (Berkovich indenter or Cube-corner indenter) and h Vickers indenter

66

3 Experimental Techniques

Table 3.2 Parameters B and ε for axisymmetric indenters (Akono et al., 2012)

Indenter type

ε

B

Conical

1

cot α

Spherical

2

1/(2R)

Flat

→∞

1/(Rε−1 )

General axisymmetric probe The scratching depth of an axisymmetric probe (such as conical indenter, spherical indenter, and flat indenter) can be defined by a monomial function of the form: z = Br ε

(3.53)

where B is the height at unit radius, ε is the degree of the homogeneous function, and r is the radius of the probe indented into the material surface. The values of B and ε are listed in Table 3.2 for axisymmetric indent probe (Akono et al., 2012). The projected contact area A(d) and parameter p(d) are then given by (Akono et al., 2012): ( )1 2Bε d ε +1 A(d) = ε+1 B

(3.54)

( ) 1ε d p(d) = β(d) B

(3.55)

where β is a dimensionless parameter defined by: ⎡ 21 ( )− 2ε ⎧1 ⎡ 2ε−2 2 d x β(d) = 2 1 + (εd) B

(3.56)

0

The scaling of the horizontal force F T and the variation of function p(d)A(d) with d then read (Akono et al., 2012): FT ∝ 2K C

( ) 1ε + 21 ( ) 21 ε d β(d)B B ε+1

p(d)A(d) =

( )2 2Bε d ε +1 β(d) ε+1 B

(3.57)

(3.58)

Conical indenter For a conical indenter with a half-apex angle α (as shown in Fig. 3.15a), by referring to Table 3.2, Eqs. (3.54)–(3.58) become:

3.5 Nanoscratch Technique

67

d2 cot α

(3.59)

d β(d) cot α

(3.60)

2 sin α

(3.61)

A(d) = p(d) =

β(d) =

1

3 (sin α) 2 KC d 2 FT ∝ 2 cos α

p(d)A(d) = 2

sin α 3 d (cos α)2

(3.62) (3.63)

Spherical indenter For a spherical indenter with radius R (as shown in Fig. 3.15b), Eqs. (3.54)–(3.58) become: A(d) =

2 (2Rd)3/2 3R

(3.64)

p(d) =

√ d 2Rdβ( ) R

(3.65)

) ( √ √ 1 d = 1 + 2x + √ arcsin( 2x) β x= R 2x ⎡ ( )⎤ 21 d 1 1 FT ∝ 4 β KC d R 2 3 R ( ) d 8 d2 R p(d)A(d) = β 3 R

(3.66)

(3.67) (3.68)

Flat indenter For a flat indenter with radius R (as shown in Fig. 3.15c), Eqs. (3.54)–(3.58) become: A(d) = 2Rd p(d) = 2R(1 + β(d) = 2(1 +

(3.69) d ) R

d ) R

(3.70) (3.71)

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3 Experimental Techniques

/ FT ∝ 2R 2d(1 +

d )K C R

p(d)A(d) = 4R 2 d(1 +

d ) R

(3.72) (3.73)

Triangular pyramid indenter (Berkovich indenter and Cube-corner indenter) For the non-axisymmetric indenter (as shown in Fig. 3.15d and e), the parameter p(d)A(d) depends much on the scratching direction due to the geometrically nonsymmetric characteristic (Hodge & Nieh, 2004; Zhao et al., 2016). As shown in Fig. 3.15g, when the angle between scratching direction and BD is θ, θ ∈ [0°, 60°], p(d)A(d) can be expressed as: tan β × [sin θ + sin(60◦ − θ )] · d 2 (3.74) cos α ⎫ ⎧/ / ⎨ 4 × tan2 β × cos2 θ 4 × tan2 β × sin2 (30◦ + θ ) ⎬ )+ [1 + ] (1 + p(d) = d × ⎭ ⎩ 3 × cos2 α 3 × cos2 α A(d) =

(3.75) tan β × [sin θ + sin(60◦ − θ )] (3.76) cos α ⎫ ⎧/ / ⎨ 2 2 2 2 ◦ 4 × tan β × cos θ 4 × tan β × sin (30 + θ ) ⎬ )+ ] Q= (1 + [1 + ⎭ ⎩ 3 × cos2 α 3 × cos2 α W=

(3.77) FT ∝

√

WQK C d 3/2

p(d) A(d) = WQd 3 ∝ d 3

(3.78) (3.79)

Vickers indenter Similar to the Berkovich indenter and the Cube-corner indenter, p(d)A(d) of a Vickers indenter (as shown in Fig. 3.15f) is related to the scratching direction (Fig. 3.15h), for the angle between the scratching direction and BE, θ ∈ [0°, 90°], p(d)A(d) can be expressed as: A(d) =

√ 2 tan β cos(45◦ − θ ) 2 ·d cos α

(3.80)

3.5 Nanoscratch Technique

69

/

2 tan2 β cos2 (45◦ − θ ) +1 cos2 α √ 2 tan β cos(45◦ − θ ) W= cos α / 2 tan2 β cos2 (45◦ − θ ) +1 Q=2 cos2 α √ FT ∝ WQK C d 3/2

p(d) = 2d

p(d)A(d) = WQd 3 ∝ d 3

(3.81)

(3.82)

(3.83) (3.84) (3.85)

where W and Q are the dimensionless parameters corresponding to the geometry of the indenter and the scratching direction.

3.5.4 Redefining Vertical Loading Rate The original vertical loading rate (V V-original ) is defined as (in N/s): VV−original = FV−max /t

(3.86)

where F V-max and t are the maximum vertical load (in N) and the transverse scratching time (in s) applied during the nanoscratch test, respectively. The transverse scratching time (t) is a variable which equals to the scratching length (L) divided by the transverse scratching speed (V T ), as shown in Eq. (3.87). L is normally set as a constant, such as 100 µm in Wei et al. (2021) and Kong et al. (2021). t = L/ VT

(3.87)

By combining Eqs. (3.86) and (3.87), the original vertical loading rate can be calculated as VV-original = F V-max × V T /L, which is affected by variables of both the maximum vertical load (F V-max ) and the transverse scratching speed (V T ). Therefore, the V V-original is not beneficial for analyzing the effect of the single parameter (F V-max or V T ) on the measured results. With the above concerning, in this book, the vertical loading rate is redefined as the V V = F V-max /L (in N/m), which is a constant once the F V-max and L are fixed during the test, and this can avoid the influence from the variable transverse scratching speed.

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3 Experimental Techniques

3.5.5 Calibration of Scratching Probe From Sect. 3.5.3, the p(d)A(d) is one of the most important parameters to calculate the fracture toughness of scratched materials, which is determined by the shape of the indenter probe. Once the probe used for scratching test is selected, the indented materials have no effect on this parameter for the same indenter probe. However, for the Berkovich probe with the three-side pyramidal feature used in this book, the value of p(d)A(d) depends greatly on which side of the probe faces scratching direction (Hodge & Nieh, 2004; Zhao et al., 2016), as the angle θ shown in Fig. 3.15g varies with the probe scratching direction. Therefore, it is recommended that the probe calibration should be conducted prior to the scratching test under the same probe scratching direction. A method proposed by Hoover and Ulm (2015) is used for the probe calibration by performing a scratching test on a material with a known fracture toughness. From Eqs. (3.50) and (3.79), the parameter p(d)A(d) can be easily deduced as: p(d)A(d) =

FT2 ∝ d3 2K C2

(3.88)

For the same probe in the same batch of scratching tests, the relationship between d 3 and p(d) A(d) remains constant even for different scratched materials. Therefore, the/same p(d)A(d) versus d 3 /relationship can be used / to calculate the parameter FT2 2K C2 for material A (FT2 A 2K C2 A ) and B (FT2 B 2K C2 B ). If a scratching test is conducted on the material A with a known fracture toughness (K CA ), the transverse scratching force F TA and the penetration depth d A can be recorded and then used to obtain the relationship between p(d)A(d) and d 3 from Eq. (3.88). Finally, the fracture toughness of the unknown material B (K CB ) can be calculated from the p(d B ) A(d B ) and the F TB obtained from the scratching test on material B. Fused silica with fracture toughness of 0.6 MPa m½ in room temperature is normally used as the reference material (material A) to calibrate p(d)A(d) versus d 3 . The relationship between p(d) A(d) and d 3 is obtained first by scratching the reference fused silica, as shown in Fig. 3.16a, and the relationship between p(d)A(d) and d can be obtained (Fig. 3.16b). This relationship can then be used to calculate the fracture toughness of the unknown materials (material B) in the same batch of scratch tests.

3.5.6 Start Point of Scratching The surface roughness of the polished clinker phase with denser microstructure is usually less than that of the hydrates phase with soft and porous microstructure. The clinker phase should be selected as the start point for scratching, such that a greater vertical load can be applied to the clinker phase at the beginning as well as to the

3.6 Scanning Electron Microscope (SEM) Test and Associated Techniques

p(d)A(d) (×109nm3)

20 15 Fitted line Scratching #1 Scratching #2 Scratching #3

10 5

(b)

25

p(d)A(d) = 2.2808 (d3) p(d)A(d) (×109nm3)

(a)

25

20

71

p(d)A(d) = 2.2808 (d)3 Fitted line Scratching #1 Scratching #2 Scratching #3

15 10 5 0

0 0

2

4

6 8 10 d3 (×109nm3)

12

14

0

0.5

1 1.5 d (×103nm)

2

2.5

Fig. 3.16 Calibration results of fused silica for a p(d)A(d) versus d 3 and b p(d)A(d) vs. d (Wei et al., 2021)

Fig. 3.17 SEM image of the scratched path with the total length of 100 µm

Scratching path (100μm)

Start point Clinker (46μm)

Hydrates (54μm) ITZ

porous hydrates phase at the later stage of scratching to obtain more accurate data of the hydrates phase which may require greater vertical load than that of the clinker phase. For example, Wei et al. (2021) conducted scratching tests with scratching length of 100 µm and selected the clinker phase as the starting point which is about 50 µm away from the inner boundary of ITZ around the clinker to ensure that all the three phases (including the clinker, the ITZ, and the hydrates) can be scratched within one scratching path (Fig. 3.17).

3.6 Scanning Electron Microscope (SEM) Test and Associated Techniques 3.6.1 Development of SEM Technique The idea of SEM was proposed in the 1930s, and the earliest demonstration is attributed to Knoll who obtained the first SEM images of the surface of a solid (Knoll, 1935; Knoll & Theile, 1939). And then, the first true SEM was developed in 1938

72

3 Experimental Techniques

by Von Ardenne who established the underlying principles of SEM, including the formation of the electron probe and its deflection, the positioning of the detector and the ways of amplifying the very small signal current (Von Ardenne, 1938a, 1938b). In 1942, Zworykin et al. (1942) upgraded Von Ardenne’s SEM, and the Zworykin’ SEM became the prototype of the modern SEM, although its minimum resolution was 50 nm. Since 1948, the SEM was upgraded five times by researchers (Broers, 1965; Everhart & Thornley, 1960; Oatley & Everhart, 1957; Pease & Nixon, 1965; Smith, 1956; Smith & Oatley, 1955; Wells, 1957), and the first commercial SEM was produced with a resolution of 10 nm in 1965. The advances of SEM promote the major breakthroughs in the study of the microstructure, composition, and properties of bulk materials. Because of the short wavelengths of electrons and their ability to be focused using electrostatic and electromagnetic lenses, SEM creates magnified images which reveal microscale information on the size, shape, composition, crystallography, and other physical and chemical properties of samples. Meanwhile, the strong interaction of electrons with matter produces a wide variety of useful signals that reveal all kinds of secrets about matter at the microscopic and even nanoscopic level.

3.6.2 Principle of SEM To understand how SEM works, it is necessary to first look at what happens when a beam of electrons collides a material. Under the high voltage, the electron beam will be launched from the cathode (usually tungsten wire), which will then be accelerated and focused by accelerating voltage and lens, respectively, resulting a very small electron beam when it reaches the sample surface. The electron beam that reaches the sample surface is called the primary electrons. When the primary electrons collide with the sample surface, a wide range of useful interactions can occur, such as the elastic and the inelastic collisions with the atoms of the materials, causing the generation of various charged particles and photons. These collisions produce three main products, the secondary electrons (SEs), the backscattered electrons (BSEs), and the characteristic X-ray, which can be detected by the microscope to form the image. The process is shown schematically in Fig. 3.18. In all cases, images are formed by scanning the electron beam over the surface in a raster and the signal is detected at each point to give the intensity of the corresponding point in the image. Secondary electrons (SEs) are the result of the inelastic collision between the electron beams and the electrons in the sample material atom. The electrons in the material atom will escape from the atom, which are called the secondary electrons. The angle between the incident electron beam and the sample surface determines the energy of the secondary electrons. The SEs have much lower energy than that of the incident electrons, so they can escape only from the near surface of the specimen and can be detected to form the image of the sample surface morphology. When the number of the detected secondary electrons is large in one region, more electron

3.6 Scanning Electron Microscope (SEM) Test and Associated Techniques

73

Electron beam Secondary electrons

Characteristic X-ray

Backscattered electrons Sample surface SEs generated zone BSEs generated zone Characteristic X-ray generated zone Continuum X-ray generated zone Fluorescent X-ray generated zone Interaction volume (1 ~ 3 μm)

Fig. 3.18 Schematic representation of the interaction of electrons with matter and the signals generated (after Scrivener et al., 2016)

Fig. 3.19 SEM image of cement paste with brighter pore edge

Brighter pore

50 μm

energy can be detected, and the brightness of the region in the image will be higher than that of other regions. The surface morphology, such as cracks and pores, is the main factor affecting the brightness of the secondary electronics images, because they have greater amounts of surface from which the weak SEs can escape, causing brighter edges and tips in the image, as shown in Fig. 3.19.

3.6.3 Principle of BSE When the electron beam collides with the sample surface, part of the incident electrons will be reflected by the atoms in the solid sample, and the reflected electrons are called the backscattered electrons. Backscattered electrons (BSEs) are the result of elastic collisions, and the energy of the BSEs is similar to that of the incident electrons; therefore, they can escape from the greater depth in the specimen. However, their energy and direction will inevitably be disturbed by other atoms, which will

74

3 Experimental Techniques

Table 3.3 Atomic numbers and backscattered coefficient of chemical components in cement-based materials (Zhao & Darwin, 1992)

Element

Atomic number (C i )

Backscattering coefficient (ηi )

H

1

0.000

O

8

0.091

Mg

12

0.141

Al

13

0.153

Si

14

0.164

S

16

0.186

Ca

20

0.227

Fe

26

0.280

reduce the imaging accuracy and the resolution of the BSE images. On the other hand, SEs are less disturbed as they escape from the near surface of the sample, resulting higher resolution. After polishing and obtaining the smooth sample surface, the contrast of the BSE images depends on the average atomic number of different phases represented by the backscattered coefficient ηi (Eq. 3.89). ηi = −0.0254 + 0.016Ci − 1.86 × 10−4 Ci2 + 8.3 × 10−7 Ci3

(3.89)

where i represents the type of element, C i is the atomic number of element i. Equation (3.89) is not suitable for H element (ηH = 0). The atomic numbers and the backscattered coefficients of the common elements in cementitious materials are summarized in Table 3.3. The backscattered intensity (η) of a compound can be calculated from the weighted average (wi ) of the elements in the compound: η=

∑

wi ηi

(3.90)

The region with high backscattered intensity reflects strong backscattered electron signal and shows a large gray level in BSE images. Gray level divides the color from black to white into 256 levels, and the range of the gray level is 0~255. The larger the gray level, the higher the brightness. Therefore, the region with higher backscattered intensity shows brighter phase in the BSE image (Fig. 3.20), and the different components of a cement paste can be easily identified.

3.6.4 Principle of EDS When an incident electron collides with the atom of the sample, the inner layer electron will escape from the collided atom, resulting in an unstable state for the collided

3.7 X-Ray Computed Tomography (CT) Technique

75

Inner product CSH Unreacted clinker Outer product CSH Pore 50 μm Fig. 3.20 Typical BSE image of cement paste

atom due to the lack of inner electron. To maintain the stable state, the electrons in the outer layer of the collided atom will migrate to the inner layer and release energy in the form of photons, producing characteristic X-rays with characteristic energy (E). Each element has its own unique characteristic X-ray which can be captured by X-ray detectors for elements analysis. For the characteristic X-ray generated from collision between the incident electron and the collided atom, two methods can be used for element analysis: energy dispersive X-ray spectroscopy (EDS) and wavelength dispersive X-ray spectroscopy (WDS). EDS with high detection efficiency is often used combined with SEM. Each characteristic X-ray has its own unique characteristic energy which can be detected by EDS for distinguishing the various elements. When the X-ray photons are detected by the Si-Li crystal detector that equipped in the EDS system, electron– hole pairs can be generated in the Si-Li crystal. At low temperature, the energy consumed for generating each electron–hole pair is 3.8 eV. The number of electron– hole pairs generated by characteristic X-ray photons with energy E is N = E/3.8. The N electron–hole pairs can produce corresponding voltage pulse signal, which can be detected by the signal receiver to judge the type of X-ray and to determine the type of element.

3.7 X-Ray Computed Tomography (CT) Technique 3.7.1 Development of X-Ray CT As a non-destructive technique, X-ray computed tomography (CT) can be used to visualize the cement-based material and to obtain its microstructure in three dimensions. The fundamentals of this imaging technique can be traced back to John Larton, who proved in 1917 that an n-dimensional object could be reconstructed from its (n1)-dimensional projection. And then, Cormack (1963, 1964) set up the mathematical foundation for the CT image reconstruction from 1963 to 1965. About 10 years later, by distinguishing the strong contrast of density among solid, liquid, and air in the human body, Hounsfield was the first one to invent the X-ray CT and described

76

3 Experimental Techniques

the results of the brain examination at the British Radiology Annual Conference (Ambrose, 1973). X-ray CT was used in visualization of human skeleton and brain at the beginning of its invention (Gawler et al., 1974; Lcdley et al., 1974; Paxton & Ambrose, 1974). Later, the use in soil, rock and geology increased gradually. High-resolution computed tomography, also known as micro-CT, was proposed in the 1980s, with the X-ray, the gamma-ray, and the X-ray synchrotron radiation as the radiation sources. However, the gamma-ray has been eliminated gradually due to its low brightness. Since 1980s, the X-ray source and the synchrotron radiation micro-CTs have been developed rapidly. The high brightness of the synchrotron radiation makes it irreplaceable in terms of high spatial resolution and low noise (Baruchel et al., 2006), but the cost of the synchrotron radiation CT equipment is much higher compared to the X-ray CT and requires a smaller sample size as well. Currently, the resolution of the X-ray CT equipment has been improved close to that of the synchrotron radiation CT and therefore has been more widely used.

3.7.2 Principle of X-Ray CT X-ray is electromagnetic wave, which is also known as Röntgen ray. The energy of each photon in the X-ray is proportional to the frequency (ν) of the X-ray, that is E = hν = hc/λ. h is Planck’s constant and equals to 6.6621 × 10–34 J·s; ν is frequency of the X-ray; c is the speed of light and equals to 3 × 108 m/s; λ is the wavelength of X-ray. In most CT equipment, the generation of X-ray depends on the Coolidge tube, also known as a hot cathode tube. In the Coolidge tube, the thermionic effect is exploited for electron production. Under the high voltage, the cathode (usually tungsten) will generate heat and release electrons, and the electrons will collide with the anode (usually molybdenum) under the acceleration of voltage. When electrons collide with the anode, X-rays are generated. When the X-ray passes through the sample, part of the photon will be absorbed by the sample. Due to the variation of the number, energy and direction of photons, the intensity of the X-ray decreases exponentially, which is usually called attenuation. The X-ray detector can capture the X-ray that has passed through the sample to measure the attenuation and convert it into electrical signals, which is then converted into binary coding information. The binary coding information can be visualized on the computer and displayed as CT images, which can be further processed by the corresponding software to extract the interested phase structure.

3.7.3 Commonly Used CT Scanners Two types of CT are normally used to meet various types of purposes.

3.7 X-Ray Computed Tomography (CT) Technique Fig. 3.21 Illustration of the commonly used clinical CT scanner

77

X-ray source X-ray Patient (or sample)

X-ray detector

Clinical CT Scanners The design of clinical CT scanners is based on the needs of patients. The X-ray source and the X-ray detector in clinical CT are designed to rotate around patients at high speed, because it is not feasible to rotate patients. The schematic diagram of clinical CT scanner is shown in Fig. 3.21, which is the most popular one in clinical practice. The X-ray source and the X-ray detector are located in the same circular plane and are placed on both sides of the circle center. The scanning images of the patient or the whole sample can be achieved by moving the X-ray source and X-ray detector, and the patient (or sample) needs to be placed in the fan-shaped area of the X-ray and remain absolutely static. The design of clinical CT usually needs to consider the scanning time and the X-ray dosage, because they are closely related to the time that the patient can bear, resulting in the lack of scanning accuracy and the lower resolution. However, the clinical CT can satisfy the scanning at macro scale. Industrial CT Scanners Compared to the clinical CT, the industrial CT can use a higher intensity X-ray source to help achieve higher-precision scanning without considering the X-ray dosage and the scanning time, but the scanning time will increase significantly. The most fundamental difference between the industrial CT and the clinical CT is that the X-ray source and X-ray detector in industrial CT remain absolutely static, while the sample is rotating at a high speed. The schematic diagram is shown in Fig. 3.22. This improvement enables the industrial CT system to have higher accuracy and stability (Kruth et al., 2011). There are two types of commonly used industrial CT, as shown in Fig. 3.22. Industrial fan beam CT scanner is similar to the clinical CT, in which the X-ray source only emits one 2D fan-shaped X-ray. The sample can be raised in a spiral way or stepped way, so as to make the X-ray detector obtain the slice data of the whole sample. Cone beam CT can emit three-dimensional X-ray to scan the whole sample. However, when the sample is too large, the industrial cone beam CT cannot scan the whole sample at one time. The sample can also be raised in a spiral way or stepped way to complete the scanning of the whole sample (De Chiffre et al., 2014).

78

3 Experimental Techniques X-ray detector

(a)

Sample

Sample X-ray source

X-ray detector

(b)

X-ray source

Fig. 3.22 Illustration of a industrial fan beam CT scanner, and b industrial cone beam CT scanner

Obviously, the scanning efficiency of cone beam CT is much higher than that of the fan beam CT. However, the cone beam CT will produce image artifacts during the imaging process, and the CT images obtained from fan beam CT have higher accuracy than that obtained from the cone beam CT. Generally, the fan beam CT becomes the first choice when measuring the microstructure of the cement-based materials. For Testing Cementitious Materials Both clinical CT and industrial CT can be used to measure the microstructure of the cementitious materials. The clinical CT can test larger samples, but its resolution is not as good as the industrial CT. The industrial CT can achieve resolution in µm level, but it requires smaller-size sample. It is necessary to select the appropriate CT equipment according to the size of the tested sample. Kong et al. (2020) reviewed the application of the CT equipment in the field of cement-based materials and found that the clinical CT is more suitable for testing the macroscopic properties of the cementbased materials, and the industrial CT is more suitable for measuring the propagation of internal cracks of concrete under compression (Zhou et al., 2009), tension (Tian et al., 2015) or fatigue load (Dang et al., 2015), and the change of micropore structure of cement paste under freeze–thaw condition (Wei et al., 2019) or alkali aggregate reaction (Marinoni et al., 2012), for observing carbonation depth under carbonation condition (Yang et al., 2018), and for characterizing the permeability (Zhang, 2017). In Wei et al. (2019), a Skyscan 1172 X-ray Micro-CT was used to scan the paste sample for pore size distribution. The sample size was ϕ5 mm × 10 mm. A power setting of 100 kV and 50 µA was used for the fan beam scan with 1247 projection views. The resolution was 2.97 µm. Each scan was made every 2.97 µm along the height of the sample. A 2D microstructure image was then pictured per scan based on the X-ray absorption value that correlates to the density of different phase, from which a 3D digital image was reconstructed (Fig. 3.23). The cubic volume of interest of 3373 voxels was extracted from the middle of the cylindrical sample, because there is abnormal increase in the gray level close to the boundary of the sample due to the ray attenuation, the scattering, and the noise which can cause misinterpretation of the scanned image (Hu et al., 2015). Considering the characteristic length of the individual phase in the hardened cement paste is about 10–6 –10–5 m (Olivier et al.,

3.8 Mercury Intrusion Porosimetry (MIP) Technique

Micro-CT specimen

Scanned length 3.7mm

79

Focus area 337 2 pixels

Series of slices

Slice

Diameter 5mm Extracted cube from the Reconstructed middle of the sample 3D volume

Cube volume: 3373 voxels

1247 slices

337 slices

Fig. 3.23 X-ray CT scanning and image processing of paste sample (Wei et al., 2019)

2003), the reconstructed cubic volume with size of 1001 × 1001 × 1001 µm can represent the homogeneous property of the tested sample. The image acquisition time for each sample, that includes both scanning and reconstructing time, was about two hours.

3.8 Mercury Intrusion Porosimetry (MIP) Technique 3.8.1 Development of MIP The theoretical basis of Mercury intrusion porosimetry (MIP) can be traced back to 1921, when Washburn (1921a) first discovered the theory of flow dynamic through the capillary. Meanwhile, Washburn assumed the pores in material as an aggregation of microcylindrical capillaries and obtained the permeability of the solid object by measuring the volume of mercury immersed in the object over a period of time. Twenty years later, the MIP technique has been used to measure the pore distribution in porous materials, such as the diatomaceous earth, the fritted glass, and the silica-alumina gel (Drake, 1949; Ritter & Drake, 1945; Ritter & Erich, 1948). It has also been used to measure the pore distribution in cement-based materials in 1970s (Bager & Sellevold, 1975; Diamond & Dolch, 1972; Sellevold, 1974; Winslow & Diamond, 1969).

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3 Experimental Techniques

Fig. 3.24 Illustration of ink-bottle pores with narrow entrance

Mercury

Narrow entrance of pores Isolated pores Ink-bottle pores

Although the pore size detected by MIP technique is related to the type of material, the maximum pressure exerted on mercury is the decisive factor affecting the minimum pore size that can be measured. In the earlier research, the minimum pore size in cementitious materials measured by Bager and Sellevold (1975) and Diamond and Dolch (1972) is 12 nm. In the current studies, the minimum pore size of cementitious materials measured by MIP technique can be as low as 3 nm. However, the mercury can only be intruded into the connected pores, which means that the isolated pores cannot be detected, as shown in Fig. 3.24. Moreover, Wills et al. (1998) considered that the results obtained by MIP technique only represent the size distribution of pore entrance, rather than the pore size distribution, as shown in Fig. 3.24. Only after the mercury penetrates through the pore entrance, it can intrude into the pore. On the other hand, the assumption of the connected cylindrical geometry of pores may lead to the inaccurate interpretation of the pore structure and the pore size distribution if ink-bottle pores exist in cement paste (Diamond, 2000). However, due to its rapid testing and good reproducibility features, MIP is still widely used for pore size characterization of cementitious materials.

3.8.2 Principle of MIP Mercury is regarded as the most suitable intrusion liquid for MIP test because of its non-wetting property; that is, the contact angle (θ mercury ) between the mercury and the solid surface is larger than 90° (Fig. 3.25a). The non-wetting liquids can hardly penetrate pores by capillary pressure, unless an external pressure that is proportional to the surface tension is applied to induce the mercury intrusion. The surface tension of the mercury is high enough (about 0.485 N/m at 25 °C (Lide, 2003)) for intrusion, resulting in that a large external pressure is required and convenient to be measured.

3.8 Mercury Intrusion Porosimetry (MIP) Technique

(a)

81

(b)

θmercury

θwetting

Fig. 3.25 Contact angle of a mercury, and b wetting liquid

However, many common wetting liquids can spontaneously penetrate pores by capillary pressure due to their lower contact angle θ wetting (Fig. 3.25b) and surface tension and thus cannot be used as the intrusion liquid. For example, the surface tension of water is 0.0072 N/m at 25 °C, and water can spontaneously penetrate the pores without external pressure. Mercury injection instrument can apply pressure on mercury to make mercury intrude into the pores of cement paste, and the pressure and the volume of intruded mercury are monitored during the test. In the MIP test, the cement paste sample is placed into a chamber, and then the chamber is vacuumized to exhaust the air inside the sample pores so that mercury can intrude into the pores. Figure 3.26 shows the typical relationship between the applied pressure on the mercury and the corresponding size of the intruded pore. When the pressure increases from P1 to P2 , the intruded pore size decreases from D1 to D2 . The mercury volume intruded during this process can be detected, which is assumed equal to the volume of the pores with size of D2 . When the pressure is changed continuously, the volume of mercury intruded into pores with different pore sizes can be measured, and the pore size distribution can be obtained. However, the maximum pressure of around 400 MPa is normally used for cementitious materials measurement (Scrivener et al., 2016), because compressive deformation in paste might be generated by the pressure greater than 400 MPa and be recorded mistakenly as the pore volume. This means that the minimum pore size in cementitious materials that MIP can detect is about 2 nm (as shown in Fig. 3.26), and the pores with size less than 2 nm cannot be measured. 800 Applied pressure (MPa)

Fig. 3.26 Relationship between the applied pressure and the intruded pore size

600

Maximum pressure for cenemtitious materials

400 (D2, P2)

200

(D1, P1)

0 1

10 100 Intruded pore size (nm)

1000

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3 Experimental Techniques

3.8.3 Calculation Method As mentioned in the previous section, due to the high surface tension of mercury, an external pressure P needs to be applied to intrude the mercury into the pores. This pressure is usually called pore pressure, and it can be calculated by Eq. (3.91). P = γC

(3.91)

where γ is the surface tension of mercury, C depends on the pore geometry as well as the contact angle between the mercury and the porous material. The parameter C can be defined as: C=

2 cos θ r

(3.92)

where θ is the contact angle between the mercury and the porous material, and r is the radius of capillary or the pores. The combination of Eqs. (3.91) and (3.92) is known as the Washburn equation (Washburn, 1921b): P=

2γ cos θ r

(3.93)

And the diameter of the pores (D) can be defined as: D=

4γ cos θ P

(3.94)

Although the surface tension of mercury and the contact angle between mercury and cement paste are susceptible to chemical composition and material surface contamination, γ = 0.485 N/m (Lide, 2003) and θ = 140° (Cook & Hover, 1991) under 25 °C are usually used to calculate the pore size. Figure 3.27 shows a typical MIP curve for a cement paste sample. The MIP results are generally displayed as the cumulative pore volume (or cumulative mercury intrusion) normalized by sample volume or mass versus the intruded pore size. It is recommended to use the logarithmic coordinate for the x-axis where the pore size is plotted. Two parameters characterizing the pore structure can be determined from the MIP curve, one is the total percolated pore volume, and the other one is the critical pore size. The total percolated pore volume is the maximum pore volume recorded throughout the MIP test, which represents the total connected pores volume at the maximum applied pressure and should not be confused with the total pore volume (Scrivener et al., 2016). The critical pore size is the pore size at the maximum slope of the cumulative pore volume curve, which can be used in permeability prediction.

3.9 Summary

83

Fig. 3.27 Typical MIP cumulative pore volume and derivative curves of cement paste by the MIP with a maximum pressure of 400 MPa (after Scrivener et al., 2016)

Total percolated pore

Critical pore

Intruded pore size

3.9 Summary Instrumented indentation techniques, including NI and MI, are the most widely used techniques for small-scale mechanical property measurement, through which the mechanical properties (such as indentation modulus, hardness, and contact creep compliance) can be measured at the micrometer and nanometer scales. In indentation test, the testing scale is determined by the ratio of the indentation depth (h) to the characteristic size of the heterogeneity in the composite (D). Dynamic modulus mapping by SPM has been used to investigate the mechanical properties of multiphase materials with a much smaller contact force and indentation depth than that of the conventional NI, and the different phases in hardened paste can be clearly distinguished by modulus mapping due to its capability to quantify the phase size at the nanometer scale under a non-destructive testing condition. The thickness of the inner CSH (IP) layers in hardened OPC and the slag-blended pastes can be quantified from the abrupt changes of the continuously measured storage modulus, such that the inner and the outer CSH can be separated meaningfully. CSM is another dynamic method based on indentation technique, capable of recording the force–indentation depth developments under various strain rates, which is a promising technique to illustrate strain rate effect on the time-dependent deformation of cement paste. However, unlike the SPM mapping technique, the applied harmonic force is much smaller than the quasi-static force, which make the indentation force–indentation depth curve obtained from CSM look like that obtained from the quasi-static method based on indentation technique. Compared to NI technique, nanoscratch technique can perform continuous mechanical properties measurement. A new loading mode of constant vertical loading rate, defined as the F V-max /L (in N/µm), was proposed for phase identification and applied for scratching test, which avoids the influence from the variable transverse

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scratching speed. Meanwhile, it was proved that the relationship of p(d) A(d)∝ d 3 is still valid for the Berkovich and the Vickers tip. In addition to measuring the small-scale mechanical properties, the study on the microstructure of cementitious materials is of significance for the multiscale characterization of concrete. The techniques of SEM, BSE, and EDS can be used in conjunction with NI and nanoscratch techniques. By observing the position of indentation and scratching in the SEM images, the different phases can be identified, and the mechanical properties of the single phase can be obtained more accurately. Meanwhile, based on the gray threshold analysis, BSE images are also used to identify the pore structure and to analyze the degree of hydration. Utilizing X-ray CT scanning to obtain the microstructure information is considered more convenient and realistic, because the delicate grinding and polishing processes which are normally applied to obtain the flat surface before the microstructural observation by the SEM and EDS techniques are not required, and thus, the original microstructure of sample can be preserved without disturbance. The MIP method is one of the most commonly used techniques for testing nanopores. This technique is based on the intrusion of a non-wetting fluid (mercury) into porous structures under increasing external pressure. The maximum pressure of 400 MPa applied in MIP test enables to measure the minimum pore size down to 2 nm in cementitious materials.

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Chapter 4

Phase Quantification by Different Techniques

Abstract It is of significance to identify different phases and quantify the smallscale mechanical properties and microstructure morphology of individual phases in cementitious materials in order to effectively characterize and predict the macroscale response of concrete structures. This chapter illustrates the existing methods of identifying individual phases and quantifying their size and micromechanical properties by using the advanced techniques introduced in Chapter 3. The SPM technique can characterize the mechanical property and distinguish phases as the static NI technique does, but with more attractive features in the form of SPM images. A new method for quantifying ITZ thickness is proposed based on the fracture toughness distribution tested by the nanoscratch technique. The key to identifying phases by the X-ray CT and the BSE techniques is segmenting the phases by the gray level threshold method. This chapter provides detailed and useful methods for quantifying individual phases in cementitious materials Keywords Degree of hydration · Discrete indentation · Phase identification · Phase segmentation · Thickness of individual phase

4.1 Introduction Cementitious material is a typical multiscale and multiphase material. Identifying different phases and quantifying the mechanical properties and microstructure morphology of individual phases can help to effectively characterize and predict the multiscale response of cementitious materials. For example, utilizing the modulus and the fractions of individual phases measured at microscale as the input parameters, the effective modulus of the hardened cement pastes with different water-to-cement (w/c) ratios, hydration degrees, and curing conditions (Haecker et al., 2005; Lin & Meyer, 2008; Sanahuja et al., 2007; Stefan et al., 2010) can be homogenized by adopting the finite element methods or the micromechanics-based schemes, as will be discussed in Chapter 8. The mechanical properties and the microstructure morphology of individual phases can be characterized by different techniques, such as the scanning electron microscope and the backscattered electron (SEM and BSE) image analysis © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_4

91

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4 Phase Quantification by Different Techniques

(Scrivener, 2004), the energy dispersive spectrometer (EDS) technique (McWhinney et al., 1990), the nanoindentation (NI) (Acker, 2001; Constantinides & Ulm, 2004), the scanning probe microscopy (SPM) mapping (Li et al., 2015), the nanoscratch technique (Hoover & Ulm, 2015), and the X-ray computed tomography (CT) technique (Morgan et al., 1980). In this chapter, the methods of identifying individual phase and characterizing micromechanical properties by these techniques are introduced.

4.2 Phase Quantification by Backscattered Electron (BSE) Mapping Technique BSE image analysis has been widely used in characterizing the microstructure of cementitious materials. It is able to not only qualitatively investigate the microstructure but also quantitatively calculate the porosity and degree of hydration of the measured materials.

4.2.1 Identifying Phase The BSE image reflects the backscattered intensity of different phases, as described in Sect. 3.6.3. The phase with the highest backscattered intensity appears brightest. Figure 4.1 shows the BSE image of the unreacted cement clinker and the slag grain which are the brightest, followed by the hydrates phase, and the pores are the darkest. Meanwhile, the BSE image can be used to distinguish phases by their morphology. In Fig. 4.1, the shape of C2 S is circular with characteristic striations of crystal twinning. The C3 S appears lath-shaped and is surrounded by a fairly even rim of CSH (Kocaba, 2009; Scrivener, 2004). Compared to the unreacted cement clinker, the microstructure of slag grains is denser.

Pore

Slag C2S 100μm

Hydrates

C3S Fig. 4.1 Morphology of unreacted cement clinkers and slag grains in BSE image (Wei et al., 2018b)

4.2 Phase Quantification by Backscattered Electron …

93

C2S CH C4AF Pore C3S HD CSH

Volume fractions

LD CSH

CSH CH

156

50 μm

Gray level

(a) 3O BSE image

(c) 3S50 BSE image

(b) 3O gray level histogram Volume fractions

Pore C3S HD CSH LD CSH C4AF Slag CH C2S 20 μm

Cement

Pore

CSH

Pore

Cement CH Slag 136 182 Gray level

(d) 3S50 gray level histogram

Fig. 4.2 BSE images and gray level histogram of Portland cement (3O) and slag-blended pastes (3S50) (Wei et al., 2016)

The typical BSE images are given in Fig. 4.2a and c for pure cement pastes (0.3 w/c, 3O) and cement paste with slag (0.3 w/c with 50% slag content, 3S50), respectively. The accelerating voltage used is 15 kV during the BSE imaging process. It can be seen that the unreacted cement grains are identified as the bright gray, and the slag grains are identified as the dark gray. The mass phases distributed around the unreacted grains are the hydrate phase and the composite phase. The BSE image analysis enables to distinguish and quantify the area fractions of the unhydrated phase, the hydrated phase, and the capillary pore by applying morphological filters in different steps of the segmentation. It normally generates a gray level histogram with scale extending from black (0) to white (255) and with distinct peaks corresponding to different phases as shown in Fig. 4.2b and d. The slag grain has gray level brighter than CH and darker than cement grains (C3 S, C2 S), which is represented by the separate peak in between the CH and the cement (Fig. 4.2d). The area fractions of each phase are considered equal to their volume fractions, as long as phase quantification is based on a large number of fields in BSE images to take into account the variations from one field to the other (Haha et al., 2010).

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4 Phase Quantification by Different Techniques

4.2.2 Phase Segmentation from BSE Images In the gray level histogram shown in Fig. 4.2b and d, the peaks of CSH phase, slag phase, and the unhydrated cement phase can be easily found. The gray level of the trough can be considered as the boundary of these phases, as shown in Fig. 4.2b and d. For example, in the 3O sample, the threshold gray level between the CH phase and the unhydrated cement phase is 156. In the 3S50 sample, the threshold gray level between the CH and the slag is 136, and that between the slag and the unhydrated cement is 182. However, the boundary between pore and CSH, and the boundary between CSH and CH are not easy to be distinguished. Therefore, an “overflow” method was explored for identifying their boundaries (Wong et al., 2006). Taking the pores identification as an example, a mortar with the w/c ratio of 0.35 was tested by BSE technique at the age of 28 days (Wong et al., 2006). The BSE image of the paste (aggregates are removed from the original image) and its gray level histogram are shown in Fig. 4.3a. Figure 4.3b shows a series of images of capillary pore with area segmented (white pixels) at different threshold levels. When the threshold levels between 80 and 110 are selected, a sudden increase in the segmented pore area can be observed. This is analogous to filling up a pore with a fluid. As the liquid arrives near the boundary of the pore, a critical threshold gray level is reached when the liquid will overflow to the surrounding areas and this will lead to a sudden increase of the area covered with the fluid. Therefore, the critical threshold gray level where the area segmented starts to “overflow” can provide a good estimate for the pore threshold level. The cumulative segmented area versus the gray level is shown in Fig. 4.3c. The critical overflow level (with threshold gray level of 88.1) can be estimated from the intersection between the two tangent lines. The gray level at this intersection can be used as the upper threshold gray level for porosity. However, this value may slightly overestimate the true overflowing point which can be revised by multiplying the gray level with a factor of, such as 0.9. The segmented BSE image of pore (black pixels) with threshold gray level of 79.3 (88.1 × 0.9) is shown in Fig. 4.3c. The same method can be used to determine the threshold gray level of the boundary between CSH phase and CH phase (the yellow mark in Fig. 4.3c). After determining the boundary between different phases, the volume content of a phase can be obtained by dividing the number of pixels in the gray level range for each phase by the total number of pixels.

4.2.3 Quantifying Degree of Hydration Estimation of hydration degree from BSE results can be achieved by comparing the total fraction of the unhydrated phase at the age of t with its original value: α(t) = 1 −

V (t) V (0)

(4.1)

4.2 Phase Quantification by Backscattered Electron … (a)

95

(b)

Threshold = 20

Threshold = 30

Threshold = 40

Threshold = 60

Threshold = 80

Threshold = 90

Threshold = 100

Threshold = 110

Threshold = 120

Fig. 4.3 a BSE image of paste regions from the mortar sample at the age of 28 days and its gray level histogram, b change in area segmented (white pixels) of pore structure at different threshold levels, and c the application of the overflow criteria to determine the threshold gray level for porosity (after Wong et al., 2006)

where α(t) is the degree of hydration, V (t) is the volume fraction of the unhydrated phase at time t, V (0) is the initial volume fraction of the unhydrated phase. The initial volume fraction of the unhydrated phase in Portland cement system V (0) can be calculated from Eq. (4.2). While for slag-blended system, the initial fractions of cement and slag are calculated from Eqs. (4.3) and (4.4), respectively. V (0)OPC =

1 1 + r × ρOPC

V (0)OPC in blended paste =

V (0)slag in blended paste =

m OPC ρOPC

m OPC ρOPC

(4.2)

+

m OPC ρOPC m slag ρslag

+

m water ρwater

+

m slag ρslag m slag ρslag

+

m water ρwater

(4.3)

(4.4)

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4 Phase Quantification by Different Techniques

where mOPC , mslag , mwater , ρ OPC , ρ slag , and ρ water are the initial mass contents and the specific gravity of Portland cement, slag, and water, respectively; r is w/c ratio. Then, the hydration degree of the slag-blended systems can be determined as the weighted summation of the separate hydration degree of OPC and slag. The results of estimating the degree of hydration from BSE images will be illustrated in Sect. 4.4.6.

4.3 Phase Quantification by Energy Dispersive Spectrometer (EDS) Technique Energy dispersive spectrometer (EDS) can be used to analyze the types and contents of elements in materials, because each element has its own characteristic X-ray wavelength and energy which can be analyzed by EDS. EDS has become a feasible method to examine the chemical composition of cementitious materials (McWhinney et al., 1990) and rocks (Zhu et al., 2007) over the past years. With the information of the microvolumes probed by EDS, the chemical composition of cementitious materials can be analyzed. It has been reported by Chen et al. (2010) that CSH/Ca(OH)2 nanocomposites exist in cement paste at the nanometer scale by using the coupled method of EDS, NI, and micromechanics. Durdzinski et al. (2015) have also developed a new method based on EDS to quantify the fly ash composition and to evaluate the reaction of its individual components in early-age cementitious materials.

4.3.1 Variation of Chemical Composition by EDS Line Scanning Across the Featured Phases The EDS line scanning was conducted to assess the chemical composition of the featured phases in both OPC and slag-blended cement pastes. The w/cm ratio of the two tested pastes was 0.3, which were sealed cured under the temperature of 20 °C for 180 days. The slag content was 50% by mass of the total cementitious materials in slag-blended paste. Figure 4.4 shows the chemical composition results. As shown in Fig. 4.4a, the EDS line was scanned starting from a C3 S grain to the matrix in the OPC paste. While the EDS line was scanned starting from a slag grain to a C3 S grain in the slag-blended paste, across the matrix, as shown in Fig. 4.4c. By using the EDS technique, the element concentrations of Si, Ca, Al, and Mg can be determined along the lines. The modulus mapping technique was used to measure the storage modulus along the lines prior to EDS line scanning. Figure 4.4b and d displays the chemical composition of the featured phases in terms of the atom percentage of Ca and Mg as well as the storage modulus for OPC and the slag-blended pastes, respectively. As shown in Fig. 4.4b, the Ca content is roughly above 22% for both C3 S grain and the CSH matrix in the OPC paste, while it is below 22% for both slag grain and the CSH matrix surrounding the slag grain in the slag-blended paste (shown in Fig. 4.4d). A Ca-rich zone where the Ca content is up to 60% detected by the EDS

4.3 Phase Quantification by Energy Dispersive Spectrometer … (a)

(b)

EDS scanning line

C3S

Atom percentage (%)

Scanning direction

50 40 30 20

Ca Mg 14

(d)

60 40

4

0

Atom percentage (%)

30

Slag

EDS scanning line

20

80

Ca = 22%

6

0

40

100

8

2

50

OP-O Mg atom percentage Ca atom percentage Storage modulus

20

Mg = 4%

0

Ca Mg 14

5

10

15 20 25 Distance (μm) Slag OP-S OP-S IP-S IP-S IP-O[S]

30

35

C3S

100

12

80

10 60

8 Ca = 22%

6

40

4

10

2

0

0

20 Mg = 4%

Storage modulus (GPa)

C3S Scanning direction

IP-O

Ca-rich zone

10

10

60

ITZ

12

0 (c)

C3S

Storage modulus (GPa)

C2S

60

97

0 0

4

8

12 16 Distance (μm)

20

24

28

Fig. 4.4 a EDS line scanning across an unreacted C3 S grain in OPC paste, b variation of calcium and magnesium contents and storage modulus along the scanned line in OPC paste, c EDS line scanning across an unreacted slag and C3 S grains in slag-blended paste, and d variation of calcium and magnesium contents and storage modulus along the scanned line in slag-blended paste (Wei et al., 2018a)

line scanning coincides with the location of ITZ between C3 S and its IP layer in the OPC paste. This Ca-rich zone may be related to the formation of calcium hydroxide in the ITZ around C3 S clinker, which is similar to the ITZ around the aggregate where the calcium hydroxide crystals are formed. However, as shown in Fig. 4.4d, no calcium-rich zone is found in the slag-blended pastes, the reason of which may be related to the pozzolanic reaction consuming calcium hydroxide in the slag-blended paste. As shown in Fig. 4.4b, the Mg content in the OPC paste is constantly below 4%, which cannot be used to distinguish phases in the OPC paste. However, as shown in Fig. 4.4d, the Mg content on different phases is highly variable in the slag-blended; i.e., the Mg content in the slag grain and the inner product of slag (IP-S) is high, while it is significantly lower in the OP-S than that in IP-S, which can be used to distinguish different phases in the slag-blended paste. It can be seen from Fig. 4.4d that a zone with higher Mg content and storage modulus than that of the surrounding OP-S zones exists in between the two OP-S zones, which may be the inner product of a completely reacted small slag grain since no visible unreacted slag core is observed in this zone judged from the gray level contrast in the BSE image (Fig. 4.4c). This zone is labeled as IP-S in Fig. 4.4d.

98

4 Phase Quantification by Different Techniques

As shown in Fig. 4.4d, the thin layer surrounding C3 S grain in the slag-blended paste is recognized as the inner CSH layer of the C3 S grain, which is labeled as IP-O[S]. The Mg content in IP-O[S] is similar to that in the C3 S grain; however, the storage modulus and Ca content are lower for IP-O[S]. Based on the chemical composition and the storage modulus, there is no outer CSH for this C3 S grain. The possible reason may be that the reaction of the adjacent small slag grain with particle size of less than 2 mm is so fast (Gruskovnjak et al., 2006) that the hydration products (especially the OP-S) fill the narrow interstitial space between the C3 S and the slag grains; therefore, the OP of C3 S could not be fully produced.

4.3.2 Chemical Composition of IP and OP in OPC and Slag-Blended Systems Based on the EDS results shown in Fig. 4.4, five types of CSH can be detected in the OPC (denoted as OP-O and IP-O) and the slag-blended pastes (denoted as OP-S, IPS, and IP-O[S]). Prior to the NI tests to measure the mechanical property, 82 discrete EDS testing points were made on different CSH phases to examine the chemical composition of the five CSH gels. As shown in Fig. 4.5a and b, the five types of CSH detected in the OPC and the slag-blended pastes have different chemical composition. In addition to the lower content of Ca, the content of Al in the CSH gel of the slag-blended paste (3–5%) is much greater than that of the OPC paste (1.5–2%). It can be seen from Fig. 4.5a that the Al content in IP of the slag cores is highest. The reason may be that the aluminarich supplementary cementitious materials (SCMs) contribute to increasing the Al uptake in CSH (Lothenbach et al., 2011); therefore, the main reaction product in the slag-blended paste is a highly Al-substituted calcium silicate hydrate (CASH) where aluminum substitutes for silicon (Bernal et al., 2013; Kocaba, 2009; Lecomte et al., 2006; Nicolas et al., 2014; Richardson & Groves, 1993). Besides, it can be seen from Fig. 4.5a that the Mg content in OP-S is much lower than that in IP-S, which can be used to distinguish the inner and outer CSH in the slag-blended paste. Moreover, the chemical composition of the inner CSH around the clinker in the slag-blended paste (IP-O[S]) lies in between those of the IP-O and the IP-S. Since the highly Al-substituted calcium silicate hydrate (CASH) is one of the main reaction products in the slag-blended paste, the Ca/(Si + Al) ratio and the Ca/Si ratio were calculated and compared for the five CSH gels, which are shown in Fig. 4.5b. It can be seen from Fig. 4.5b that the Ca/Si and the Ca/(Si + Al) atomic ratios are almost stable for both inner and outer CSH in the OPC paste; however, they vary with the grain type in the slag-blended paste. The Ca/Si ratio in the OPC paste is 1.6–1.7, while it is 1.0–1.3 in the slag-blended paste. In fact, a lower Ca/Si molar ratio of 0.7 has been reported in the alkali-activated slag matrix that is catalyzed by potassium hydroxide (Lecomte et al., 2006). As shown in Fig. 4.5b, the Ca/(Si + Al) atomic ratio in the slag-blended system is 0.8–1.1, which is much lower than that in OPC system where the Ca/(Si + Al)

4.4 Phase Quantification by Nanoindentation Technique

(a) 45

Ca 1.83 1.38

35 30

14.65

Si Al Mg 2.07 2.48 6.34 3.53 3.93 5.60

13.85 15.78

14.30

20 15 10

Ca/Si 1.8 1.62 1.6

Ca/(Si+Al)

1.68 1.52

1.45 1.33

1.4

14.92

25

(b)

1.12

1.2

Ratio

Atomic percentage (%)

40

0.87 1.73

99

1.07 1.01

1 23.66

23.15

19.54

15.79

15.99

IP-O[S] OP-S

IP-S

5

0.80

0.80

0.8 0.6

0 OP-O

IP-O

OP-O

IP-O

IP-O[S]

OP-S

IP-S

Fig. 4.5 Difference on a atomic percentage, and b Ca/Si and Ca/(Si + Al) ratios of inner and outer CSH in OPC (IP-O, OP-O) and slag-blended (IP-S, IP-O[S], and OP-S) pastes (the error bars represent one standard deviation from 82 measurements) (Wei et al., 2018a)

atomic ratio is about 1.5. This is comparable with the existing findings; for example, Kocaba (2009) showed that the Ca/(Si + Al) atomic ratio is 1.8–1.9 for the CSH in OPC paste. The Ca/(Si + Al) ratio of CSH in water active cement–slag pastes (0–100% slag) has been reported to range from 0.7 to 2.4 (Richardson & Groves, 1993).

4.4 Phase Quantification by Nanoindentation Technique In the NI test, an indenter with a sharp probe is pushed into a well-prepared smooth surface of a hardened cement paste, and the force acting on the indenter and the penetration depth of the probe are recorded during the test. Based on the measured indentation force–indentation depth curve, the elastic modulus of the tested sample can be obtained. Due to the multiphase and multiscale nature of cementitious materials, a NI grid test to cover large testing areas is usually conducted for cementitious materials. Although there are still several disputes on data analysis and interpretation of the results by the NI technique for cementitious materials (DeJong & Ulm, 2007; Vandamme et al., 2010), NI technique proves a powerful tool to evaluate the mechanical properties of cementitious materials at micro scale (Constantinides & Ulm, 2007; Zadeh & Bobko, 2013).

4.4.1 Categorization and Mechanical Property of Phase by Discrete NI Based on the morphology and indentation modulus (M) measured by discrete NI, there are four solid phases including the low-density calcium silicate hydrate (LD

100

4 Phase Quantification by Different Techniques

CSH), the high-density calcium silicate hydrate (HD CSH), the unhydrated, and the composite (of hydrate and unhydrated phases) phase can be identified in the cement paste. Five cementitious materials were prepared for the test, which are denoted as 3O (w/c = 0.3, 0% slag content), 3S50 (w/cm = 0.3, 50% slag content), 3S70 (w/cm = 0.3, 70% slag content), 4O (w/c = 0.4, 0% slag content), and 5O (w/c = 0.5, 0% slag content). All the samples were sealed cured under the temperature of 20 °C for 180 days. In the discrete NI test, there were 88 indents made on 3O and 3S50 samples. The tested locations were chosen intentionally on the unreacted grains and at different distances from the grains; therefore, all the phases are possible to be measured by the discrete NI test. Figure 4.6 and Table 4.1 show the measured indentation modulus of the 88 indents and the number of indents located at different phases. As shown in Table 4.1, the measured indentation modulus ranges from 15 to 115 GPa. The indentation modulus of the unhydrated clinker grains, the slag grains, HD CSH, LD CSH, the composite of hydrate and unhydrated phases are 75–114 GPa, 81–115 GPa, 28–4 GPa, 15–31 GPa, and 45–66 GPa, respectively, and the corresponding number of indents is 10, 5, 23, 43, and 7. Based on the NI measurements, no single portlandite or calcium hydroxide (CH) (M = 44.2 ± 4.1 GPa in Nˇemeˇcek et al., 2016) phase was detected in both the 3O and the 3S50 samples. The possible reason may be that CH is intermixed with the LD CSH or the HD CSH, which will result in a similar indentation modulus as CSH (Chen et al., 2010; Hughes & Trtik, 2004). This indicates that the NI technique fails to distinguish 120 LD CSH HD CSH Composite

100 80 M (GPa)

10 µm

60 40 Cement Slag

20

10um

0 LD CSH

HD CSH

Composite Cement Slag

Fig. 4.6 Indentation modulus of different phases in OPC and slag-blended paste samples measured by discrete NI (Wei et al., 2016)

Table 4.1 Indentation modulus of single phase by discrete NI (Wei et al., 2016) LD CSH

HD CSH

Composite

Cement

Slag

Total indents 88

Number of indents

43

23

7

10

5

M (GPa)

15–31

28–41

45–66

75–114

81–115

Average (GPa)

22.9

34.2

53.0

85.5

93.5

4.4 Phase Quantification by Nanoindentation Technique

101

different phases with similar indentation modulus in cementitious materials. The identified composite phase could be a mixture of CSH and the unhydrated phase or a mixture of CH and the unhydrated phase. Since CH normally precipitates in the water-filled pores and CSH deposits mainly around the unreacted grains (Scrivener, 2004), the identified composite phase mainly consists of CSH and the unhydrated clinker. In addition, the discrete NI results reveal that whether the tested samples contain slag or not doesn’t affect the measured mechanical properties of HD CSH and LD CSH. Although HD CSH and LD CSH share similar chemical compositions, they differ in the gel porosity (Jennings, 2008). The difference in the CSH solid and the gel porosity will result in different mechanical properties for the two types of CSH. In fact, it has also been reported by Zadeh and Bobko (2013) that the measured nanoscale mechanical properties of hydration products in slag-blended paste are similar to those in Portland cement paste.

4.4.2 Deconvolution of Grid Indentation Data To obtain the mechanical properties of each individual phase in the tested sample, deconvolution on the large number of grid indentation data in terms of indentation modulus and contact hardness is required after the NI test. It is widely accepted that the statistical features of the measured results in the whole sample can be represented by the randomly selected indentation areas. Therefore, the deconvoluted results including the phase distribution, the mechanical properties, and the volume fraction of each phase can be taken as the phase properties of the material (Constantinides & Ulm, 2007). More details about the deconvolution method can refer to Sect. 6.3.

4.4.3 Fraction of Single Phase with Different w/cm Ratios and Slag Contents After the deconvolution of grid indentation data, the volume fraction of the four phases (pore, hydrate, composite, and unhydrated phases) in different tested samples can be obtained, which are shown in Fig. 4.7. It is noted that the phase possessing an indentation modulus of less than 15 GPa is classified as the pore phase. Table 4.2 summarizes the measured data. It can be seen from Table 4.2 that the volume fraction of each phase in hardened paste is influenced by the w/cm ratio and the slag content. The volume fraction of pore phase increases with increasing w/cm ratio and is higher in the pure cement paste than that in the slag-blended paste. The volume fraction of composite in the slag-blended pastes is 0.35 (3S50) and 0.41 (3S70), which is much greater than that (0.10) in the 3O cement paste with the same w/c ratio. Hughes and

102

4 Phase Quantification by Different Techniques (b) 0.7 3O 4O 5O

Volume fraction

0.6 0.5 0.4 0.3 0.2 0.1

0.6 Volume fraction

(a) 0.7

3O

0.5

3S50

0.4

3S70

0.3 0.2 0.1

0

0 Pore

Hydrate

Composite Unhydrated

Pore

Hydrate

Composite Unhydrated

Fig. 4.7 Volume fraction of the four (pore, hydrate, composite, unhydrated) phases in a Portland cement pastes (3O, 4O, 5O), and b slag-blended pastes (3S50, 3S70) (Wei et al., 2016)

Trtik (2004) have also detected the composite phase in a w/cm ratio = 0.45 paste, and the volume fraction is about 0.27. When there exists a large amount of composite phases in cementitious materials such as the slag-blended paste and the low w/cm ratio paste, it is important to quantify the exact content of the hydrate and the unhydrated phases for further property analysis, e.g., quantifying degree of hydration.

4.4.4 Phase Distribution and Mechanical Properties of CSH After filtering and deconvoluting the grid NI results, the mechanical properties as well as the phase distribution of the hydration products such as LD CSH and HD CSH can be obtained. It has been reported that the mechanical behavior of HD CSH can be measured by using an indentation depth of about 100–200 nm, while a larger indentation depth is required to measure the mechanical property of LD CSH due to its greater size (Mondal et al., 2007; Richardson, 2004). However, it should be noted that a significant decrease of the obtained mechanical properties will be seen when the indentation depth is greater than 500 nm (Davydov et al., 2011), which may be related to the different defects in the samples (Constantinides & Ulm, 2007). The indentation data which satisfy both hmax < 500 nm and M < 40 GPa can be identified as hydration products and used for deconvolution to obtain the properties of LD and HD CSH. Figure 4.8 shows the deconvoluted results of the hydrated phases. It can be seen from Fig. 4.8 that a bimodal trend with two different peaks is found for the probability distributions of the indentation modulus of each paste. The two peaks correspond to the LD CSH and the HD CSH, and the mean indentation modulus as well as the volume fraction (V f ) are shown in Table 4.2, which are in good agreement with the existing findings (Constantinides & Ulm, 2004; DeJong & Ulm, 2007; Mondal et al., 2007; Venkovic et al., 2014). It can also be seen from Fig. 4.9a that the w/c ratio and the slag content have little effect on the indentation modulus of HD and LD CSH

Indent spacing (μm)

20

20

20

20

20

Grid size

15 × 15

15 × 15

15 × 15

15 × 15

15 × 15

Sample

3O

3S50

3S70

4O

5O

450

450

450

450

450

Number of indents

Table 4.2 Deconvolution results based on grid NI (Wei et al., 2016)

0.62

0.46

0.07

0.04

0.29

V pore

0.34

0.50

0.48

0.55

0.60

V hydrate

0.04

0.02

0.41

0.35

0.10

V composite

0.01

0.02

0.04

0.06

0.01

V unhydrated

21.5 ± 5.7

18.9 ± 2.9

22.0 ± 4.3

19.4 ± 3.8

0.64

0.61

0.54

0.52

0.57

34.9 ± 5.7

28.8 ± 6.3

35.0 ± 4.5

32.1 ± 5.0

31.5 ± 5.8

M (GPa) 19.6 ± 3.2

HD CSH

M (GPa)

Vf

LD CSH

0.36

0.39

0.46

0.48

0.43

Vf

4.4 Phase Quantification by Nanoindentation Technique 103

104

4 Phase Quantification by Different Techniques

since they are relatively constant in different cementitious materials. In addition, the measured indentation modulus by the grid NI test over large areas of hardened pastes is in good agreement with those measured by the discrete NI test on the single CSH phase as the dashed lines shown in Fig. 4.9a. The volume fraction of LD and HD CSH is slightly affected by the w/c ratio and the slag content, as shown in Figs. 4.9b and c. 0.2

0.25

(a) 3O

0.15

(c) 5O

0.1

0.1

0.05

PDF

0.15

PDF

PDF

0.15

(b) 4O

0.2

0.1

0.05

0.05 0

0

0 15

25 M (GPa) 0.15

15

35 (d) 3S50

15

25 35 M (GPa) 0.15 (e) 3S70

25 M (GPa)

35

0.1

PDF

PDF

0.1

0.05

0.05

0

0 15

25 M (GPa)

15

35

25 M (GPa)

35

Fig. 4.8 Probability distribution of indentation modulus for LD CSH and HD CSH in hydrate phase of Portland cement pastes (3O, 4O, and 5O) and slag-blended pastes (3S50 and 3S70) by grid indentation (Wei et al., 2016)

HD

LD

(b)

1.2

HD

LD

3S50

3S70

4O

5O

1.2

1

1

0.8

0.8

0.6 0.4 0.2 0

3O

(c)

Volume fraction

45 40 35 30 25 20 15 10 5 0

Volume fraction

M (GPa)

(a)

HD

LD

0.6 0.4 0.2 0

3O

4O

5O

3O

3S50

3S70

Fig. 4.9 a Indentation modulus of LD and HD CSH measured by grid NI (columns) and by discrete NI (dashed lines), volume fraction of LD and HD CSH in hydrate for b Portland cement pastes (3O, 4O, and 5O) and c slag-blended pastes (3S50 and 3S70) (Wei et al., 2016)

4.4 Phase Quantification by Nanoindentation Technique

105

4.4.5 Nanoindentation (NI)-Based Degree of Hydration The degree of hydration of cementitious materials can be quantified by the universal equation as shown in Eq. (4.1), and it can also be evaluated based on the NI results. When the volume fraction of the unhydrated phases in both the OPC paste and the slag-blended paste at time t is denoted as V (t), it can be estimated as: u+ V (t) =

n Σ i=1

i Vunhydrated in composite

(4.5)

N

where u and n are the number of indents directly made on the unhydrated phase and i the composite phase, respectively; Vunhydrated in composite is the calculated fraction of the unhydrated phase in composite based on homogenization method; N is the total number of grid indents.

4.4.6 Comparison of Different Techniques in Estimation of DOH Table 4.3 and Fig. 4.10 show the measured degree of hydration by different techniques including NI, BSE, and TGA. It should be noted that a range rather than a single value is obtained by the NI-based method due to the fact that the same effective composite modulus can be achieved within a range of volume fraction of the unhydrated phase. The degrees of hydration of the slag-blended paste measured by the three techniques are much lower than those of the OPC paste. Feng et al. (2004) also reported a lower degree of hydration of slag-blended paste by other techniques such as the point counting method based on SEM, the total degree of hydration of the paste with w/c ratio of 0.4 and slag content of 30% is 68% after 90-day curing. It has been found by Pane and Hansen (2005) that the total degree of hydration of the paste with w/c ratio of 0.45 and slag content of 25% is 66% after 180-day curing. In addition, the degree of hydration of 3S50 paste is similar to that of the 3S70 paste, as shown in Table 4.3. Table 4.3 Comparison of NI, BSE, and TGA-based degree of hydrations for different paste samples at the age of 180 days Sample

by NI (%)

by BSE (%)

by TGA (%)

ΔNI−TGA (%)

ΔBSE−TGA (%)

3O

83–87

73

81

2.5–7.4

−9.9

3S50

46–56

55

54.6

−15.8–2.6

0.7

3S70

43–53

48

55

−22.2–4.2

−13.2

4O

94.7–95.4

–

89

6.0–6.8

–

5O

92–93

–

93

−0.5–0.5

–

106

100 DOH by NI or BSE (%)

Fig. 4.10 Comparison of degree of hydration (DOH) determined by NI, BSE, and TGA techniques (Wei et al., 2016)

4 Phase Quantification by Different Techniques

by NI 3S50 by BSE 3O 3S70 by BSE 3O by BSE

90 80

4O 5O

70 3S50

60 50

3S70 40 40

50

60 70 80 DOH by TGA (%)

90

100

It has been reported that the degree of hydration of cementitious materials can be reliably evaluated by the chemically bound water (Darquennes et al., 2013); therefore, the NI- and BSE-based degrees of hydration are compared to the TGA-based results. As shown in Table 4.3, the NI- and the BSE-based degree of hydrations is in a reasonable agreement with the TGA-based results. The symbol of Δ in Table 4.3 denotes the deviation of the measured results by the NI/BSE-based method from that by the TGA-based method. It is noted that the measured degree of hydration of 4O paste by the NI-based method is greater than that of 5O paste, which contradicts the expected. The possible reason may be that the number of indents belonging to the hydrate phase made in 4O paste is greater than that in 5O paste, which indicates the importance of selecting representative areas for the NI test. Overall, the NI-based method can provide a better prediction of the degree of hydration of the OPC paste than that of the slag-blended paste, and the difference between the NI-based method and the TGA-based method is within ± 10%.

4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique The main advantage of SPM is that it can map the local variation of the mechanical properties without causing plastic deformation to the sample at nanoscale. The equipment for SPM quantitative modulus mapping is the same as the NI, but conducted under different operating model. Recently, the frequency response of polymer materials has been widely studied by the modulus mapping technique with in situ SPM imaging, the microstructural features and the viscoelastic behavior of solid such as the human teeth have also been successfully evaluated by the modulus mapping technique (Asif et al., 2001; Herbert et al., 2008). However, limited researches using the SPM-based modulus mapping technique have been made on cementitious materials

4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique

107

(Li et al., 2015; Xu et al., 2015b), and it is promising to characterize the multiscale and multiphase cementitious materials by the powerful the SPM-based modulus mapping technique.

4.5.1 Methods of Data Extraction from Mapping Line data extraction Normally, the variation of the mechanical properties at desired positions can be reflected by the line data extracted from the mapping images (Li et al., 2015; Wei et al., 2017; Xu et al., 2015a). The distinct differences of the mechanical properties of different phases shown in the line data can be used to distinguish phases in heterogeneous materials (Gao et al., 2018). The storage modulus images of the target areas in the OPC and the slag-blended pastes are shown in Fig. 4.11. In each storage modulus image, four vertical lines are extracted and the data are also plotted in Fig. 4.11. It can be seen that obvious changes of storage modulus would exist at the boundary between the C3 S clinker or the slag grain and their IP, as well as at the boundary between IP and OP. The thickness of IP rims around the C3 S clinker and the slag grain is 3.7–6.0 μm and 0.6–1.3 μm, respectively. Based on the BSE images analysis, Gruskovnjak et al. (2006) found that the IP rims around the slag grain were also narrower than that of the C3 S clinker; however, the accurate measurement of the thickness of IP rims wasn’t provided by Gruskovnjak et al. (2006), since the color contrast between the IP and OP in BSE images fails to give a precise result of the thickness. Area data extraction A new method based on the mapping images is proposed to extract the area data in cementitious materials. The amount of data in one area is at least an order of magnitude larger than that in single lines, the detailed method to calculate the area of individual phase is illustrated by the following steps, which is displayed in Fig. 4.12a. Step 1 Since the amount of pixels in mapping images along each dimension is 256, the mapping images are first meshed into 256 × 256 elements. Step 2 Based on the contrast of color for different phase, the boundaries between different phases can be determined. And then the properties within the boundaries can be assigned to a specific phase. Step 3 Based on the measured properties (e.g., the storage moduli) of each phase, the average values and the standard deviation are calculated. According to the above three steps, the storage moduli of OP and IP in the OPC paste can be calculated, which are 24 ± 7 GPa and 35 ± 6 GPa, respectively, while they are 25 ± 6 GPa and 39 ± 9 GPa in the slag-blended paste, respectively. The measured storage modulus of the C3 S clinker is 62 ± 13 GPa and it is 69 ± 16 GPa for the slag grain. As shown in Fig. 4.12b, the storage moduli extracted from the area data are in good agreement with those evaluated from the line data for different

4 Phase Quantification by Different Techniques

C3S

2μm

20 IP≈4.6μm 0 5

IP

Slag 1μm

C3S ITZ IP OP

40 20 IP≈4.1μm 0 5

10 15 Distance (μm)

20 Slag IP OP

80 60 40 Line 1

20

IP≈0.8μm 0 0

(b')

Storage modulus (GPa)

(a’)

Storage modulus (GPa)

OP

x direction 2 3 4

20

60

0

1

10 15 Distance (μm) Line 3

80

(d)

Slag-blended

(d')

60 40 20 IP≈3.7μm 0

0

(b) Storage modulus (GPa)

(a)

40

C3S ITZ IP OP

Line 2

80

Storage modulus (GPa)

IP

60

1

2

3 4 5 Distance (μm)

6

7

Slag IP OP

80 60 40 20

Line 3 IP≈0.6μm

0 0

1

2

3 4 5 Distance (μm)

6

7

0

(c) Storage modulus (GPa)

OP

C3S ITZ IP OP

Line 1

80

5

10 15 Distance (μm)

20

Line 4

80

C3S ITZ IP OP

60 40 20 IP≈6.0μm 0 0

(e) Storage modulus (GPa)

x direction

1 2 3 4

5

10 15 Distance (μm)

20 Slag IP OP

80 60 40 Line 2

20

IP≈1.3μm 0 0

(c')

Storage modulus (GPa)

OPC

Storage modulus (GPa)

108

(e')

1

2

3 4 5 Distance (μm)

80

6

7

Slag IP OP

Line 4

60 40 20 IP≈1.0μm 0 0

1

2

3 4 5 Distance (μm)

6

7

Fig. 4.11 Line data extracted from the modulus images of the OPC and slag-blended paste: a and a’ the modulus image and the locations of the selected lines; b, c, d, e and b’, c’, d’, e’ phase identification by the variation of storage moduli along the four vertical lines in a and a’ and the measured thickness of IP (inner hydration product) rims (Gao et al., 2018)

phases, which indicates that the color contrast in modulus images can be used to identify the individual phases.

4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique Step 1: Meshing 20μm

OPC

20μm

(a)

256 rows

Step 2: Boundary drawing 20μm

Ave.=24GPa Std.=7GPa

IP

Ave.=35GPa Std.=6GPa

C3S

7μm

Slag-blended

Step 3: Modulus calculation

OP

256 columns 7μm

109

Ave.=62GPa Std.=13GPa

7μm OP

Ave.=25GPa Std.=6GPa

IP

Ave.=39GPa Std.=9GPa

Slag

Ave.=69GPa Std.=16GPa

256 rows

256 columns

(b)

Fig. 4.12 a Steps for extracting the area data from modulus images; b storage modulus of individual phases calculated based on the four sets of line data in Fig. 4.11 and one set of area data of a (Gao et al., 2018)

4.5.2 Mapping Images Based on Different Measuring Parameters SPM with a sinusoidal force of 4 ± 3.5 μN at the frequency of 200 Hz was conducted on the randomly selected testing spots on the polished samples by using a Berkovich tip. The static force and the dynamic force were 4 μN and ± 3.5 μN, respectively. During the SPM test, the parameters including the contact force at the indenter tip, the amplitude of indenter displacement under the dynamic load, and the phase between the force applied and the displacement at each of the pixels were recorded. Figure 4.13a and b shows the optical image and the BSE image of a selected area with C3 S and slag grains in the slag-blended paste. Figure 4.13c–g shows the corresponding measured mapping parameters, which include the storage modulus, the

110

4 Phase Quantification by Different Techniques

loss modulus, the contact force, the amplitude, and the phase, respectively. It is seen that the storage modulus is much greater than the loss modulus. Although both the storage modulus and the loss modulus can be used to analyze the phase properties, it is recommended to use the storage modulus for phase identification for cementitious materials due to the fact that the individual phase could not be identified clearly by loss modulus with great scatter. The high scattering of the loss modulus measured under a low loading force of 4μN may be originally related to the low signal-to-noise (s/n) ratio. However, the measured loss modulus also shows scatters for modulus mapping tests done using the loads of 4 ± 1.5 and 8 ± 3.5 μN. The load magnitude of μN level will not cause low s/n ratio when it is compared to the load level of nN. In fact, the existing results show that the materials type would affect the scattering of the loss modulus: The loss modulus of the human teeth is not scattered (Balooch et al., 2004). (c)

150 120

(b) slag C3S

Slag

Slag C3S

10μm

20μm 10 8

slag C3S

60 30 0 0

(e)

Loss modulus (GPa)

(d)

90

6

slag C3S

4 2

Contact force (uN)

C3S

Storage modulus (GPa)

(a)

0 0

Amplitude (nm)

slag C3S

(g)

3

slag C3S

2

10 20 30 Distance (um)

40

20

40

60 40 Phase

(f)

7 6 5 4 3 2 1 0

10 20 30 40 Distance (um)

10 20 30 40 Distance (um)

20 0 0

1 0

10 20 30 Distance (um)

40

-20 Distance (um)

Fig. 4.13 Images and mappings of slag-blended paste a optical image, b SEM image, c storage modulus mapping and its variation along the line from C3 S to slag grain, d loss modulus mapping and its variation along the line, e contact force mapping and its variation along the line, f amplitude mapping and its variation along the line, g phase mapping and its variation along the line (Wei et al., 2017)

4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique

111

Table 4.4 Testing parameters in the SPM technique SPM

Applied force

Indentation depth

Pixel spacing

Resolution

4 ± 3.5 μN

0.5–1 nm

137 nm

40 nm

4.5.3 Resolution of Modulus Mapping Unlike the NI with the probing depth of 100–300 nm, the probed depth of modulus mapping is usually about 0.5–1 nm (shown in Table 4.4), which is small enough to ensure that the deformation of material under the SPM modulus mapping load is within the elastic domain. Since SPM only probes the elastic region, the resolution of SPM is mainly influenced by the radius of curvature of the indenter tip and the tested material. As a result, the contact radius can be used to assess the resolution of SPM, which can be calculated as (Balooch et al., 2004): √ ' a = 3FR/4E ' (4.6) where F is the nominal contact force and F = 4 μN here; R is the tip radius of the Berkovich tip and R = 400 nm; E ’ is the measured storage modulus, which will increase with increasing resolution (small value). Figure 4.14 shows the resolution calculated for phases from soft CSH to stiff unreacted grains, which will be used to assess the range of resolution of SPM measurement on the hardened cement paste. According to Eq. (4.6), the calculated resolution of CSH with storage modulus of 20–40 GPa is 30–40 nm. However, the resolution on the unreacted grains is 20–25 nm due to their high storage modulus. Overall, the measured resolution range of SPM for the hardened cement paste is 20–40 nm. Therefore, the resolution of SPM is taken as 40 nm (shown in Table 4.4) for conservation, which is much less than the pixel spacing (i.e., 137 nm) selected here for modulus mapping. The low resolution of SPM for hardened cement paste can ensure that there are no overlapping indents made on the sample during the modulus mapping. However, it is noted that a great pixel spacing may underestimate the thickness of the phases in hardened cement paste, especially for those with small thickness. The optimal experimental setting may be that the value of pixel spacing is similar to the resolution, and both of them should be small enough as compared to the thickness of the phase in cement paste.

4.5.4 Quantifying Thickness of CSH Layer It has been shown in the previous sections that modulus mapping in the form of SPM images can precisely quantify the size of the featured phases in hardened cement paste by following the steps: The SPM image was first obtained in terms of storage modulus by scanning the selected areas; lines were then drawn on the SPM image by passing through the unreacted phase and the surrounding phases of interest. It is

40 35 30 25 20

range of resolution: 24 ~ 37 nm

Fig. 4.14 Resolution of SPM modulus mapping for hardened paste (Wei et al., 2017)

4 Phase Quantification by Different Techniques

Resolution (nm)

112

OP-O OP-S IP-S IP-O C2S C3S slag

range of average storage modulus :23 ~ 91 GPa

15 0

20 40 60 80 100 Storage modulus (GPa)

noted that the lines should be normal to the surface of the unreacted phase in order to accurately determine the phase thickness. Figure 4.15 shows the variation of the storage modulus with the measuring distance along the lines. The color change in the SPM image as well as the abrupt drop or increase of the storage modulus can be used to identify the border of each phase, and the phase thickness can be taken as pixel spacing × number of pixels on the line segment located in the phase of interest. In order to make the results more representative, the thickness of the typical phases including IP of C3 S and C2 S in the OPC and the slag-blended pastes, and IP of slag grain in the slag-blended paste was quantified by 46 measurements on 31 SPM images. Figure 4.16 shows the quantified results of the total 46 measurements. In Fig. 4.16, the column denotes the average thickness value of each phase while the measured single value is plotted as the data point for both OPC and the slag-blended pastes. It can be seen that the thickness of inner CSH layer is influenced by the unreacted phase as well as the supplementary cementitious material used in the cementitious materials. The average IP layer thickness surrounding C3 S in the OPC paste is 6.3 μm, which is thicker than those of other types of IP layer (1.0–2.6 μm). The measured IP layer thickness surrounding C3 S in the OPC paste is consistent with the existing results where the measured IP thickness is 6 μm around a partially hydrated C3 S in the OPC paste by the BSE technique (Scrivener, 2004). However, there is no much difference in the thickness of IP layer surrounding C2 S grain between the OPC and the slag-blended pastes. The average thickness of IP-C2 S in the OPC paste is 2.6 μm, and it is 1.5 μm in the slag-blended paste. The average thickness of IP layer surrounding slag grain is 1.0 μm, which is smaller than that of any other types of IP layer. This may explain the different properties between OPC and the slag-blended systems. Gruskovnjak et al. (2006) have performed research to investigate the microstructures of alkali-activated slag (AAS) system and found that a thick shell of inner product will protect the unhydrated cores of the cement grains in the OPC paste, while the thickness of the protective layers of slag grain in the AAS paste is very small. However, the exact thickness of the IP layers in hardened

4.5 Phase Quantification by Scanning Probe Microscopy (SPM) Technique (a)

C3S IP thickness ≈ 3 ~ 6 μm

(b)

(c)

(d) Slag

C3 S

113

Slag

C3S

C3S Slag

Impurity

10μm

IP Impurity C3S ITZ

80 60

(f) 100

Line 1

IP≈10μm

40 20

ITZ≈2μm

Storage modulus (GPa)

Storage modulus (GPa)

(e) 100

0

ITZ≈ 0.3μm

80 Slag IP≈ 1.0μm

60

OP IP Slag ITZ C3S IP >4.9μm

IP≈ 8.6μm

40 ITZ≈ 2.5μm

20 0

(g) 100

10

20 30 Distance (um)

Line 3

80 60

C3S ITZ Slag IP≈ 11μm

0

40

OP IP

(h) 100

Slag IP≈ 0.5μm

40 20 ITZ≈1.4μm

0

Storage modulus (GPa)

0

Storage modulus (GPa)

Line 2

10

20 30 Distance (um) IP

Line 4

ITZ

80

C3S C3S IP >12μm

60

C3S IP >6μm

40 20 ITZ≈2.7μm

0 0

10

20 30 Distance (um)

40

40

0

10

ITZ≈ 1.4μm

20 30 Distance (um)

40

Fig. 4.15 Quantifying thickness of inner product in slag-blended paste a optical image of target area, b quantifying IP thickness from BSE image, c quantifying IP thickness from modulus mapping image, d gradient image, storage modulus variations along e Line 1, f Line 2, g Line 3, and h Line 4 (Wei et al., 2017)

OPC and AAS pastes was not provided in the work conducted by Gruskovnjak et al. (2006).

4.5.5 Quantifying Thickness of ITZ Between C3 S and Matrix The accurate evaluation of the thickness of the tiny ITZ between C3 S and the surrounding matrix is difficult to be done by the conventional quasi-static NI due

114

10

OPC paste Ave.=6.3

Slag-blended paste

8

Thickness (μm)

Fig. 4.16 Thickness of inner product (IP) layers around C3 S, C2 S, and slag grains in OPC and slag-blended pastes based on 46 measurements on 31 SPM images (Wei et al., 2018a)

4 Phase Quantification by Different Techniques

6 Ave.=2.6 Ave.=2.4

4

Ave.=1.5 Ave.=1.0

2 0

IP-C3S

IP-C2S

IP-C3S

IP-C2S IP-Slag

to the relatively large indentation force used in the NI test. To deal with this issue, Wei et al. (2018a) used the SPM mapping to quantify the ITZ thickness and the modulus of ITZ between different phases. Figures 4.17 shows the identified ITZ between the unreacted C3 S grain and the surrounding IP layer in both the OPC and the slag-blended pastes by modulus mapping technique. It can be seen from Fig. 4.17 that the identified thickness of the ITZ between C3 S and the matrix is 1–2 μm in both the OPC and the slag-blended pastes. The influence of adding nano-SiO2 on the properties of the interface between CSH gel and the cement grains has been studied by Xu et al. (2015b), and the modulus mapping results reveal that the thickness of the interface between the unreacted core and the CSH gel was about 200 nm, which is much less than that (1–2 μm) reported by Wei et al. (2018a). Xu et al. (2015b) didn’t provide the mineralogy of the cement grain used in their study and whether the cement grain is C3 S is unclear. A gap of around 1 μm has been reported by Scrivener (2004) to exit between the large grain and the surrounding CSH layer. However, the gap will only exist during the early ages and disappear after about 7–14 days, which is mainly caused by the formation of the denser CSH inside the gap (Scrivener, 2004). The ITZ between C3 S and the surrounding matrix may become a weak zone in cementitious materials as the ITZ between the aggregate and the matrix in concrete does, which may have a negative effect on the mechanical property of cementitious materials.

4.6 Phase Quantification by Nanoscratch Technique As compared to the NI technique, nanoscratch technique can perform continuous measurement of the fracture properties. Akono and Ulm (2014, 2017) derived the formula based on the linear fracture mechanics theory for calculating fracture toughness from the scratching results and applied it to various types of materials including cement, shales, and amorphous polymers. Wei et al. (2021) proposed a new method

4.6 Phase Quantification by Nanoscratch Technique

115

OP IP ITZ C3S

5μm

Storage modulus (GPa)

(a') 80

(a)

60

ITZ≈1.5μm

40 20 IP≈6μm

0 0

5

10 15 Distance (μm)

C3S

C2S 10μm

Storage modulus (GPa)

(b') 80

(b)

C3S 10μm

Storage modulus (GPa)

slag

20

C2S ITZ IP OP

60 40 20 IP≈2μm

IP≈5.5μm

0 0

10

ITZ≈1μm

20 30 Distance (μm)

(c') 150

(c)

C3S ITZ IP OP

100 ITZ≈2μm

40

C3S ITZ OP Slag

50 0 0

5

10 15 20 25 30 35 40 Distance (um)

Fig. 4.17 Thickness of ITZ between unreacted grain and the surrounding hydrates in OPC and slag-blended pastes (Wei et al., 2018a)

to identify the individual phase by using the fracture toughness distribution and improved the ITZ thickness quantification method.

4.6.1 Scratching Results To investigate the applicability of the scratching method with the redefined vertical loading rate (V V ) as discussed in Sect. 3.5.4, different transverse scratching speeds (V T ) of 1, 2, 4, and 8 μm/s were used. The maximum vertical forces were selected as 6, 8, 10, and 14 mN, and thus, the vertical loading rates (V V = F V−max /L) were

116

4 Phase Quantification by Different Techniques

(a)

(b) Scratching direction

Scratching direction

0.14mN/μm

8μm/s

0.10mN/μm

4μm/s

0.08mN/μm

2μm/s

0.06mN/μm

1μm/s 50μm

50μm

Fig. 4.18 SEM image of scratched path, a under various vertical loading rate and fixed transverse scratching speed of 4 μm/s, and b under various transverse scratching speed and fixed vertical loading rate of 0.1 mN/μm (Wei et al., 2021)

calculated as 0.06, 0.08, 0.10, and 0.14 mN/μm, given the scratching length of 100 μm. Ordinary Portland cement was used as cementitious material to prepare cement paste with w/c ratio of 0.3. The samples were then sealed cured in a room with temperature of 20 °C until the age of 28 days. The scratching path can be clearly seen from the SEM image, as shown in Fig. 4.18. All the scratching was started from the clinker phase. The green ellipse represents the ITZ between the clinker and the hydrates, and the yellow ellipse represents the clinker on the scratching path. The variations of the transverse scratching force (F T ) and the square of the penetration depth (d 2 ) along the scratching path (L) during the test are plotted in Figs. 4.19 and 4.20, respectively, for varying vertical loading rate (V V ) but fixed transverse scratching speed (V T ) of 4 μm/s, and varying transverse scratching speed (V T ) but fixed vertical loading rate (V V ) of 0.1 mN/μm. The ellipses in Figs. 4.19 and 4.20 are corresponding to those in Fig. 4.18. Three stages are clearly seen from the curves in Figs. 4.19 and 4.20 by the slope of the curve, which correspond to the phases of clinker, hydrates, and ITZ between the clinker and the hydrates. This suggests that different phases can be identified by the slope variation of the curve of either F T versus L or d 2 versus L along the scratching path. This feature is independent of the vertical loading rate as well as the transverse scratching speed.

4.6.2 Phase Identification Based on the simple definition in tribology, the force of friction or transverse scratching force (F T ) and the scratching hardness (H) can be calculated as: FT = COF × FV

(4.7)

4.6 Phase Quantification by Nanoscratch Technique

1.5

FT

-0.15

1

Clinker

-0.2

0.5

Hydrates

-0.1 -0.15

0

20

40 60 L (μm)

80

1 Clinker

-0.25

0 0

FT

1.5 1

Hydrates

-0.4 40 60 L (μm)

80

-d 2 (×10 6 nm2)

2

FT (mN)

-d 2 (×10 6 nm2)

d2

20

40

60 L (μm)

80

100

3

-0.1

2.5

-0.1

0

20

0

3

Clinker

0.5

(b) VV = 0.08mN/μm

0

-0.3

Hydrates

-0.3

100

(a) VV = 0.06mN/μm

-0.2

1.5

-0.2

0

-0.25

2

d2 FT

2.5

-0.3

d2 FT

-0.4

Clinker

-0.2

0.5

-0.5

0

-0.6

100

2 1.5 1

Hydrates

FT (mN)

-0.1

-d 2 (×10 6 nm2)

d2

2.5

-0.05

2

-0.05

3

0

FT (mN)

2.5

FT (mN)

-d 2 (×10 6 nm2)

0

117

0.5 0 0

(c) VV = 0.10mN/μm

20

40

60 L (μm)

80

100

(d) VV = 0.14mN/μm

Fig. 4.19 Variation of square of penetration depth (d 2 ) and transverse force (F T ) with scratching length (L) under fixed transverse scratching speed of 4 μm/s and vertical loading rate (V V ) of a 0.06 mN/μm, b 0.08 mN/μm, c 0.10 mN/μm, and d 0.14 mN/μm (Wei et al., 2021)

H=

FT A(d)

(4.8)

where COF is the coefficient of friction, F V is the vertical force, A(d) is the projected area of the probed volume to the plane perpendicular to the transverse scratching direction, as illustrated in Sect. 3.5.2. According to tribology theory, COF is related to the roughness of sample surface (Shaha et al., 2011). A new method can be proposed for phase identification based on the scratching technique. For one phase with constant COF, the first derivative of Eq. (4.7) can be expressed as: dFV dFT = COF × dL dL

(4.9)

According to the redefined vertical loading rate V V , the term of V V = dF/dL is a constant. And thus, the slope of F T versus L curve along the scratching length is proportional to the COF of each phase (dF/dL ∝ COF). Xu and Yao (2011) found that the COF of clinker and hydrates are different due to their distinct material properties with the COF of clinker (0.306) less than that of hydrates (0.403). Therefore, according to Eq. (4.9) the slope of F T versus L curve of clinker should be less than

4 Phase Quantification by Different Techniques

Clinker

1

Clinker

-0.3

1.5

-d 2 (×106 nm2)

2

d2 FT

-0.2

FT (mN)

-0.1

0.5

Hydrates -0.4 20

40 60 L (μm) (a) VT = 1μm/s

80

d2 FT

-0.2

Clinker

1

Clinker

-0.3

1.5

-d 2 (×106 nm2)

2

-0.4 0

20

40 60 L (μm) (c) VT = 4μm/s

1

Hydrates

0 20

40 60 80 L (μm) (b) VT = 2μm/s

100

3 2.5

Clinker

-0.1

2

d2 FT

-0.2

1.5 1

Clinker

-0.3

Hydrates

Hydrates

-0.4

0 80

1.5

0.5

0.5 Hydrates

Clinker

0

2.5

Clinker

Clinker

-0.3

0

3

-0.1

2

d2 FT

-0.2

100

0

2.5

-0.1

-0.4

0 0

-d 2 (×106 nm2)

3

0

2.5

0.5 0

0

100

FT (mN)

3

FT (mN)

-d 2 (×106 nm2)

0

FT (mN)

118

20

40 60 L (μm) (d) VT = 8μm/s

80

100

Fig. 4.20 Variation of penetration depth (d 2 ) and transverse force (F T ) with increasing scratch length (L) under fixed vertical loading rate of 0.1mN/μm and transverse scratching speed (V T ) of a 1 μm/s, b 2 μm/s, c 4 μm/s, and d 8 μm/s (Wei et al., 2021)

that of the hydrates. This is verified by the slope of the solid lines in Figs. 4.19 and 4.20, that the slope of the solid line fitting the data of clinker is significantly less than that of the hydrates. In these two figures, the black solid line and the purple solid line represent the F T versus L curve of clinker and hydrates, respectively. The slope was calculated from the F T versus L curve for various vertical loading rates and transverse scratching speeds, as listed in Table 4.5 and plotted in Fig. 4.21. It is seen that the slope of clinker phase is one-half of that of the hydrate phase, regardless of either the value of the vertical loading rate or the transverse scratching speed. This demonstrates the applicability of F T versus L curve method to a wide range of scratching parameters. Table 4.5 Slope of F T versus L curves to distinguish clinker and hydrates phase under various transverse scratching speed (V T ) and vertical loading rate (V V ) (Wei et al., 2021) Slope (mN/μm) for clinker/hydrate

V V (mN/μm) 0.06

0.08

0.10

0.14

V T (μm/s)

0.0117/0.0204

0.0122/0.0301

0.0165/0.0372

0.0233/0.0418

1 2

0.0119/0.0211

0.0144/0.0290

0.0176/0.0351

0.0221/0.0424

4

0.0122/0.0224

0.0131/0.0303

0.0170/0.0368

0.0229/0.0395

8

0.0120/0.0201

0.0128/0.0296

0.0154/0.0381

0.0215/0.0421

4.6 Phase Quantification by Nanoscratch Technique

Slope (mN/μm)

0.04 0.03

(b) 0.06

Hydrates

VT = 1μm/s VT = 2μm/s VT = 4μm/s VT = 8μm/s

0.05 Slope (mN/μm)

(a) 0.05

0.02 Clinker 0.01 0 0.04

119 VV = 0.06mN/μm

VV = 0.10mN/μm

VV = 0.08mN/μm

VV = 0.14mN/μm

0.04 Hydrates

0.03 0.02 0.01 0

0.06

0.08 0.1 0.12 VV (mN/μm)

0.14

0.16

Clinker 1

2

4 VT (μm/s)

8

Fig. 4.21 Slope of F T versus L curves for clinker and hydrate under various a vertical loading rates (V V ), and b transverse scratching speeds (V T ) (Wei et al., 2021)

It is observed from Fig. 4.21a that the change of vertical loading rate (V V ) will affect significantly the slope of F T versus L curve. Increasing V V will lead to an increased slope of F T versus L curve. The change of transverse scratching speed (V T ) does not affect the slope of F T versus L curve significantly, as shown in Fig. 4.21b. Under the same vertical loading rate, the slope of F T versus L curve remains a relatively constant value for the varying transverse scratch speed. It is thus recommended to adopt a constant vertical loading rate during scratch test when aiming to identify phases by the slope of F T versus L curves. On the other hand, from Eqs. (3.74), (3.76), (4.7), and (4.8), dd 2 COF dFV = · dL W × H dL

(4.10)

where W is a constant related to the shape of the indenter tip (as shown in Sect. 3.5.3). It is seen that the slope of the square of the penetration depth (d 2 ) versus L is proportional to the ratio of the COF and to the scratching hardness (H) of the material, since the vertical loading rate dF V /dL is a constant during a scratching test 2 and W is constant for the same indent ( 2 tip. The ) slope of the d versus L curve is proportional to COF of each phase dd ∝ COF . H dL H The slopes of d 2 versus L curves for the clinker and the hydrate phases under various vertical loading rates and transverse scratching speeds are demonstrated in Table 4.6 and Fig. 4.22. Similar to Fig. 4.21, the slope is proportional to the vertical loading rate (V V ) and does not vary much with the transverse scratching speed (V T ). Moreover, the slope of d 2 versus L curve of clinker is one order less than that of the hydrates, and it is thus more beneficial to use the slope of d 2 versus L curve for phase identification compared to that of the F T versus L curve. However, it should be noted that the slope method cannot distinguish LD CSH from HD CSH due to their very close mechanical properties.

120

4 Phase Quantification by Different Techniques

Table 4.6 Slope of d 2 versus L curves to distinguish clinker and hydrates phase under various transverse scratching speed (V T ) and vertical loading rate (V V ) (Wei et al., 2021) Slope (10−3 nm2 /μm) for clinker/hydrate V T (μm/s)

V V (mN/μm) 0.06

0.08

1

0.285/2.416

0.351/3.220

0.461/4.101

0.651/5.042

0.269/2.391

0.371/3.315

0.483/4.366

0.632/5.693

4

0.291/2.491

0.342/3.204

0.471/3.979

0.626/5.719

8

0.289/2.434

0.339/3.197

0.493/4.272

0.647/5.552

(b) 8

4

Hydrates VT = 1μm/s VT = 2μm/s VT = 4μm/s VT = 8μm/s

3 2 Clinker

1 0 0.04

VV = 0.06mN/μm VV = 0.08mN/μm

7 Slope (×10-3 nm2/μm)

Slope (×10-3 nm2/μm)

5

0.14

2

(a) 7 6

0.10

VV = 0.10mN/μm VV = 0.14mN/μm

6 5

Hydrates

4 3 2

Clinker

1 0

0.06

0.08 0.1 0.12 VV (mN/μm)

0.14

0.16

1

2 4 VT (μm/s)

8

Fig. 4.22 Slope of d 2 versus L curve for clinker and hydrates under various a vertical loading rates (V V ) and b transverse scratching speeds (V T ) (Wei et al., 2021)

4.6.3 Probability Distribution of Fracture Toughness Due to the heterogeneous nature of cement paste, sufficient data are required for statistic analysis to obtain the fracture properties of individual phase. It was found that 100 data are sufficient for the statistical analysis with an error within ± 5% (Sorelli et al., 2008). In this section, total 101 scratching data with space of 1 μm were obtained from each of the 100 μm-long scratched paths as shown in Fig. 4.18. The fracture toughness was calculated at the 101 locations on each scratching path. The probability density function (PDF) of the calculated fracture toughness for different phases is plotted in Figs. 4.23 and 4.24. Different phases can be clearly identified based on the PDF curve of the fracture toughness of each phase, as shown in Figs. 4.23 and 4.24. It can be concluded that the fracture toughness of clinker is the highest, followed by the hydrates, and the fracture toughness of ITZ is the smallest. In addition, there are two peaks in the PDF curves for hydrates, and these two peaks can be considered as the contribution from the LD CSH and the HD CSH. It was previously concluded that the calcium hydroxide (CH) cannot be distinguished statistically from the CSH phase through the scratching depth as well as the coefficient of friction (Xu & Yao, 2011), because there is minor difference in physical and mechanical properties between CSH and CH phase. However, it is found

4.6 Phase Quantification by Nanoscratch Technique 0.25

0.25

(a)

0.2

PDF

PDF

0.1

(b)

0.2

Clinker

HD CSH 0.15 LD CSH

0.05 ITZ

121

0.1 ITZ

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88 KC (MPa•m½) 0.25

0.1 ITZ

CH

(d)

0.2

Clinker PDF

HD CSH 0.15 LD CSH

HD CSH 0.15 LD CSH 0.1 0.05

Clinker

ITZ CH

0 0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

0 KC (MPa•m½)

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

PDF

KC (MPa•m½)

(c)

0.2

0.05

CH

0

0

0.25

Clinker

0.15 LD CSH

0.05

CH

HD CSH

KC (MPa•m½)

Fig. 4.23 Distribution of fracture toughness of different phases under the fixed transverse scratching speed (V T = 4 μm/s) and the vertical loading rate (V V ) of a 0.06 mN/μm, b 0.08 mN/μm, c 0.10 mN/μm, d 0.14 mN/μm (Wei et al., 2021)

that the fracture toughness of CH is 0.2–0.3 MPa m½ higher than that of CSH by NI test (Nˇemeˇcek & Hrbek, 2016; Nˇemeˇcek et al., 2016). Accordingly, the CH phase is identified in Figs. 4.23 and 4.24 from the PDF curves of the fracture toughness. It is seen that the mean fracture toughness of CH is 1.06 MPa m½ , which is 0.19 MPa m½ higher than that of CSH, and is consistent with the findings by Nˇemeˇcek et al. (2016). This suggests that the distribution of fracture toughness is more accurate for phase identification compared to the method of scratching depth and the coefficient of friction. One criterion about the applicability of phase identification by the scratching technique is to ensure that the scratching depth to the characteristic size of the phase (d/D) should be less than 1/10 (Durst et al., 2004), this is to avoid the phase overlap effect on the measured data. Considering the characteristic size of hydrates and the clinker is at the scale of × 10−6 m, and the maximum scratching depth achieved in this study is about × 10−7 m under the maximum vertical load of 14 mN, it is thus concluded that the overlap effect of different phase on the measured data can be avoided, and the phase identification by the scratching method proposed in this section is valid. Although the slope of the scratch curve can be used to identify different phases, the slope of curve varies proportionally with the vertical loading rate. Different from

122

4 Phase Quantification by Different Techniques

0.25

0.25

(a)

0.2

0.2 LD CSH HD CSH

Clinker

0.1

0.05

ITZ

0.15

PDF

PDF

0.15

CH

0.05

ITZ

CH

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

KC (MPa•m½)

KC (MPa•m½) 0.25

(c)

(d)

0.2 HD CSH LD CSH

Clinker

0.1

0.05 ITZ

CH

0.15

PDF

PDF

Clinker

0

0.2 0.15

HD CSH LD CSH

0.1

0

0.25

(b)

HD CSH LD CSH

Clinker

0.1

0.05 ITZ

CH

0 0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

0.64 0.75 0.86 0.98 1.09 1.20 1.32 1.43 1.54 1.66 1.77 1.88

0 KC (MPa•m½)

KC (MPa•m½)

Fig. 4.24 Distribution of fracture toughness of different phases under the fixed vertical loading rate (V V = 0.10 mN/μm) and the transverse scratching speed (V T ) of a 1 μm/s, b 2 μm/s, c 4 μm/s, d 8 μm/s (Wei et al., 2021)

the slope method, the calculated fracture toughness of each phase and the probability distribution of their fracture toughness does not vary with different vertical loading rate or transverse scratching speed, and thus a reliable alternative for phase identification.

4.6.4 Fracture Toughness of Individual Phases Based on the PDF curve of the fracture toughness as shown in Figs. 4.23 and 4.24, the fracture toughness of each phase can be obtained, which can serve as the input parameters for mechanical property prediction through upscaling schemes. The fracture toughness of each phase is plotted in Figs. 4.25 and 4.26 under the fixed transverse scratching speed of V T = 4 μm/s and the various vertical loading rate (V V ), and the fixed vertical loading rate of V V = 0.10 mN/μm and the various transverse scratching speed (V T ), respectively. The boxplot shows the statistical indicators of the fracture toughness of each phase, such as the mean value and the maximum and the minimum values.

4.6 Phase Quantification by Nanoscratch Technique Fig. 4.25 Fracture toughness of different phases under the fixed transverse scratching speed (V T = 4 μm/s) and various vertical loading rates (V V ) (Wei et al., 2021)

Fig. 4.26 Fracture toughness of different phases under the fixed vertical loading rate (V V = 0.10 mN/μm) and various transverse scratching speed (V T ) (Wei et al., 2021)

123

VV = 0.06mN/μm

VV = 0.10mN/μm

VV = 0.08mN/μm

VV = 0.14mN/μm

Mean 25% ~ 75%

Median Min. ~Max.

VT = 1μm/s

VT = 4μms

VT = 2μm/s

VT = 8μm/s

Mean 25% ~ 75%

Median Min. ~Max.

From Figs. 4.25 and 4.26, the fracture toughness of each phase in cement paste ranges between 0.60–1.88 MPa m½ . Clinker shows the highest fracture toughness of 1.44–1.88 MPa m½, followed by CH (0.98–1.09 MPa m½ ), HD CSH (0.86– 0.98 MPa m½ ), LD CSH (0.71–0.87 MPa m½ ), and ITZ (0.60–0.65 MPa m½ ). ITZ has the lowest fracture toughness among all the phases, which further verifies that ITZ is the weakest phase in cement paste (Königsberger et al., 2014a and 2014b). Therefore, ITZ should be the critical phase that determines the fracture properties of cement paste. The protocol of scratching test has minor effect on the measured fracture toughness of different phases. The fracture toughness remains relatively constant under different vertical loading rate as well as the transverse scratching speed. The scratching process is not a relaxation process in cementitious materials which is different from the organic polymer materials. The transverse scratching speed has significant effect on the fracture toughness of organic polymer materials due to the sliding and entanglement between molecules, leading to a logarithmic growth of fracture toughness with the transverse scratching speed (Akono & Ulm, 2017; Gittus, 1975). Cementitious materials do not have such behavior, and the effect of transverse scratching speed (V T ) on fracture toughness of each phase is limited.

124

4 Phase Quantification by Different Techniques

In summary, both the vertical loading rate and the transverse scratching speed show minor effect on the measured fracture toughness of each phase and thus provides a wide range of scratching test protocol.

4.6.5 Thickness of Clinker and Hydrates Assessed from Fracture Toughness From the scratching path as shown in the SEM images (Fig. 4.18), the clinker and the hydrates phases can be clearly distinguished. The thickness (h) of each phase can be quantified by counting the number of pixels of each phase in the SEM image, which is calculated as: Np h = N100 100

(4.11)

where N p is the number of pixels in one phase, N 100 is the number of pixels within a distance of 100 μm. Pixel is the basic unit of the SEM image, and the number of pixels in each phase can be counted by using the Image-Pro Plus software which is equipped with a function of distance measurement. By drawing a straight line that coincides with the scratching path, and the number of pixels can be counted automatically. It is seen that the thickness of clinker quantified by the SEM image (Fig. 4.27a and c) and that by the distribution of fracture toughness (Fig. 4.27b and d) are very similar, the difference is less than 1 μm. For example, for the scratching path under the vertical loading rate (V V ) of 0.10 mN/μm and the transverse scratching speed (V T ) of 1 μm/s as shown in Fig. 4.27a, the thickness of clinker is 35.3 μm and 35 μm quantified by the SEM image and by the distribution of fracture toughness, respectively, and the thickness of hydrates is 64.7 μm and 64 μm quantified by the SEM image and the fracture toughness distribution, respectively. In Fig. 4.27c, for the scratching path under the vertical loading rate (V V ) of 0.10 mN/μm and the transverse scratching speed (V T ) of 4 μm/s, the thickness of hydrates by the two methods is 21.2 μm versus 19 μm, 6.3 μm versus 6 μm and 5.5 μm versus 5 μm, respectively. Apparently, the thickness of hydrates phase quantified by the SEM image analysis is greater than that quantified from the fracture toughness distribution. This is because that the ITZ phase cannot be clearly identified through the SEM images where the ITZ phase is mixed with the hydrates phase, leading to a thicker hydrates phase by the SEM image.

4.6 Phase Quantification by Nanoscratch Technique (b)

(a)

125

64.7μm

0.10mN/μm

KC (MPa•m½)

E 35.3μm

(c)

1.7 A 1.4 35μm D 0.8

G F

C 0

(d)

B 64μm

1.1

0.5

50μm

Clinker ITZ Hydrates

2

20

40 60 L (μm)

80

100

2

4μm/s

15.8μm 15.0μm

21.2μm 6.3μm 5.5μm

50μm

KC (MPa•m½)

1.7 36.2μm

1.4

37μm

19μm 15μm 6μm 5μm

1.1

15μm

0.8 0.5 0

20

40 60 L (μm)

80

100

Fig. 4.27 Quantifying thickness of different phase by using a SEM image analysis of scratching path, and b fracture toughness distribution under vertical loading rate of 0.10 mN/μm; c SEM image analysis of scratching path, and d fracture toughness distribution under transverse scratching speed of 4 μm/s (Wei et al., 2021)

4.6.6 Thickness of ITZ Assessed from Fracture Toughness The thickness of ITZ between the clinker and the hydrates has been quantified by the coefficient of friction (COF) method based on the nanoscratch test (Xu et al., 2017), in which the boundaries of ITZ were determined based on the discontinuity of the COF curve (Fig. 4.28). In this method, a tangent line through the inflection point of the fitting curve intersects with the two lines representing the mean COF values of the adjacent phases, and the region between the two intersections is regarded as the region of the ITZ phase. However, COF method brings uncertainty to the quantification of ITZ thickness due to the influence of surface roughness of sample on the measured COF. It has been proved that the mechanical properties (such as the dynamic modulus or the hardness) can be used for ITZ thickness quantification (Xu et al., 2015a and 2015b, Wei et al., 2018a). However, fracture toughness as an important mechanical property has barely been used for ITZ thickness quantification. From Figs. 4.23 and 4.24, there are big differences between fracture toughness of ITZ, hydrates, and clinker, which is the base for ITZ thickness calculation (Habelitz et al., 2001; Hodzic et al., 2000). Figure 4.27b shows the distribution of fracture toughness along the scratched path. Clearly, the distribution is categorized into three parts. The fracture toughness

126

Tangent line Coefficient of friction (COF)

Fig. 4.28 Illustration of coefficient of friction (COF) method to determine the thickness of ITZ (T ) between clinker and the hydrates (Wei et al., 2021)

4 Phase Quantification by Different Techniques

T

Inflection point Fitting curve of COF

Mean value of COF of hydrates

Mean value of COF of clinker Scratch length (L)

maintains at a constant highest value within AB distance, and the phase in AB stage is considered to be clinker. After B point, there is an abrupt drop of fracture toughness to point C and then jumped to point D after a distance of 1 μm. The fracture toughness of DE stage maintains at a relative constant range with an average value slightly greater than the fracture toughness of point C. The phase in DE stage is considered to be the hydrates phase. Therefore, the ITZ is in between points B and D and including point C. Based on the distributions of the measured fracture toughness as shown in Fig. 4.29a (obtained by enlarging Fig. 4.27b), a new method is proposed to quantify the thickness of ITZ between the clinker and the hydrates. According to the concept of COF method, a “S” shape curve should be first constructed around the ITZ to obtain the tangent line at point C. However, the fracture toughness distribution around the ITZ appears like “V” shape due to the sparse data point in the ITZ phase, it is not possible to construct the “S” shape curve using the concept of COF method. Thus, it is proposed that the thickness of ITZ (T ) is divided into two parts segmented by point C. One part with thickness of T 1 is close to the clinker phase, and the other part with thickness of T 2 is close to the hydrates phase, and thus, we have ITZ thickness of T = T 1 + T 2 . Draw a line passing the average fracture toughness of the clinker phase and denoted as M1, and then find the symmetric point of B, E, and line M1 about the point C. These symmetric points and lines are correspondingly denoted as B’, E’, and M1 ’, respectively, as shown in Fig. 4.29b. Similarly, D’, F’, and M2 ’ can be obtained. Finally, the “S” shape curves can be constructed passing points E, B, M1 , E’, B’, M1 ’ for calculating T1, and D, F, M2 , D’, F’, M2 ’ for calculating T 2 , respectively, as shown in Fig. 4.29b. Then, the tangent line was drawn passing point C for “S” shape curve EBCB’E’, which is intersected with the average fracture toughness line of M1 and M1 ’, respectively, and the transverse distance between the intersected points is denoted as T 3 (Fig. 4.29c). Similarly, T 4 can be obtained (Fig. 4.29d). By symmetry, T1 = T3 /2, T2 = T4 /2, and thus, the thickness of ITZ between clinker and hydrates is T = T3 /2 + T4 /2.

4.6 Phase Quantification by Nanoscratch Technique

T1

B

1.5

T2

M1

CSH

Clinker

1

D

F

0.5

M2

KC (MPa•m½)

E

C

0 45

46

47

48 49 L (μm)

50

51

(c) E

KC (MPa•m½)

2.5 2 1.5 1 0.5 0 -0.5 -1

2.5 (b) 2 E 1.5 1 F' 0.5 0 M 2' -0.5 -1 45 46

ITZ thickness = T1 + T2

(a)

2

B

M1 C B'

E' M 1'

T3=2T1 45

46

47

48 49 L (μm)

50

2.5 2 1.5 1 0.5 0 -0.5 -1

KC (MPa•m½)

KC (MPa•m½)

2.5

51

127

B

M1 D

F M2

C D'

M 1' E'

B' 47

48 49 L (μm)

50

51

(d)

T4=2T2 F' M 2'

45

C

M2 D

F

D'

46

47

48

49

50

51

L (μm)

Fig. 4.29 Illustration of fracture toughness-based (K C ) method to determine the thickness of ITZ (T ) between clinker and hydrates, a fracture toughness distribution around ITZ, b constructing “S” shape curve of fracture toughness, c demonstration of T 1 calculation, and d demonstration of T 2 calculation (Wei et al., 2021)

The calculated thickness of ITZ between the clinker and the hydrates by the COF method and the K C method proposed in this section is compared and the results are shown in Fig. 4.30. It is seen that both the vertical loading rate (V V ) and the transverse scratching speed (V T ) have minor influence on the quantified ITZ thickness by either the COF method or the fracture toughness (K C ) method. This suggests that nanoscratch as a continuous measuring technique can quantify the thickness of ITZ between clinker and hydrates without special consideration of the scratch test parameters. It should be noted that the effect of surface roughness causes the slight difference on the quantified ITZ thickness by the COF method and the K C method. The calculated average ITZ thickness under different V V and V T by the COF method is 0.926 μm, and it is 0.884 μm by the K C method. Meanwhile, the standard deviation of ITZ thickness calculated by K C method is 0.0124 μm and that calculated by COF method is 0.0151 μm.

128

4 Phase Quantification by Different Techniques

Thickness of ITZ (μm)

1

VV = 0.06mN/μm

VV = 0.08mN/μm

VV = 0.10mN/μm

VV = 0.14mN/μm

0.95 0.9 0.85 0.8 0.75 COF

KC KC

VT 1μm/s = 1μm/s

COF

KC KC

VT 2μm/s = 2μm/s

COF

KC KC

COF

KC KC

VT 4μm/s = 4μm/s VT 8μm/s = 8μm/s

Fig. 4.30 Comparison of thickness of ITZ between clinker and hydrates obtained from the COF method and the fracture toughness method (K C ) under different vertical loading rate and transverse scratching speed (Wei et al., 2021)

4.7 Phase Quantification by X-Ray Computed Tomography X-ray computed tomography (CT) allows nondestructive examination of the composition, structure, and distribution of defects in cement-based materials by threedimensional (3D) reconstruction of samples from CT images. It has attracted wide attentions in the detection of the internal structure of cement-based materials.

4.7.1 Reducing Background Noise of Original CT Image by Image Filtering Image filtering should be conducted on the original CT image before image segmentation to reduce the background noise introduced during scanning. Background noise can affect the gray level of a phase, causing misidentification. Figure 4.31 shows the typical CT images of the slag-blended and the Portland cement pastes. In these images, the gray level ranges from 0 to 255, the higher-density phase appears white with higher gray level, the lower-density phase appears black with lower gray level. However, background noise can blur a phase, forming a mixed gray level even in a single phase. It can be observed from Fig. 4.31 that the color of the single clinker or slag phase is not uniform due to the background noise. This leads to miscalculation of phase fractions. Many image filtering methods have been developed to reduce the background noise of gray level images (Hildebrandt and Polthier 2004; He et al., 2013; Milanfar, 2013). The curvature flow algorithm is used here because it effectively maintains the edges of different phases in cement pastes. To reduce the computing time, a small

4.7 Phase Quantification by X-Ray Computed Tomography

129

5 mm

444 μm

(b)

(a)

Focus area

444 μm (d) 5 mm

1001 μm

(c)

Focus area 1001 μm Fig. 4.31 Typical CT images of a slag-blended cement paste, and b its focus area (300 × 300 pixels); c Portland cement paste, and d its focus area (337 × 337 pixels) (Wu et al., 2020)

cube region in the middle of the image was selected for the subsequent processing. For the slag-blended cement pastes, the central 300 × 300 pixels with resolution of 1.48 μm are selected, and 300 images along the vertical direction are analyzed, as shown in Fig. 4.31b. For the Portland cement pastes, the central 337 × 337 pixels with resolution of 2.97 μm (the border length 1000 μm) are selected and 337 images along the vertical direction are analyzed, as shown in Fig. 4.31d. Figure 4.32 shows the curve of gray level distribution before and after filtering using the curvature flow algorithm. It seems that the irregular abrupt points from the background noise are reduced after filtering and the overall trend is still consistent with the original one. The background noise appears significant in the slag-blended system, as many irregular abrupt points are located in the curve. This is because the scanning resolution is higher for this system, which amplifies the effect of background noise. The large fluctuations at the tail of the filtered curve for the slag-blended system are caused by the fluctuation of a few points from the amplifying effect of the log scale used in the y-axis; the very small number of such points means that the entire trend of the filtered curve is not affected. The filtered image is used for segmentation to identify the different phases.

130

OPC-original OPC-filtered slag-blended-original slag-blended-filtered

1.E+07 1.E+06 Number of pixels

Fig. 4.32 Histogram of CT image gray level before and after filtering by curvature flow for slag-blended and Portland cement pastes (Wu et al., 2020)

4 Phase Quantification by Different Techniques

1.E+05 1.E+04

image noise

1.E+03 1.E+02 1.E+01 1.E+00 1.E-01

0

50

100

150 200 Gray level

250

300

4.7.2 Determining Gray Level Range of Phases for Threshold Segmentation Threshold image segmentation is widely used to identify different phases solely based on gray level. Gray level thresholds can be chosen manually or automatically from the gray level histogram. According to differences in phase gray levels, the range of gray level of the unreacted particle, the hydration product, and the pore phases can be determined. In fact, determination of the gray level threshold of each phase in cement-based materials is a complex task, since there is no standard method for it. Huang et al. (2013) determined the range of gray level for the unhydrated cement particles and the hydration products in hardened cement paste by judging the wave valley of the gray level histogram. The gray level threshold between the clinker and the hydration products can vary from 114 to 162 with changes in the w/c ratio and the ages of the samples. Segmentation of the Portland cement paste sample with w/c ratio of 0.5 at the age of 28 days (5O28d) is illustrated in Fig. 4.33. The ranges of gray level of the clinker and the pore phases are determined first, and the range of gray level between them is taken as the gray level range of the hydration product. An arrow line with length of 172 μm runs through a randomly selected clinker (Fig. 4.33a), the distribution of gray level along the arrow line is plotted as the solid curve shown in Fig. 4.33b. The highest peak is observed in the middle length of the curve, corresponding to the position of the clinker passed by the line in the CT image. Because the upper gray level threshold of the clinker is normally specified as 255 for its brightest color, segmentation determines the lower gray level threshold by the surrounding peaks of the gray level distribution of the hydration product, which is about 80 from the curve shown in Fig. 4.33b. The lower threshold of the clinker phase is recorded as A1 , and it also serves as the upper threshold of the hydration product phase. Similarly, to determine the gray level threshold of the pore phase, an arrow line runs through a randomly selected pore and the distribution of the gray level along this line is seen

4.7 Phase Quantification by X-Ray Computed Tomography

131

as the dashed curve shown in Fig. 4.33b. A lower peak with a distance of 30–90 μm from the beginning of the arrow line corresponds to the gray level distribution of the pore, and thus, the peak value of 29 is recognized as the upper gray level threshold of the pore and denoted as B1 . Five typical clinker regions are selected to repeat this procedure to obtain an average value of A1 . Table 4.7 shows the scatter of the thresholds for the clinker phase, the average threshold of A1 is 81, and the coefficient of variation is limited to 1.8%. Similarly, the average value of B1 is obtained from five typical pore regions. In summary, the gray level range of each phase in 5O paste at the age of 28 days is as follows: clinker [81, 255], hydration product [29, 80], and pore [0, 28]. Segmentation of the slag-blended cement paste is similar to that of the Portland cement paste. The segmentation of the 3O50S specimen at the age of 7 days is shown in Fig. 4.34. The gray level of the slag is between that of the clinker and the hydration product due to the difference in density of these phases. In the image, the unreacted particles with higher gray level are clinkers, the lower ones are slags. Similarly, the gray level range of each phase can be obtained for the slag-blended system at the age of 7 days: clinker [73, 255], slag [53, 72], hydration product [7, 52], and pore [0, 6]. Bentz et al. (2002) found that the mix proportion, w/c ratio, and the curing age are all important factors affecting the gray level when the setting parameters of the (a)

Line 1 Line 2

(b) 120

Line 1

Gray level

1001 μm

100

Clinker

60 40

B1: 29

20

Line 2

clinker A1: 80

80

Pore

hydration product pore

0 0

1001 μm

30

60 90 120 150 180 Length (μm)

Fig. 4.33 Determining threshold for Portland cement pastes with w/c ratio of 0.5 at the age of 28 days: a original CT image; and b identifying threshold gray level of each phase (Wu et al., 2020)

Table 4.7 Lower thresholds of clinker phase in 5O from the five typical regions at the age of 28 days paste (Wu et al., 2020)

Region No.

Lower threshold of clinker

1

82

2

80

3

79

4

81

5

82

Average

81

Standard deviation

1.50

Coefficient of variation

1.8%

132

4 Phase Quantification by Different Techniques (b)

(a)

(c)

126μm

444 μm

41μm

Slag

444 μm

Clinker

444 μm

Pore

174μm 444 μm lower threshold = 73 for clinker

80 Clinker

40

160

(e) lower threshold = 53 for slag

120 Gray level

120 Gray level

160

(d)

80 40

Slag

0

0 0

10

20 30 40 Length (μm)

50

(f)

120 Gray level

160

444 μm

444 μm

upper threshold = 6 for pore

80

Pore

40 0

0

45 90 135 Length (μm)

180

0

30

60 90 120 150 Length (μm)

Fig. 4.34 Determination of threshold of slag-blended cement pastes at the age of 7 days: a clinker in CT image; b slag in CT image; c pore in CT image; d lower threshold for clinker; e lower threshold for slag; and f upper threshold for pore (Wu et al., 2020)

CT equipment are unchanged. The gray level range of each phase is summarized in Table 4.8 for both the OPC and the slag-blended pastes at the age of 1, 7, and 28 days. It is seen that the clinker in the OPC pastes had a smaller upper threshold compared with that in the slag-blended pastes. In addition, the upper gray level threshold of the pore in the OPC pastes is between 16 and 45, whereas it is less than 15 in the slag-blended pastes. This may be attributed to the different CT resolutions used in the two tests. The slag-blended cement system is scanned with a higher CT resolution, leading to a more sensitive image. Different gray level thresholds are also found for pores in Berea sandstone by CTs with different resolutions (Peng et al., 2012). The adjustment of contrast and brightness before CT scanning and the distance between the X-ray source and the specimen influence the gray level as well. The threshold-segmented CT images for both the OPC and the slag-blended pastes are shown in Fig. 4.35. For clear demonstration, the clinker is set to be white, the hydration product is set to be gray, and the pore is set to be black in the segmented image (Fig. 4.35b). It can be seen that different phases in the CT image after segmentation are clearly identified compared with the original image (Fig. 4.35a). The segmented images can be used for 3D reconstruction to quantify the fraction of each phase. However, the threshold segmentation is not suitable for the unreacted slag particles. Figure 4.35c and d shows the slag-blended paste before and after threshold segmentation, respectively. For a clear demonstration, the unreacted clinker is colored red, the slag is colored yellow, the hydration product is colored blue, and the pore is colored green. From Fig. 4.35d, it is seen that the red clinker is often mixed with the yellow unreacted slag, which is not the case in real slag-blended paste.

4.7 Phase Quantification by X-Ray Computed Tomography

133

Table 4.8 Range of gray level of different phases determined from CT images using the threshold method for the four pastes at the age of 1, 7, and 28 days (Wu et al., 2020) Mixture

Age (days)

Pore

HP

UP

Slag

Clinker

3O

1

[0, 45]

[46, 69]

–

–

[70, 255]

7

[0, 16]

[17, 37]

–

–

[38, 255]

28

[0, 26]

[27, 58]

–

–

[59, 255]

1

[0, 31]

[32, 52]

–

–

[53, 255]

7

[0, 25]

[26, 59]

–

–

[60, 255]

28

[0, 28]

[29, 80]

–

–

[81, 255]

1

[0, 1]

[2, 37]

[37, 255]

[38, 59]

[60, 255]

7

[0, 6]

[7, 52]

[48, 255]

[53, 72]

[73, 255]

28

[0, 1]

[2, 47]

[49, 255]

[48, 74]

[75, 255]

1

[0, 1]

[2, 34]

[44, 255]

[35, 56]

[57, 255]

7

[0, 10]

[11, 62]

[67, 249]

[63, 76]

[77, 255]

28

[0, 11]

[12, 67]

[65, 255]

[68, 92]

[93, 255]

5O

3O50S

5O50S

Note HP = hydration products, UP = unhydrated particles

(b) Clinker

1001 μm

(a)

Hydration product Pore

1001 μm (d) 444 μm

(c)

Clinker Slag Hydration product Pore

444 μm

Fig. 4.35 Results of phase identification of OPC and slag-blended cement pastes by threshold segmentation: a original OPC CT image, b OPC phase identification, c original slag-blended CT image, d slag-blended phase identification (Wu et al., 2020)

4.7.3 Determining Phase Boundary Using Edge Detection Similar gray levels might exist for two phases due to the limited contrast between phases as well as the local gray level variations for the same material, which makes the threshold segmentation be a challenge in distinguishing phases, such as the case in

134

4 Phase Quantification by Different Techniques

Fig. 4.35d. Gradient-based edge detection overcomes these drawbacks, and it detects the outline of an object and the boundary between the object and the background in a CT image (Sharifi et al., 2002). Edge detection has been used in remote sensing, medicine, automation, aerospace, transportation, and the like, but has not been used in processing cementitious materials. An effective edge detector reduces a large number of data points while preserving the most important features of the image. Many mature edge detector algorithms have been developed, such as the Sobel (Vincent & Folorunso, 2009), the Roberts (Al-amri et al., 2010), and the Prewitt (Seif et al., 2010). The Sobel edge detector algorithm, one of the most common ones, uses 3 × 3 neighborhoods for gradient calculations and thus avoids calculating the gradient of an interpolated point between pixels. The Sobel edge detector algorithm is used in this section. The gray level of a pixel in a 2D gray image is recorded as f (x, y), which is a function of the horizontal (x) and vertical (y) coordinates of that pixel. The separate gradient components of the gray level can be quantified as the partial derivatives S x and S y for horizontal and vertical orientation, respectively. They are combined to find the magnitude of the gradient at the pixel. Figure 4.36a shows the gray level distribution of pixels in the 3 × 3 neighborhood surrounding the target pixel, and Fig. 4.36b shows the real 3 × 3 neighborhood surrounding f (x, y) in a CT image. The horizontal partial derivatives (S x ) and the vertical partial derivatives (S y ) can be implemented using convolution masks as shown in Fig. 4.36c and d. Sx = [ f (x + 1, y − 1) + 2 f (x + 1, y) + f (x + 1, y + 1)] − [ f (x − 1, y − 1) + 2 f (x − 1, y) + f (x − 1, y + 1)]

(4.12)

S y = [ f (x − 1, y + 1) + 2 f (x, y + 1) + f (x + 1, y + 1)] − [ f (x − 1, y − 1) + 2 f (x, y − 1) + f (x + 1, y − 1)]

(4.13)

The gray level gradient G(x, y) at location (x, y) can then be calculated as G(x, y) =

/

Sx2 + S y2

(4.14)

The calculated gray level gradient G(x, y) reflects the gray level change in the target pixel (x, y) in the horizontal and the vertical directions. The value of G(x, y) is then compared with a value of T to identify the boundary between the target pixel (x, y) and the surrounding pixels. If G(x, y) > T, the target pixel (x, y) is considered the boundary and so the gray level of that pixel is set as 255, which appears white in the image. Otherwise, the gray level of that pixel is set as 0, which appears black. By repeating the process across the entire image, the boundaries between phases can be identified. The value of T is selected according to different image types. A small T value indicates a very gentle change in gray level, which requires precise edge detection.

4.7 Phase Quantification by X-Ray Computed Tomography

135

f(x-1,y-1)

f(x,y-1)

f(x+1,y-1)

-1

0

1

f(x-1,y)

f(x,y)

f(x+1,y)

-2

0

2

f(x-1,y+1)

f(x,y+1)

f(x+1,y+1)

-1

0

1

(a)

(c)

pixel f(x,y)

(b)

-1

-2

-1

0

0

0

1

2

1

(d)

/

Fig. 4.36 Sobel gradient G(x, y) = Sx2 + S y2 operator used in edge detection to determine whether pixel f (x, y) belongs to boundary: a 3 × 3 neighborhood surrounding f (x, y); b actual 3 × 3 neighborhood of f (x, y) in CT images; c horizontal convolution mask for partial derivative S x ; and d vertical convolution mask for partial derivative S y (Wu et al., 2020)

A large T indicates a dramatic change, which is favorable in edge detection as the boundary can be easily detected. However, the selection of T depends largely on what type of image is analyzed. Gradient-based edge detection with the Sobel algorithm is used to identify the unreacted phases. It should be noted that the results of edge detection will be quite different for different limit values of T. For cement paste, it is determined that T should be in the range of 0–1. When it is greater than 1, all phases in the cement paste would be recognized as the inner part and no boundary would be detected, which is of course not realistic. The results of edge detection with different T values as well as the original CT images are shown in Fig. 4.37 for the OPC and the slag-blended pastes. In the segmented images, white represents the detected phase boundary and black represents the phase’s inner region. It is obvious that different T values give very different segmented images. In the case of T = 0.3, too many regions are identified as boundaries, even the interior of the unreacted particles. In the case of T = 0.5, the detected boundary is somewhat reduced. At T = 0.7, the boundary is reduced quite a bit. Increasing T to 0.9 and 1.1, as shown in Fig. 4.37, further reduces the boundary. However, the external particle boundary is not continuous in the edge-detected images and the complete shape of the particles could not be outlined. It is seen from Fig. 4.37, the edge detection is not successful in segmenting different phases in both OPC and the slag-blended pastes. The objective of the edge detection is to find the boundary of a phase and identify the pixels inside it to realize image segmentation. However, the complex compositions in cementitious materials can introduce many boundaries from gradient-based segmentation, which is why a large number of meaningless and discontinuous boundaries are obtained.

136

4 Phase Quantification by Different Techniques Slag-blended cement paste

Ordinary Portland cement paste

Original image

T = 0.3

T = 0.5

T = 0.7

T = 0.9

T = 1.1

Edge

Inner region of phase

Fig. 4.37 Phase identification of ordinary Portland cement paste and slag-blended cement paste based on edge detection. The size of CT images of the slag-blended cement paste is 444 × 444 μm, and that of the ordinary Portland cement paste is 1001 × 1001 μm. T is the limit value for gray gradient G(x, y), a pixel is identified as edge of an object if G(x, y) > T (Wu et al., 2020)

4.7.4 3D Reconstruction of Microstructure Three-dimensional reconstruction of the CT images provided a powerful tool for monitoring and simulating changes in the microstructure of the cement pastes. Portland Cement Paste For Portland cement paste, a square of 337 × 337 pixels (length around 1001 μm) in the middle of the image is cut out, and 337 continuous scanning images along the vertical direction are selected for reconstruction. Thus, a cube model of 337 × 337 × 337 voxels is established. Table 4.9 presents the 3D microstructure models of the cement pastes with w/c ratio of 0.3(3O) and 0.5(5O) from 1 to 28 days of aging. The clinker is colored white, the hydration product is colored gray, and the pore is colored black. The hydration products form the skeleton of the model, and the white clinker and the black pores are distributed within them. At the age of 1 day, a large number of clinkers were observed in the model due to the low degree of hydration. The shape of the clinker particle is an irregular polygon, and some particles are elongated, ranging from tens to hundreds of microns. The number of pores in the 5O paste at 1 day is greater than that in the 3O paste at the same age because of the higher w/c ratio. At 7 days, the 3D model shows significant changes, with the clinker volume fraction

4.7 Phase Quantification by X-Ray Computed Tomography

137

Table 4.9 3D reconstruction of CT microstructure images for Portland and slag-blended cement pastes at 1, 7, and 28 days by threshold method (Wu et al., 2020) Age

3O

5O

3O50S

5O50S

1d

7d

28d

reducing dramatically. The volume fraction of the clinker in the 5O paste at 7 days is less than that in the 3O paste, indicating a higher degree of hydration in the 5O paste. At 28 days, the number of clinkers and pores in both pastes is further reduced. Slag-Blended Cement Paste For slag-blended cement paste, a cube model of 300 × 300 × 300 voxels is established. As shown in Table 4.9, the fractions of both the clinker and the slag particles in the slag-blended paste gradually decrease as age increased, while the fraction of the hydration products gradually increases. In addition, there are many pores at 1 day in the slag-blended paste, much more than that in the Portland cement paste at the same age. This shows that the addition of slag results in a decrease in overall hydration at 1 day compared with the OPC system. The degree of hydration in 5O50S is higher than that in 3O50S at the same age because the water supply for the 5O50S paste is relatively sufficient. It is worth noting that the clinker particles in the slagblended cement pastes are usually covered with a thin layer of slag, indicating that the boundary between the clinker and the slag particles is difficult to be distinguished by the threshold segmentation in CT images. However, the overall development of hydration is still reflected in the 3D microstructure.

138

4 Phase Quantification by Different Techniques

4.7.5 Calculation of Degree of Hydration Based on CT Images The volume fraction of each phase can be determined by counting the voxel number of that phase in the reconstructed microstructure volume. According to changes in the voxel number of the unreacted particles during the hardening process, quantitative calculation of the degree of hydration (DOH) can be carried out. The DOH of the clinker is calculated as Eq. (4.1), in which the volume fraction of the unreacted clinker and slag is given by Eqs. (4.15) and (4.16), respectively. Vclinker (t) = Vslag (t) =

Sclinker (t) Stotal

(4.15)

Sslag (t) Stotal

(4.16)

where Sclinker (t) and Sslag (t) are the number of voxels of clinker and slag at age t, respectively. Stotal is the number of voxels of the entire model, Stotal =3373 =38272753 for the ordinary Portland cement paste, Stotal =3003 =27000000 for the slag-blended cement paste in this case. The CT-based DOH of both OPC and the slag-blended pastes is shown in Table 4.10. It is seen that the overall DOH of the OPC system is greater than that of the slagblended system at the same age and for the same water-to-binder ratio, but the DOH of the slag-blended system is catching up to that of the OPC system at later age, the higher the w/b ratio, the faster the speed of the catching up. In the slag-blended system, the DOH of the clinker particle is greater than that of the slag particle. This can probably be attributed to the enhanced cement hydration in the presence of the slag (Escalante-Garcia & Sharp, 1998). Meanwhile, since the slag reacts more slowly than the cement, more water is available for reaction with the cement in the early stages of hydration (Escalante-Garcia & Sharp, 2001). The DOH of both particles is greater in high w/b ratio = 0.5 paste than in low w/b ratio = 0.3 paste. With a reduction of the replacement level of slag, the alkaline activating effect from the clinker would be greater, such that the reactivity of slag will increase (Wang et al., 2010). It should be noted that the existence of slag in the slag-blended system reduces the hydration degree of the clinker compared to that in the OPC system for both high and low w/b ratios, which agrees with Pane and Hansen (2005). The lower the water/binder ratio, the more reduction of the degree of hydration of the clinker. The degree of hydration of clinker in the 3O50S system is 61.7% of that in the 3O system at the age of 28 days, it is 80% for the high w/b ratio of 0.5. This is because in slagblended system water is adsorbed on the surface of the slag (Zhang & Wei, 2015), less water is left for the clinker hydration, which leads to lower degree of hydration of clinker. In addition, the calcium-rich phase in the slag reacts (Papadakis et al., 1992) with the water in the alkaline environment, which also consumes water and takes

0.149

0.094

0.085

7

28

0.092

0.173

0.156

1

0.214

0.160

0.150

7

28

0.183

0.245

0.211

1

–

0.155

0.119

7

28

–

–

0.190

1

V unhydrated

0.165

0.252

0.310

0.324

0.389

0.430

0.119

0.155

0.190

0.195

0.227

0.290

α clinker

0.555

0.508

0.184

0.404

0.365

0.161

0.695

0.603

0.513

0.655

0.560

0.438

α slag

0.559

0.285

0.170

0.334

0.221

0.107

–

–

–

–

–

–

α CT1

0.557

0.397

0.177

0.369

0.293

0.134

–

–

–

–

–

–

α CT2

0.587

0.370

0.224

0.384

0.260

0.182

0.695

0.603

0.513

0.655

0.560

0.438

0.517

0.396

0.233

0.387

0.257

0.155

0.761

0.692

0.418

0.707

0.628

0.490

α TGA

TGA

7.70

0.14

24.10

4.60

14.20

13.40

–

–

–

–

–

–

| | / |αCT1 − αTGA αTGA |

13.58

6.69

3.68

0.80

1.29

17.65

8.70

12.90

22.80

7.36

10.80

10.70

| | / |αCT2 − αTGA αTGA | (%)

Note α CT1 represents the overall DOH of slag-blended system which can be calculated according to the mass-weighted average of the two components as α CT1 = (αclinker + α slag )/2; α CT2 represents the DOH of the mixture; α clinker represents the DOH of clinker; α slag represents the DOH of slag

5O50S

3O50S

5O

–

–

0.227

0.195

7

V slag

28

V clinker

–

1

3O

CT

0.290

Age (days)

Mixtures

Table 4.10 Volume fractions and degree of hydration based on CT and TGA (Wu et al., 2020)

4.7 Phase Quantification by X-Ray Computed Tomography 139

140

4 Phase Quantification by Different Techniques

up the space of the hydration products. Therefore, in a composite system with large amounts of slag, the DOH of clinker decreases. Wang et al. (2010) developed a kinetic hydration model for cement–slag blends. The simulation and the experiment showed that with more slag replacement in the slag-blended cement paste, the amount of the chemically bound water is decreased, which indicates that the DOH of the system is decreased.

4.8 Summary This chapter introduces six techniques for phase quantification in terms of the data processing method and the phase identification method. The SPM can map the local variation of the mechanical properties without causing plastic deformation to the sample at nanoscale. Also, this technology makes up for some inherent disadvantages of the NI technique. For example, it is difficult by using the NI to characterize the more delicate interfacial transition zone (ITZ) between the cement grain and the CSH gel due to the relatively large volume of the contact between the nanoindenter and the sample, and some mechanical information at smaller scale cannot be acquired by NI. The mechanical properties of the CSH obtained through the grid NI over large areas of hardened pastes agree fairly well with those obtained from the discrete NI on single phases. The mechanical properties of each phase remain stable in pastes with different w/cm ratio and the slag contents. No single calcium hydroxide (CH) phase is identified by the NI, indicating the limitation of NI technique in distinguishing phases with similar mechanical properties. Similarly, the ITZ phase cannot be clearly identified by the SEM images where the ITZ phase is mixed with the hydrates phase, leading to a thicker hydrates phase by the SEM image. Nanoscratch technique can perform continuous measurement under a constant vertical loading rate to test the deformation and the mechanical properties of all phases in cementitious materials including the CH phase and the ITZ phase, which makes up for the limitation of NI and SEM technique. Both the slope characteristics of the deformation curve and the fracture toughness distribution can be used to identify different phases. A new method for quantifying the thickness of ITZ is proposed based on the fracture toughness distribution, and the ITZ thickness and the microfracture toughness are minorly affected by the testing parameters such as the transverse scratching speed and the vertical loading rate. The key to identify phases by the X-ray CT and the BSE techniques is segmenting the phases. It is found that the edge detection is not suitable for segmenting cementitious materials, as the boundaries between different phases cannot be clearly distinguished in hardened paste. The gray level threshold method, although subjective process and relies heavily on operator’s skill and experience, is a reliable method for segmenting unreacted particles and pores.

References

141

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Chapter 5

Porosity Characterization and Permeability Prediction of Cementitious Materials

Abstract In view of the difficulty and limitation of the direct experimental measurement of water permeability, particularly for high-strength modern dense concrete, this chapter summarizes the existing techniques and methods to measure the pore parameters and to predict the intrinsic permeability of cementitious materials, including the MIP, BSE, X-ray CT scanning pore measuring techniques, and the Katz-Thompson, the general effective media (GEM), and the Navier–Stokes method for permeability prediction. The predicted intrinsic permeabilities based on the measured microstructure parameters are compared to the experimental results, and the applicability of different techniques and prediction methods are discussed. It is noted that the computed permeability is more suitable as a comparison tool for selecting mixture under the conditions of using the same technique and calculating method. The content of this chapter can provide an insight into the feasibility of the measuring techniques and the predicting method for quantifying the permeability of cementitious materials. Keywords Computed permeability · General Effective Media (GEM) theory · Katz-Thompson (K-T) equation · Navier-Stokes method · Pore characteristics

5.1 Introduction Cementitious material is a typical porous material with the internal pore structure controlling its mechanical properties and durability. Permeability is normally used as one of the key indicators for the durability of concrete (Mehta & Monteriro, 2006; Soroushian & Elzafraney, 2004; Wei et al., 2019; Wu et al., 2020), which is directly affected by the pore characteristics, such as the total porosity, the pore size distribution, and the critical pore diameter of the paste phase in concrete. Laboratory-based tests have been conducted for many years to measure the permeability of concrete and paste. It is widely recognized that the direct experimental measurement of water permeability usually encounters practical difficulties for modern dense concrete due to its low permeability and complicated testing conditions. Given the limitations of experimental tests, the prediction of permeability of cementitious materials is quite helpful to assess the durability of concrete efficiently, either with analytical or numerical models. Researchers developed analytical models for permeability prediction © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_5

145

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5 Porosity Characterization and Permeability Prediction …

from the microstructure information obtained by the MIP or the BSE images. These models are established based on the mechanism that the permeability is controlled by pore size distribution and the percolation of the pore network. In this chapter, the pore characteristics of pastes are assessed by using different techniques, the permeability is then predicted based on the measured pore characteristics, and finally, the predicted permeability is compared to the experimental results. The applicability of different techniques and the prediction methods are discussed. The results of this study can provide an insight into the feasibility of different measuring techniques and the predicting method for quantifying the permeability of cementitious materials.

5.2 Pore Characteristics Quantified from Different Techniques Pore structures can be measured directly through experimental techniques or predicted indirectly using analytical methods (Kim & Choi, 2020; Sun et al., 2020) or numerical models (Bishnoi & Scrivener, 2009; Pignat et al., 2005; Ye, 2003). The X-ray CT, the BSE, and the MIP technique are all commonly used methods to measure the pore structure in cementitious materials. However, the measured pore size distribution might be different even for the same sample due to the different resolutions of the CT and BSE image as well as the maximum intrusion pressure that can be applied in MIP test. Therefore, these techniques should be combined to measure the pores with different size distributions. Henry et al. (2014) pointed out that MIP was recommended to integrate with the X-ray CT to measure the pore size across various scales. Das et al. (2015) also proved that the combination of MIP and X-ray synchrotron tomography facilitates a complete characterization of the pore structure. The following sections will discuss the measured pore characteristics by utilizing different techniques.

5.2.1 Pore Characteristics from MIP Technique Wei et al. (2020) investigated the porosity and the pore size distribution of the frozen cement paste with different water-to-cement ratios at the age of 40 days by using MIP. Cement pastes with w/c ratios of 0.3 and 0.5 are denoted as 3P and 5P, respectively. “1d-FT” and “7d-FT” denote samples subject to freeze–thaw cycles at the curing ages of 1 and 7 days for 12 days, respectively. “Control” denotes the sample that was not subject to freeze–thaw cycles and was under the sealed curing conditions. By differentiating the intruded volume, the characteristics of pore size distribution can be seen in Fig. 5.1. For 3P mixture, the pore size distributions of the frozen and the unfrozen samples are similar, with one single peak within the pore size range of

5.2 Pore Characteristics Quantified from Different Techniques 3P-1d-FT 3P-7d-FT 3P-control

(a) 0.4

0.2

0.3

102 101 Pore size (nm) 3P-1d-FT 3P-7d-FT 3P-control

dV/dlogD (mL/g)

0.2

(d) 0.3

Lc range 30 ~ 40nm

0.1

100

101 102 Pore size (nm)

0

103

103

dV/dlogD (mL/g)

100

(c) 0.3

0

0.2 0.1

0.1 0

5P-1d-FT 5P-7d-FT 5P-control

(b) 0.4

Porosity

Porosity

0.3

147

0.2 0.1 0

100

101 102 Pore size (nm)

103

5P-1d-FT 5P-7d-FT Lc (5P-1d-FT): 5P-control 800nm Lc (5P-7d-FT): 90nm Lc (5P-control): 60nm

100

101 102 Pore size (nm)

103

Fig. 5.1 Measured pore size distribution by MIP and determination of critical pore size (L c ) in 3P and 5P pastes at the age of 40 days after sealed curing (Control) and subject to freeze–thaw cycles (FT) at the age of 1 and 7 days (Wei et al., 2020)

30–40 nm. This indicates that frozen does not alter the pore size distribution for the low w/c ratio paste 3P. The pore size corresponding to the peak of curves is known as the critical pore diameter (L c ), which relates to the pore diameter at the inflection point on a cumulative volume versus pressure diagram measured from the MIP test (Cui & Cahyadi, 2001). For the high w/c ratio paste 5P, multiple peaks were found for both pastes with and without freeze–thaw cycles because of the greater pore size distribution and porosity in such high w/c ratio paste. As shown in Fig. 5.1d, the paste frozen at the age of 1 day shows the most significant large size pore with the L c ranging between 600–800 nm. It is 90 nm for the paste frozen at the age of 7 days and 60 nm for the control sample. The pore size distribution measured at the age of 40 days by MIP for the 3P and 5P pastes is shown in Table 5.1. It is clear that the MIP measures the pores with sizes greater than 3 nm and less than 1 µm, which includes both gel pore (5 nm). Jehng and Sprague, (1996) found through NMR measurement that the gel pore is not affected by the freeze–thaw cycles, because of the very low freezing point of the gel pore water. For 3P paste with a low w/c ratio = 0.3, the porosity of the sample froze at the age of 1 day is similar to that of the sample frozen at the age of 7 days. And their porosities are slightly greater than those of the control sample. The above findings suggest that the low w/c ratio paste is not sensitive to the early-age freeze–thaw cycling, and the porosity at a later age is not affected very much by the early-age frozen compared to the control sample.

2.00

26.00

1 µm–30 µm 41%

> 10 µm

BSE

X-ray CT

34%

0.68

3 nm–1 µm

5P-1d-FT

φ (%) L c (µm)

37%

MIP

Technique Range of pore size

16.50 µm 13.00 µm

10

> 10 µm

X-ray CT

24

40.30 nm

19.50

1.60

0.55 0.47

0.57

0.85 27

33

31

1.65 µm

32.40 nm

L max 0.74 0.47

1.85 µm

50.35 nm

22

45.30 nm

0.51

0.83

18

27

23

1.65 µm

36.35 nm

L max

0.47

0.74

S(L max )

1.65 µm

31.40 nm

S(L max ) 0.44

0.83 12.50 µm 12.50 µm 0.16

1.65 µm

32.40 nm

L max

16.50 µm 13.00 µm 0.21

2.15 µm 5P-control

9

20

S(L max ) φ (%) L c

19.50 µm 19.50 µm 0.47

2.35 µm

62.50 nm

L max

3P-control S(L max ) φ (%) L c

16.50 µm 13.00 µm 0.21

2.15 µm

5P-7d-FT

9

21

L max (µm) S(L max ) φ (%) L c

0.19

0.51

0.64

32.40 nm 1.60 µm

40.30 nm

2.00 µm

22

3 nm–1 µm

1 µm–30 µm 24

MIP

3P-7d-FT S(L max ) φ (%) L c

L max

3P-1d-FT

φ (%) L c

BSE

Technique Range of pore size

Table 5.1 Range of pore size, porosity (φ),critical pore size (L c ), pore size at which the electrical conductivity reaches maximum (L max ), and the fraction (S(L max )) of connected pores with size ≥ L max , measured by the three techniques (MIP, BSE, and X-ray CT) for 3P and 5P pastes under the conditions of sealed curing and subject to freeze–thaw (FT) cycles at the age of 1 (1d-FT) and 7 (7d-FT) days (Wei et al., 2020)

148 5 Porosity Characterization and Permeability Prediction …

5.2 Pore Characteristics Quantified from Different Techniques

149

Due to the existence of the “ink-bottle pores” (described in Sect. 3.8.1) in cement paste, the mercury needs greater intrusion pressure to pass through the throat pores to reach the ink-bottle pores, and thus, the pore size distribution can be mistakenly quantified. To eliminate the “ink-bottle effect”, Zhou et al. (2010) proposed a pressurization–depressurization cycling mercury intrusion porosimetry (PDC-MIP) method, as illustrated in Fig. 5.2a. Within each PDC, the pressurization and the depressurization are recorded as Pnin and Pnex , and the mercury intrusion and extrusion volumes are recorded as Vnin and Vnex . According to Zhou et al. (2010), Vnex equals to the volume of the throat pores with a diameter of Dn , and Vnin − Vnex equals to the volume of the corresponding ink-bottle pores. Figure 5.2b illustrates a comparison of the pore size distributions measured by the PDC-MIP, the standard MIP, and the BSE image analysis. The total porosities measured by PDC-MIP and standard MIP are higher than that measured by BSE image analysis. PDC-MIP and standard MIP give total porosities from 41 to 22% at the ages of 1–28 days, while BSE image analysis gives total porosities from 33 to 16%. Due to the resolution limit of BSE images, the pores with a diameter smaller than 0.25 µm are not detected. Regardless of the curing age, the result of PDC-MIP shows a critical pore diameter larger than that of the standard MIP and smaller than

3 days, PDC-MIP

3 days, BSE image analysis

3 days, standard MIP

dV/dlogD (ml/ml)

(b)

Cumulative pore volume (ml/ml)

(a)

28 days, PDC-MIP 28 days, BSE image analysis

28 days, standard MIP

Pore diameter (μm)

3 days, PDC-MIP 3 days, standard MIP

Pore diameter (μm)

dV/dlogD (ml/ml)

Cumulative pore volume (ml/ml)

Pore diameter (μm)

3 days, BSE image analysis

28 days, BSE image analysis 28 days, PDC-MIP 28 days, standard MIP

Pore diameter (μm)

Fig. 5.2 a Illustration of the PDC-MIP testing sequence with pressurization–depressurization cycles, b pore size distributions of Portland cement pastes with w/c ratio of 0.4 at the ages of 3 and 28 days obtained by PDC-MIP, standard MIP, and BSE image analysis (Zhou et al., 2010)

150

5 Porosity Characterization and Permeability Prediction …

1.2

0.8 0.6 0.4 0.2 0 1

Incremental pore volume (mL/100g)

Incremental pore volume (mL/100g)

Plain-7d Plain-28d Plain-91d

1

1.2

10 100 Pore size (nm) F50-7d F50-28d F50-91d

1 0.8 0.6 0.4 0.2 0 1

10 100 Pore size (nm)

1.2

1000

F35-7d F35-28d F35-91d

1 0.8 0.6 0.4 0.2 0

1000

1 Incremental pore volume (mL/100g)

Incremental pore volume (mL/100g)

that of the BSE image analysis. For example, for the 3-day sample, the standard MIP measurement allocates most pores below 1 µm with the critical pore size of 0.32 µm, and the PDC-MIP measurement allocates most pores between 0.1 µm and 50 µm with the critical pore size of 1.2 µm, and BSE image analysis allocates most pores bigger than 0.25 µm with the critical pore size of 4.5 µm. Park and Choi (2021) measured the pore size distribution of the fly ash (FA) cement paste by using MIP. Figure 5.3 shows the incremental pore volume. All samples with w/c ratio of 0.3 were cured for 7, 28, and 91 days. Plain sample does not exhibit a significant variation in the pore size distribution with age when compared with the FA cement paste. The main pore sizes of plain sample are in the range of 30–40 nm regardless of the curing age. In the case of the FA cement paste, as the FA replacement ratio increases, the peak appears at a larger pore. This trend is more distinct at the relatively early age (7 days) because the pozzolanic reaction of FA has not yet started. Furthermore, as the curing age increases, the diameter of the pore corresponding to the peak decreases. This implies that the pozzolanic activity of FA increased with curing age. Figure 5.4 shows the pore volume distributions of the samples in Fig. 5.3. The measured pores are classified into three size ranges of the meso pores (100 nm). The volume ratio of the meso pores is the highest for all specimens. For F50, the volume ratio of the middle capillary pores and the meso pores decreases as the age increases. For F65, the volume ratio of the large capillary pores (>100 nm) is approximately

1.2

10 100 Pore size (nm)

1000

F65-7d F65-28d F65-91d

1 0.8 0.6 0.4 0.2 0 1

10 100 Pore size (nm)

1000

Fig. 5.3 Incremental pore volumes of the OPC paste (Plain) and the fly ash cement paste (F) with w/c ratio of 0.3 at the age of 7, 28, and 91 days; F35–65 represents the 35–65% of the mass of cement is replaced by fly ash (Park & Choi, 2021)

5.2 Pore Characteristics Quantified from Different Techniques

151

Fig. 5.4 Volume and fraction of meso pore (100 nm) measured by MIP for w/c ratio = 0.3 OPC paste with 0%, 35%, 50%, and 65% cement replaced by fly ash (Park & Choi, 2021)

50% at 7 days. However, at 28 days, most of the large capillary pores are changed to middle capillary pores and meso pores, and the volume fraction of the meso pores increases to 60%. This suggests that the pore structure became more complex as the degree of reaction of FA increased with age owing to the formation of additional products. Therefore, it can be seen that the pores of the FA cement paste are improved over the long term by the pozzolanic reaction of FA. Liu et al. (2022) investigated the effect of recycled concrete powder (RCP) and nano silica (NS) on the pore structure of cement paste by using MIP. The results are shown in Fig. 5.5, and the pure cement paste is denoted as CON. Regarding the different effects of pore size on cement paste performance, the pore in cement paste is classified as harmless pore (200 nm) (Zhang & Li, 2011). It can be seen from Fig. 5.5 that the introduction of RCP made the proportion of the harmless pores and the less-harmful pores decrease significantly, while the more-harmful pores increased significantly. With the addition of NS, the volume of the harmless pores and the less-harmful pores increased, and the volume of the harmful pores and the more-harmful pores decreased. The increase in the proportion of the harmless pores contributes to strength improvement.

5.2.2 Pore Characteristics from BSE Technique The MIP cannot measure the volume of pores larger than 1 µm in size. Since Lange et al. (1994), BSE image analysis has been a popular supplemental technique for quantifying the fraction of pores in the hardened paste. The BSE images of the 3P

152

5 Porosity Characterization and Permeability Prediction …

Fig. 5.5 The volume distribution of harmless pore (200 nm) of CON, RCP, and RCP-2%NS sample (Liu et al., 2022)

and 5P samples, measured at the age of 40 days in Wei et al. (2019), are shown in Fig. 5.6. The microstructure of cement paste is shown to be considerably impacted by the early-age freeze–thaw cycles, especially for pastes with high w/c ratios (i.e., 5P sample). For the 5P paste subject to freeze–thaw cycles at the age of 1 day and a duration lasting for 12 days, the severely damaged microstructure of the hydration products is observed even after sealed curing at normal temperature for 27 days. The damage caused by the early-age freeze–thaw cycles will not be healed by the later-age curing since there are significant voids surrounding the residual clinkers and no tightly compacted structure in the 5P-1d-FT sample. The 5P-7d-FT paste also exhibits this phenomenon, but in a milder form. Compared to the 5P sample, the situation with the 3P sample is substantially better because the 3P-1d-FT sample shows no obvious damage.

3P-1d-FT

3P-7d-FT

3P-control

5P-1d-FT

5P-7d-FT

5P-control

Fig. 5.6 BSE images taken at the age of 40 days for 3P and 5P pastes under sealed curing (control) and subject to freeze–thaw cycles at the ages of 1 and 7 days (1d-FT and 7d-FT) (Wei et al., 2019)

5.2 Pore Characteristics Quantified from Different Techniques

(b)

(a)

153

(c)

50μm

50μm

(e) 2 Pore Frequency (%)

(d)

50μm

50μm HP

Clinker

1.5 1 0.5 0

0

50 100 150 200 250 Gray level

Fig. 5.7 a BSE image, b binary image after segmentation of pore, c binary image after segmentation of hydration products (HP), d binary image after segmentation of clinker; and e distribution of gray level of pore, HP, and clinker phases of 3P paste (Wei et al., 2019)

It is possible to calculate the proportions of the pore, hydration products (HP), and clinker phases by using gray level-based segmentation (Fig. 5.7). As seen in Fig. 5.7e, a histogram may be produced using a gray level that ranges from black (0) to white (255). While the boundary between the pore phase and the HP phase cannot be clearly seen and must instead be determined by the “overflow” method (Wong et al., 2006), which is covered in Sect. 4.2.2, the boundary between the HP phase and the clinker phase can be distinguished by an obvious trough. The area fraction in each segmentation step may be used to compute the proportion of the particular phase. The gray level distribution pattern for various pastes is an interesting finding from the BSE image segmentation. The 3P samples had a major HP distribution peak as well as two smaller peaks for porosity and clinker (Fig. 5.8). However, the 5P samples lack significant peaks, especially in the 5P-1d-FT sample, which only shows one peak. This phenomenon suggests that the contrast of the gray level between distinct phases in the 5P-1d-FT sample is not significant due to the broken microstructure and consequently the relatively uniform reflection of the signals during the BSE imaging process. The porosity assessed from the pictures in Fig. 5.8 is shown in Fig. 5.9. It has been discovered that the early freeze–thaw cycles harm the high w/c ratio paste’s microstructure to the point where subsequent sealed curing is unable to fully restore the pore structure by continuous hydration and the filling of the pores with hydration products. In contrast to the control sample, the porosity of the low w/c ratio paste (i.e., 3P sample) does not significantly alter after being subject to freeze–thaw cycles, even at extremely young ages. Based on the stereological principle, which holds that the volume fraction in 3D space may be represented by the area fraction on a 2D picture, the pore size

154

5 Porosity Characterization and Permeability Prediction … 3P-1d-FT

HP

Clinker

1

0

Frequency (%)

Clinker

0 0

5P-1d-FT

HP

1

50 100 150 200 250 Gray level Pore

Pore

2

Pore

HP

1

0 0

50 100 150 200 250 Gray level

Clinker

50 100 150 200 250 Gray level 5P-control

2 Clinker

HP

0 0

5P-7d-FT

2

Pore

1

50 100 150 200 250 Gray level Frequency (%)

0

2

3P-control

3P-7d-FT

2

Clinker

1

0

Frequency (%)

HP

Frequency (%)

Pore

Frequency (%)

Frequency (%)

2

Pore

HP

Clinker

1

0 0

50 100 150 200 250 Gray level

0

50 100 150 200 250 Gray level

Fig. 5.8 Variation of gray level distribution for 3P (w/c = 0.3) and 5P (w/c = 0.5) pastes under sealed curing and subject to freeze–thaw cycles at the ages of 1 and 7 days (based on Fig. 5.6) (Wei et al., 2019) 60

3P pastes

5P pastes

50 Porosity (%)

Fig. 5.9 Porosity of 3P and 5P pastes at the age of 40 days under the conditions of sealed curing and subject to freeze–thaw cycles at the ages of 1 and 7 days (calculated from Fig. 5.8)

41

40 30

33 24

27 21

20

20 10 0 1d-FT 7d-FT Control 1d-FT 7d-FT Control

distribution can also be estimated from the BSE image (Lange et al., 1994. The shape of the pore identified in the 2D image is equivalent to be round, and then the effective pore diameter (d) can be calculated from the area of the pore (S): √ d = 4S/π . Figure 5.10 shows the predicted pore size distribution based on the BSE images for several paste samples. The detectable pore size in BSE measurement relies on the predetermined resolution and the BSE image field size. The lower limit of the observable pore size for the 0.27 × 0.27 µm pixel size attained in this work is around 0.86 µm. Based on the BSE approach, it is shown that pores with sizes ranging from 1 to 30 µm are measured. Due to the BSE image analysis’s inability to identify pores smaller than 0.86 µm in this research, as previously mentioned, there are essentially no pores with a size less than 1 µm (see the enlarged figure in Fig. 5.10). Similar to the MIP results, the early-age freeze–thaw cycles have little impact on the pore size

5.2 Pore Characteristics Quantified from Different Techniques

155

12% 6%

3P-1d-FT

4%

3P-7d-FT

Porosity

Porosity

10% 8%

3P-control

2%

5P-7d-FT

0%

6%

5P-7d-FT

0~1 4~5 2~3 Pore diameter (µm)

4%

5P-control

2% 0% 0~3

6~9

12~15 18~21 Pore diameter (µm)

24~27

Fig. 5.10 Pore size distribution quantified by stereology method based on BSE for 3P and 5P pastes (Wei et al., 2019)

of the paste with a w/c ratio of 0.3 (3P) sample. In contrast, the 5P sample exhibits increased porosity within the pore size range of 1–20 µm than the control sample. To efficiently generate a large set of high-resolution 3D pore structures from 2D BSE images, Liu (2020) and Liu et al. (2021) developed a parameterized (digital concrete) model. In this model, a deep learning system was adopted to create the 3D structure by using more than 100 images of the decomposed 2D probabilistic map. By utilizing the ImageJ software, the virtual MIP test is carried out on the recreated 3D pore structures and compared with the real MIP test. The results of the simulation closely mirror those of the MIP test. However, the isolated microscopic pores created during the rebuilding process that are inaccessible in the virtual MIP test result in somewhat lower overall porosity than that determined by the actual MIP. Figure 5.11 depicts the 2D and 3D pore structures of a cement sample after it has been hydrated for one day and stopped by immersion in ethanol for seven days. The 2D images are used to construct the 3D microstructure; hence, the pixel/voxel sizes and the segmentation of the two images into solid and pore regions are the same. In accordance with the pore size, the pores have various colors. It is possible to see how the pore structures twist in 3D space, which is not possible with the original BSE characterization methods. Similarly, Lin et al. (2022) segmented the BSE images of the graphene oxidesilica (GOS)-reinforced cement paste. The content of the GOS is 0.1% of the mass of the cement. After image processing, the pore area was segmented and colored based √ on the equivalent diameters of the pores (D) which can be calculated by D = 2 × area/π . The segmented SEM images of the cement paste samples are shown in Fig. 5.12. Control represents the cement paste, GOS-10 and −30 represent the GOS with the thickness of 10 and 30 nm, respectively. The unhydrated and the hydration products are colored black as solid. The control sample contains a number of big pore areas with equivalent diameters of 30 µm (colored green), and the rest of the sample is predominantly brilliant yellow (10–20 µm) in color. The GOS-10 and GOS-30

156 (a)

5 Porosity Characterization and Permeability Prediction … (b)

(c)

Fig. 5.11 a Decomposed 2D probabilistic map, b the 2D pore structure from the BSE image, and c the reconstructed 3D pore structure images of the 1-day hydrated cement sample (Liu et al., 2021)

Fig. 5.12 BSE images of OPC, GOS-10, and GOS-30 samples at the age of 7 days (Top) and corresponding color map based on equivalent diameter of pores (Bottom) (Lin et al., 2022)

samples, in contrast, exhibit a significantly darker overall color (less than 10 µm), indicating the pores’ overall size is significantly smaller. Further evidence that the addition of nanosheets densifies the microstructure of cement paste is provided by the complete absence of big pores (larger than 30 µm).

5.2.3 Pore Characteristics from X-Ray CT Technique Wei et al. (2020) assessed the pore size distribution for paste samples from the scanned 2D images by using X-ray CT, as shown in Fig. 5.13 and Table 5.1. The test with 1284 projection views for each sample was done with a power setting of 100 kV and 50 µA. The scans were done every 2.97 µm along the height of the sample. The resolution was 2.97 µm. It can be seen that the majority of pores are between 20 and 60 µm in size. It is important to note that the X-ray CT used in this study can only detect pores that are at least 3 µm in size. From Fig. 5.10, it looks like the best size range for pores that the BSE can see is below 20 µm (the pixel size of the BSE image is 0.27 µm in Wei et al., 2020), while the X-ray CT can detect pores that are

5.2 Pore Characteristics Quantified from Different Techniques

157

10% 3P-1d-FT

Porosity

8%

3P-7d-FT

6%

3P-control

4% 2% 0% 0~20

40~60

80~100

120~140

160~180

200~

Pore diameter (µm)

Fig. 5.13 Pore size distribution measured by CT for 3P and 5P pastes under sealed curing and subject to freeze–thaw cycles at the age of 1 and 7 days (Wei et al., 2019)

less than 200 µm in size (Fig. 5.13). For cement pastes with a high w/c ratio that are exposed to freeze–thaw cycles at a young age, X-ray CT is recommended for pore size quantification because of its capability of capturing the large pores. Instead of the reconstructed 3D microstructure of paste, the pore size distribution depicted in Fig. 5.13 is based on the analysis of the 2D photographs. This is because the commercial software (Mimics) quantifies the properties of pores from the reconstructed 3D microstructure, where the pores are assumed to be spheres filling the pore space, which typically results in an unrealistic pore size distribution compared to the real pore size distribution. Through the use of a digital model, Zheng et al. (2015) also discovered that the Mimics software’s accuracy was decreased while characterizing intricate multipore systems. To correctly measure the pore size distribution in cementitious materials or other porous media, the applicability of some commercial software has to be enhanced. Sun et al. (2014) employed the “overflow” method (Wong et al., 2006), which is described in Sect. 4.2.2, to segment pore and other phases from the 2D image for reconstructing the 3D microstructure of the cement paste. The Transmission X-ray Microscope (TXM) test was conducted to obtain the 440 images each with size of 512 pixels by 512 pixels under the 9 keV X-rays. However, the “overflow” method makes it challenging to properly separate the pores due to the poor quality of TXM images. Thus, two different pixel threshold intensity values were manually selected from the discontinuous histogram (Fig. 5.14a) and correspondingly the digital Sample 1 and Sample 2 were constructed. The volume fraction of the pore pixels was then quantified as 23.17% and 12.19% for the two digital samples, respectively (Fig. 5.14b). For subsequent permeability calculations, the TXM pictures would be transformed into binary image data, where the pore and solid phases would be assigned black and white color, respectively. The 3D microstructures shown in Fig. 5.14c were created by stacking the 2D binary pictures, where the pore pixels are white. Xue et al. (2019) utilized MIP and multiscale X-ray CT (nanoVoxel-3502E) to investigate the microstructure features of cement mortar exposed to different temperatures. They compared the effects of different voxel sizes on the pore size distribution (PSD). The voxel sizes used for reconstructing the CT images are selected as 1.5µm3

5 Porosity Characterization and Permeability Prediction … (a)

Threshold intensity factor: 20~41 (b)

Porosity: 12.19% (c)

Pixel count percentage

158

Pixel intensit

Threshold intensity factor: 41~63

Porosity: 23.17% Sample 2

Sample 1

Fig. 5.14 The computational model setup: a selected model with pixel size 200 × 200 pixels and pixel count histogram of image intensity, b generated binary image after applying threshold intensity, and c 3D digital pore structure by stacking slices with identical image processing (Sun et al., 2014)

and 4µm3 , and the resolution of each image is 2048 × 2048. The samples that were heated to 400, 600, and 800 °C are each termed as M400, M600, and M800. In Fig. 5.15, the PSD curves measured by MIP (blue lines) and those quantified by X-ray CT images with a voxel size of 4.0 µm3 (red lines) and 1.5 µm3 (green lines) are plotted together (green lines). It can be seen that the PSD curves determined by the X-ray CT images with smaller voxel sizes (1.5 µm3 ) match the right peaks of the PSD curves measured by MIP. This indicates that the smaller voxel size should be used for pore size characterization by the X-ray CT to improve the accuracy of the test. Meanwhile, due to the limitation of the CT image resolution, it is impossible to characterize the pore with a size less than the voxel size. However, various testing methods can be used combined to quantify the global pore structure of cement-based materials.

5.2.4 Comparison of Pore Size Distribution Quantified by Different Techniques Three techniques including MIP, BSE, and X-ray CT have been adopted by Wei et al. (2020) to characterize the porosity and the pore size distribution of the paste samples.

0.04

6 4

0.02

2 0.00 0 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm)

Voxel counts (×106)

dV/dlogD

Voxel counts (×106)

0.06

6

0.04

3

0.02

0.10

1.5 µm3/voxel

0.10

12

0.08

9

0.06

6

0.04

3

0.02

0.00 0 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm)

Voxel counts (×106)

dV/dlogD

Voxel counts (×106)

X-ray CT MIP

0.08

9

15

(c) 16

4.0 µm3/voxel 0.12 14 0.10 12 10 0.08 8 0.06 6 0.04 4 0.02 2 0.00 0 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm)

12

0 0.00 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm)

(b) 28

4.0 µm3/voxel 0.10 24 0.08 20 16 0.06 12 0.04 8 0.02 4 0 0.00 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm)

0.12 dV/dlogD (mL/g)

8

18 1.5 µm3/voxel 15

dV/dlogD

0.06

159

21 1.5 µm3/voxel 0.12 18 0.10 15 0.08 12 0.06 9 0.04 6 0.02 3 0.00 0 0.001 0.01 0.1 1 10 100 1000 Pore diameter (µm) X-ray CT MIP

dV/dlogD

12 4.0 µm3/voxel 10

Voxel counts (×106)

Voxel counts (×106)

(a)

dV/dlogD (mL/g)

5.2 Pore Characteristics Quantified from Different Techniques

Fig. 5.15 Comparison of the pore diameter measured by MIP and X-ray CT images with different voxel sizes for a M400, b M600, c M800 (Xue et al., 2019)

The porosity as well as the critical pore diameter captured by different techniques are different, though the same sample was measured because the pore size range that each technique can detect is different. By plotting the pore size distribution measured on different samples by the three techniques, the measuring range of each technique can be seen in Fig. 5.16. Generally, the MIP captures pores with sizes less than 1 µm, the BSE captures pores with sizes ranging from 1 to 30 µm, and the X-ray CT captures pores with sizes greater than 10 µm. There is an overlap of the measured pore size by BSE and X-ray CT if the pore size is within the range of 10–30 µm. This overlapping might be less or greater depending on the resolution of the BSE and the X-ray CT. It is seen that the BSE and the X-ray CT techniques can work as complementary methods to the MIP method to measure pores with sizes greater than 1 µm. The lower limit of the pore size that can be detected by the BSE and the X-ray CT depends on

160

5 Porosity Characterization and Permeability Prediction …

30µm captured captured by captured by by CT CT and BSE BSE

30% 20% 10% 0% 1

10

100

1000 Pore diameter (nm)

10000

100000

Fig. 5.16 Cumulative distribution of the pore volume fraction measured by MIP, BSE, and CT for 3P and 5P pastes under sealed curing and subject to freeze–thaw cycles at the age of 1 and 7 days (Wei et al., 2020)

the resolution of the technique, normally X-ray CT pictures pore with sizes slightly greater than that from BSE image analysis. It should be noted that adding together each porosity determined by a single approach will not yield the overall porosity of a sample. Instead of determining the total porosity, the objective of employing various techniques to quantify porosity is to assess the porosity within the corresponding pore size range.

5.3 Calculated Permeability Based on Katz-Thompson Equation The Katz-Thompson equation calculates the permeability based on the critical pore diameter of concrete (Katz & Thompson, 1987). It is generally believed that the KatzThompson equation works better with cementitious materials that have a higher w/c ratio and with interconnected capillary pore networks than it does for those with large amount of gel pores (Christensen et al., 1996; Cui & Cahyadi, 2001; Halamickova et al., 1995).

5.3.1 Katz-Thompson Equation Theory for Permeability Katz and Thompson used the percolation concept in terms of conductivity to create an equation connecting the permeability of rock to its microstructural characteristics (Katz & Thompson, 1986 and 1987). The connectivity of the pore (denoted by σ/σ0 , also known as the conductivity formation factor) and the pertinent length scale (L c ) together indicate the intrinsic permeability (k in m2 ):

5.3 Calculated Permeability Based on Katz-Thompson Equation

k=

1 σ 2 L 226 σ0 c

161

(5.1)

where σ is the electrical conductivity of the saturated porous materials; σ0 is the electrical conductivity of the solution in the pores; and L c is the critical pore diameter, which corresponds to the pore diameter at the inflection point on a cumulative volume versus pressure diagram measured from MIP (Cui & Cahyadi, 2001). When the pore diameter exceeds L c , a percolated path can develop across the sample. The finer the pore structure, the smaller the critical pore size. The intrinsic permeability of the sedimentary rock was initially predicted using this equation based on MIP data. It was experimentally demonstrated by measuring the electrical resistance of the sample that the threshold for the formation of a continuous pathway is located at the inflection point of the rapidly rising portion on the cumulative intrusion curve. According to an equation provided by Katz and Thompson (1986 and 1987), the electrical conductivity formation factor may be estimated by using the MIP data as follows: σ Le = max φ S(L emax ) σ0 Lc

(5.2)

where L emax is the electrical conductivity characteristic dimension that produces maximum conductance, which is defined from the mercury intrusion curve. L emax = 0.34 µm for a very broad pore size distribution in rocks (Katz & Thompson, 1986 and 1987), φ is the total porosity, and S(L emax ) is the fractional volume of the connected pore involving pores with diameter of L emax and larger. Both S(L emax ) and L emax are strictly defined in terms of the mercury intrusion curves. The intrinsic permeability k and the conductivity formation factor σ/σ0 can be predicted using the same mercury intrusion measurements without adjusting the parameters (Katz & Thompson, 1986 and 1987). By combining Eqs. (5.1) and (5.2) and considering the different weights of conduction pathways (L emax L emax versus. L c ) for the electrical conductance vs. the hydraulic conductance, the intrinsic permeability of the porous materials can be expressed as a function of porosity (φ), the critical pore diameter (L c ), the pore size that produces the maximum hydraulic conductance L max , and the fractional volume S(L max ) of the connected pores with size greater than L max : k=

( ) L max 1 2 φ S(L max ) L max 89 Lc

(5.3)

The Katz-Thompson equation, as demonstrated by Eq. (5.3), is an analytical technique for determining the intrinsic permeability of porous material from its pore properties. By using several measuring techniques, including MIP and BSE, which will be used in this work for pore structure assessment, it is possible to acquire the parameters in Eq. (5.3) for estimating permeability.

162

5 Porosity Characterization and Permeability Prediction …

5.3.2 Assessment of Katz-Thompson Equation Many studies have found successful in applying the Katz-Thompson equation to cementitious material, although it was initially developed to predict rock conductivity based on the mercury intrusion readings. El-Dieb and Hooton (1994) tested the pore structure and the water permeability of OPC and the blended cement pastes; however, by using the Katz-Thompson equation to predict the permeability yielded very different findings from the experiment, and they concluded that the KatzThompson theory could not be employed to predict the permeability of cementitious materials. Christensen et al. (1996) studied the relative conductivity of highly porous cement paste and utilized the theory to predict the water permeability of the paste. They found that the calculated permeability tracked quite closely with the one that was tested. Garboczi (1990) reported that the prediction from the Katz-Thompson equation coincides with the measured permeability of cement paste and anticipated that the equation should still work for concrete. Since cement paste contains gel pores, its pore structure differs from that of stone or rock. Powers et al. (1959) noticed that the cement paste’s permeability reduces precipitously when the sample’s capillary porosity is below a certain value. This is because the capillary pores become discontinuous from the hydration product, and the gel pores in this condition, which are more than one order smaller in size than the capillary pores, must allow water to pass through. Therefore, gel pores regulate the water permeability, particularly in the low w/c ratio cement paste, and the Katz and Thompson hypothesis which is effectively applied to the single pore structure of stone or rock, may not be applicable to predict the permeability of cementitious materials. In this chapter, the permeability of the early-age frozen cement paste is determined using the Katz-Thompson equation. The freeze–thaw cycles will result in an increase in pore connectivity and porosity. It is anticipated that the Katz-Thompson equation will apply in this situation.

5.3.3 Determination of Lc and Lmax For calculating intrinsic permeability by using the Katz-Thompson equation, the parameters, such as the porosity (φ), the critical pore diameter (L c ), and the pore diameter (L max ) which produces the maximum conductance of concrete, are necessary to be determined first. L c can be obtained by differentiating the intruded volume curve (Fig. 5.1). L max can be determined from the mercury intrusion curve by finding the diameter that corresponds to the peak of the porosity times square pore diameter vs. the pore diameter curve, as shown in Fig. 5.17, and the calculated L max for the 3O and 5O pastes subject to the sealed curing and the early-age freeze–thaw cycles are summarized in Table 5.1. It is evident that there is a significant discrepancy

5.3 Calculated Permeability Based on Katz-Thompson Equation 6.E+02 4.E+02 2.E+02 0.E+00

Porosity × Diameter2

4.E+10

3.E+10

163

Lmax(MIP)=32nm 0

20 40 60

5.E+07 4.E+07 3.E+07 2.E+07 1.E+07 0.E+00

2.E+10

1.E+10

0

1000

2000

Lmax(CT)=13000nm

Lmax(BSE)=1600nm

0.E+00 0

5000

10000 Pore diameter (nm)

15000

20000

Fig. 5.17 Determination of L max based on pore size distribution measured from MIP, BSE, and X-ray CT (Wei et al., 2020)

between L max determined from the three pore measuring techniques. L max is typically measured in nm by MIP technique and in µm by BSE and X-ray CT techniques. Other than the variation in microstructure itself, the substantial disparity is caused by the resolution of the various techniques employed to measure the pore properties. According to Table 5.1, L max = 0.34 µm for a very broad pore size distribution in rocks (Katz & Thompson, 1987) is not held in cementitious materials.

5.3.4 Results of Permeability Prediction from the Katz-Thompson Equation Wei et al. (2020) computed the intrinsic permeability of cement paste by using the Katz-Thompson equation based on the measured pore parameters (summarized in Table 5.1). The results are plotted in Fig. 5.18 and summarized in Table 5.2. The predicted permeability exhibits variations in magnitudes, as shown in Fig. 5.18, as a result of the large discrepancies in the pore characteristics determined by the three techniques. The predicted permeability ranges × 10−19 –10−16 m2 based 1.E-11 Intrinsic permeability (m2)

Fig. 5.18 Comparison of the calculated intrinsic permeability by using the Katz-Thompson equation based on the measured pore size distribution by MIP, BSE, and X-ray CT techniques (Wei et al., 2020)

1.E-12

MIP

BSE

CT

1.E-13 1.E-14 1.E-15 1.E-16 1.E-17 1.E-18 3P-1d 3P-7d

3P

5P-1d 5P-7d

5P

164

5 Porosity Characterization and Permeability Prediction …

Table 5.2 Calculated intrinsic permeability (m2 ) by different computing methods based on the pore characteristics measured by different techniques (Wei et al., 2020) Mixture

Computing Method

Pore measuring technique MIP

3P-1d

3P-7d

3P-control

5P-1d

5P-7d

5P-control

BSE

X-ray CT

K-T equation

1.33E-18

2.80E-15

2.64E-14

GEM method

4.24E-19

1.57E-15

–

Navier–Stokes

–

–

1.96E-16

K-T equation

1.69E-18

2.27E-15

2.75E-14

GEM method

6.36E-19

9.05E-16

–

Navier–Stokes

–

–

4.76E-16

K-T equation

1.94E-18

2.20E-15

2.80E-14

GEM method

5.26E-19

9.05E-16

–

Navier–Stokes

–

–

1.98E-16

K-T equation

8.85E-16

5.36E-15

5.00E-13

GEM method

5.72E-16

6.01E-15

–

Navier–Stokes

–

–

1.17E-13

K-T equation

5.93E-16

5.07E-15

4.08E-13

GEM method

3.32E-18

2.72E-15

–

Navier–Stokes

–

–

1.25E-13

K-T equation

1.82E-18

2.84E-15

4.06E-14

GEM method

5.30E-19

5.41E-15

–

Navier–Stokes

–

–

6.27E-15

on the MIP measurement, and it is about × 10−15 m2 based on the BSE measurement, and × 10−15 –10−13 m2 based on the X-ray CT measurement. The computed permeability based on the X-ray CT is about three to five orders of magnitude higher than the MIP-based permeability. Given that the MIP is able to reach significantly smaller pores than the BSE and the X-ray CT scanning, this is not surprising. The predicted permeability based on the MIP measurement provides a clear understanding of how the w/c ratio and the early-age freeze–thaw cycles affect the permeability. Inversely, the predicted permeability based on the pore characteristics obtained using the BSE and the X-ray CT methods does not reflect the impacts of the w/c ratio and the early-age freeze–thaw cycles that the pastes were exposed to. This may indicate that when the Katz-Thompson equation is used to predict permeability, the pore parameters derived from the MIP measurement are more appropriate than the BSE and the X-ray CT measurements. For pores smaller than 1 µm, the Katz-Thompson equation works effectively. Tibbetts et al. (2020) evaluated the tested and the predicted permeabilities of concrete. The pore size distribution was measured by using the recently invented MIP equipment which allows the use of the macro-penetrometer cells to handle the large concrete specimens. It is reported that the predicted permeability based on MIP

(a)

Line of equality

Measured permeability (m/s)

Predicted permeability (m/s)

Predicted permeability (m/s)

5.3 Calculated Permeability Based on Katz-Thompson Equation

165

(b)

Oven-dried specimens Saturated specimens Measured permeability (m/s)

Fig. 5.19 a Comparison between the measured permeability and the predicted permeability of concrete, b effect of drying procedure on the relationship between the measured permeability and the predicted permeability of concrete by using Katz-Thompson equation Tibbetts et al. (2020)

data is nearly three magnitudes greater than the measured permeability, as illustrated in Fig. 5.19a. They thought that the discrepancies were due to the sample preparation and the testing procedure of MIP measurement. Since the MIP samples must be ovendried before testing, the oven-dried specimens are expected to have higher porosity and hence a higher estimated permeability due to microcracking than the undried sample used for the permeability test. To be consistent with the MIP sample conditions, the permeability test was also performed on the oven-dried samples Tibbetts et al. (2020). Figure 5.19b shows the measured and the predicted permeability results, they are comparable under the same oven-dried conditions. However, for the saturation conditions, the measured permeability is nearly three orders of magnitude lower than the predicted permeability. According to Tibbetts (2020), the drying condition is the primary source of disparity between the measured permeability and the predicted permeability. Furthermore, Sakai (2020) reviewed the predicted and the measured water permeability and its relation with the total pore volume, the critical pore diameter, and the median pore diameter for cement paste, cemented soil, mortar, and concrete. Figure 5.20a compares the predicted permeability based on the Katz-Thompson equation with the measured permeability values. Although the prediction results in a slight overestimation, the values agree quantitatively to some extent. The correlation coefficient between the critical pore diameter and the measured permeability is 0.73 (see Fig. 5.20b), which is greater than the correlation coefficients of 0.63 (Fig. 5.20c) between the permeability and the median pore diameter, and 0.12 (Fig. 5.20d) between the permeability and the total pore volume. This indicates that the critical pore diameter is one of the most critical factors influencing the permeability of cementitious materials, verifying the validity of the Katz-Thompson equation for permeability prediction. Table 5.3 summarizes the permeability predicted by the Katz-Thompson equation in the literature, and it can be found that the predicted permeability varies significantly, from 2.80 × 10−14 to 0.92 × 10−19 m2 . This is mainly due to the different critical pore diameters used in the Katz-Thompson equation, which varies from 20 nm to 13 µm. The critical pore sizes are affected by the w/c ratio, curing age, and test

166

5 Porosity Characterization and Permeability Prediction …

(a)

(b)

(c)

(d)

Fig. 5.20 a Relationship between the reported and calculated (predicted) permeability of cement paste, cemented soil, mortar, and concrete by using Katz–Thompson equation; the relationship between the measured water permeability and b the critical pore diameter, c the median pore diameter, d the total pore volume (Sakai, 2020)

method. As a result, the uncertainty and variability of the Katz-Thompson equation on cementitious material permeability prediction are implied. Table 5.3 Permeability of cementitious materials predicted by Katz-Thompson equation in literature Reference

Material

Wei et al. (2020) Cement paste

Property

Critical pore diameter Predicted (measuring permeability (m2 ) technique) (nm)

w/c 0.3, 7d

32 (MIP) 1600 (SEM) 13,000 (CT)

1.94 × 10−18 2.20 × 10−15 2.80 × 10−14

Song et al. (2020)

Cement paste

w/c 0.45, 90d

20–60 (FIB/SEM)

0.92–8.23 × 10−19

Song et al. (2019)

Concrete

w/c 0.43, 90d

30–40 (FIB/SEM)

1.4–3.3 × 10−19

Care and Derkx (2011)

Cement paste

w/c 0.45, 28d

20 (MIP)

3.3–10−19

Sun et al. (2015) Mortar

w/c 0.36, 28d

260–429 (SEM)

5.7–15.6 × 10−17

Sun et al. (2014) Concrete

w/c 0.45

62.9–85.7 (Transmission X-ray Microscope)

1.79–7.24 × 10−18

5.4 Calculated Permeability Based on GEM Method

167

5.4 Calculated Permeability Based on GEM Method 5.4.1 General Effective Media (GEM) Theory for Permeability Unlike other porous materials, the solid phase of the cement paste contains gel pores. This means that even if the capillary porosity is less than the percolation threshold, the diffusion can still occur (Oh & Jang, 2004). The general effective media (GEM) theory was developed by McLachlan (1986, 1987 & 1988) and McLachlan et al. (1990) to calculate the overall conductivity of two-phase material, with the doubt of that the Katz-Thompson equation cannot directly apply to predict the water permeability of cementitious materials, because it was developed based on rocks in which the gel pore does not exist. The GEM model is capable of considering the contributions of both capillary pores and the gel pores to predict the permeability of OPC paste. In this model, cement paste is considered as a two-phase material (Fig. 5.21). One phase is the high-permeability capillary pores and the other one is the low-permeability phase consists of CSH gel, CH, and unhydrated cement particles. The volume fraction and the permeability of each phase will determine the overall permeability of the OPC paste. The overall permeability (k) of the cement paste can be calculated based on the volume fraction and the permeability of each phase: ) ( 1/t (1 − φ) kl − k 1/t 1/t

kl

+ Ak 1/t

+

) ( 1/t φ kh − k 1/t 1/t

kh + Ak 1/t

=0

(5.4)

where φ is the capillary porosity, kh is the permeability of high-permeability phase, kl is the permeability of low-permeability phase, t is the critical exponent for the permeability, and k is the permeability of the two-phase material. The value of k is always between kh and kl . To obtain k, the values of kh and kl have to be determined first. It should be noted that Eq. (5.4) is based on the analogy of the theory that was designed for conductivity. A is a constant related to φc and can be calculated by A = (1 − φc )/φc , and φc is the critical capillary porosity. Fig. 5.21 Two-phase model of cement (Cui & Cahyadi, 2001)

Low permeable phase

High permeable phase

168

5 Porosity Characterization and Permeability Prediction …

5.4.2 Calculation for kh and kl If the cement paste is very porous, water mainly flows through the capillary pores. Comparing with the contribution of capillary pores, the contribution of the lowpermeability gel pores to the overall permeability can be neglected. By substituting k l = 0 into Eq. (5.4), the following equation can be obtained. kh (φ − φc )t (1 − φc )t

k=

(5.5)

As aforementioned, the Katz-Thompson equation can also be used to predict the permeability of very porous cement paste. By substituting σ/σ0 = 1.8(1 − φc )2 proposed by Garboczi and Bentz (1991) for OPC paste into Eq. (5.1), the permeability of high-permeability phase k h can be calculated as: kh =

1 2 L (1.8)(1 − φc )2 226 c

(5.6)

Therefore, the permeability of high-permeability phase only depends on the critical pore diameter and the critical capillary porosity. For the low-permeability case, the capillary porosity is assumed to be zero. The permeability of cement paste depends on the gel pores in CSH. In this situation, the cement paste consists of three components: CSH gel, CH, and the unhydrated cement particles. Among them, only CSH gel is permeable. Therefore, the overall permeability is due to the presence of gel pores. Such kind of material can be considered as a two-phase material where CSH gel is considered as the high-permeability phase, and the CH and the unhydrated cement particles are considered as the low-permeability or the impermeable phase. According to Powers (1985), the permeability of CSH gel k CSH is equal to 7 × 10−23 m2 . Thus, by using the GEM theory the overall permeability is calculated for the low-permeability case as: ( ) 1 − φCSH 2 kl = kφ=0 = kCSH 1 − 1 − φc'

(5.7)

'

where φc is the critical volume fraction of CSH, which is chosen to be 0.17 (Bentz et al., 1996), and φC S H is the volume fraction of CSH in the solid phase, which can be calculated by using Eq. (5.8): φCSH =

VCSH Vunhyd. + VCH + VCSH

(5.8)

The volume fraction of the hydration products (V CSH and V CH ) and the unhydrated cement particles (Vunhyd. ) can be calculated by using Powers model (Hansen, 1986) for ordinary Portland cement paste (Eqs. (5.9) and (5.10)).

5.4 Calculated Permeability Based on GEM Method

VCH + VCSH = Vunhyd. =

169

0.68α (w/c) + 0.32

0.32(1 − α) (w/c) + 0.32

(5.9) (5.10)

The hydration degree α can also be calculated from the Powers model if the capillary porosity of cement paste φ is known. α=

(1 − φ)(w/c) − 0.32φ 0.36

(5.11)

The ratio of volume fraction of CSH and CH can be calculated from stoichiometry. In this chapter, V CSH /V CH = 1.66:0.63 is chosen for OPC paste (Garboczi & Bentz, 1991). By substituting the ratio of volume fraction into Eq. (5.9), the volume fraction of CSH can be obtained. Then, the overall permeability for the low-permeability case can be calculated by using Eq. (5.7). The overall permeability of OPC paste depends on how the capillary pores and the gel pores interact with each other. Mathematically, this interplay can be expressed as the GEM equation (Eq. (5.4)). Figure 5.22 describes the process to calculate the permeability of cement paste. All the symbols used are consistent with other part in this chapter. The GEM theory is shown to apply to both low- and high-permeable cement paste (Christensen et al., 1996; Cui & Cahyadi, 2001). The main path of transport in cement paste is known to be the capillary pore space, whereas, the gel pores play a minor role in transport, except at very low capillary porosity. Also note that the model itself contains time-dependent hydration effect, which is revealed by the microstructural changes in cement paste specimens.

Volume fraction of CSH Volume fraction of CH and unhydrated

and

=

1 226

)

)

+

)

=0

Fig. 5.22 Process of calculating permeability of cement paste based on the GEM method (Cui & Cahyadi, 2001)

170

5 Porosity Characterization and Permeability Prediction …

5.4.3 Results of Permeability Prediction from the GEM Method Cui and Cahyadi (2001) predicted the permeability of cement paste with different w/c ratios and curing ages by using the GEM method. The comparison between the measured permeability and the predicted permeability from the GEM method is shown in Table 5.4. The results show that the predicted permeability is well consistent with the measured results. Meanwhile, the hardened cement paste with higher w/c ratio has higher permeability, and the permeability will decrease when the curing age increases. This model can also be verified by the measured results of cement paste obtained by other researchers (Christensen et al., 1996; Hughes, 1985), as shown in Fig. 5.23. This indicates that the GEM method can accurately predict the permeability of both porous and dense OPC paste. GEM theory can also be used to explain the mechanism of permeability changes. Gao et al. (2019) studied the effect of graphene oxide—multiwalled carbon nanotube hybrid (GO/MWCNT) on the permeability of cement paste. As seen in Table 5.5, compared with the reference pastes, the reinforcing effect of GO/MWCNT on the water impermeability of cementitious composites is mainly due to the decrease of k h . Because the permeability of cementitious pastes depends on the capillary pores which Table 5.4 Comparison between the measured permeability and the predicted permeability of cement paste from the GEM method (Cui & Cahyadi, 2001) w/c = 0.3 7 days Φ

210 days

0.174

0.059

68.6

L c (nm)

w/c = 0.4 7 days

35 days

0.2069

10.1

210 days

0.129

45.3

0.091

36.6

10.1

α

0.53

0.732

0.69

0.85

0.93

Φ CSH

0.51

0.618

0.602

0.81

0.70

k l (m2 )

1.201 × 10−23

2.04 × 10–23

1.90 × 10−23

2.55 × 10−23

2.85 × 10−23

1.68 ×

10−17

5.46 × 10−19

2.86 ×

10−22

1.02 × 10−22

3.47 ×

10−22

1.22 × 10−22

k predicted

(m2 )

k measured

(m2 )

2.52 ×

10−17

5.49 ×

10–19

2.22 ×

10−21

4.33 ×

10–23

2.66 ×

10−21

Fig. 5.23 Comparison between the GEM method-predicted and the measured permeability of cement paste from Hughes (1985), Christensen et al. (1996), and Cui and Cahyadi (2001) (Cui & Cahyadi, 2001)

–

1.1 ×

10−17

1.83 ×

10−20

8.11 ×

10−21

Measured permeability (m2)

kh

(m2 )

X=Y

Predicted permeability (m2)

5.4 Calculated Permeability Based on GEM Method

171

are the key to the values of k h in the GEM theory. When the admixture contributes to the formation of a stiffer hydration product, it will lead to the decrease of k h value and improve the permeability resistance. The calculated intrinsic permeability of cement pastes with w/c ratios of 0.3 (3P) and 0.5 (5P) based on the GEM method as well as the Katz-Thompson equation in Wei et al. (2020) is plotted in Fig. 5.24 and Table 5.2. Pore parameters from the MIP and BSE measurements are used as inputs. It is apparent that the computed permeability by the GEM method shows similar trend to the permeability computed by the Katz-Thompson equation for the same pore measuring technique. In general, based on the same pore measuring technique, the computed permeability from the GEM method is less than that from the Katz-Thompson method. The reason might contribute to that the GEM method applies well for less porous media such as the cementitious materials with both gel pore and capillary pore network dominating transport, whereas the Katz-Thompson equation applies to more porous media with capillary pore network dominating transport. This can be clearly seen from the greater difference between the permeability computed from the GEM and the Katz-Thompson equation for the low w/c ratio (0.3) pastes, this paste is less porous than other pastes, and the predicted permeability by the GEM method is expected to be closer to the real permeability. Again, the MIP measurement on the Table 5.5 Predicted permeability of OPC and GO/MWCNT enhanced paste based on the GEM theory (Gao et al., 2019) Sample

w/c

k l (m2 )

OPC-1

0.35

OPC-2

0.4

GO/MWCNT-2 OPC-3 GO/MWCNT-3

0.45

k predicted (m2 )

k measured (m2 )

1.94 × 10−23

9.64 × 10−17

2.42 × 10−22

2.71 × 10−22

2.15 ×

10−23

3.41 ×

10−17

1.41 ×

10−22

9.87 × 10−23

2.38 ×

10−23

1.24 ×

10−16

3.39 ×

10−22

3.17 × 10−22

2.47 × 10−23

4.76 × 10−17

2.43 × 10−22

1.60 × 10−22

2.61 ×

10−23

9.72 ×

10−17

5.69 ×

10−22

4.15 × 10−22

2.71 ×

10−23

5.53 ×

10−17

4.30 ×

10−22

2.34 × 10−22

Fig. 5.24 Comparison of the calculated intrinsic permeability of cement pastes with w/c ratios of 0.3 (3P) and 0.5 (5P) by using the Katz-Thompson (K-T ) equation and the GEM method from measured pore parameters by MIP and BSE techniques (Wei et al., 2020)

1.E-11

Intrinsic permeability (m2)

GO/MWCNT-1

k h (m2 )

MIP-KT MIP-GEM BSE-KT BSE-GEM

1.E-12 1.E-13 1.E-14 1.E-15 1.E-16 1.E-17 1.E-18 1.E-19 3P-1d

3P-7d

3P

5P-1d

5P-7d

5P

172

5 Porosity Characterization and Permeability Prediction …

pore characteristics is more suitable for permeability prediction by using the KatzThompson and the GEM methods, as the effects of both w/c ratio and the early-age freeze–thaw cycles are revealed by the computed permeability.

5.5 Computed Permeability Based on Navier–Stokes Method With improved technologies in image acquisition, processing, and analysis, we can take advantage of the 3D information provided by X-ray CT, not only to visualize, but also to quantify more accurately the pore system, cracks, and other properties to assess the durability of cement-based materials. Navier–Stokes method was developed to calculate the permeability of cement-based materials from the microstructure information detected by the X-ray CT technique. Martys et al. (1994) used this method to simulate random packing of spheres and obtained the universal curves for permeability. The permeability solver code was developed at NIST (Garboczi, 1998) to calculate the permeability of cement paste by using finite difference method from the X-ray CT-scanned results (Bentz & Martys, 2007). This method is capable of taking into account the effect of porosity, pore size distribution, and the percolation of pore network. It has been used for predicting permeability of cementitious materials in recent years (Chung et al., 2015; Das et al., 2015; Pieralisi et al., 2017; Promentilla et al., 2016; Sun et al., 2014; Zhang, 2017).

5.5.1 Navier–Stokes Theory for Permeability The basic idea of computational analysis of permeability based on the Navier–Stokes method is to solve the Stokes equation for the average flow velocity of fluid and then use the average flow velocity to calculate permeability based on Darcy’s law. According to Darcy’s law, the flow velocity of fluid under pressure is: v=−

k p · u L

(5.12)

where v is the average fluid velocity in the direction of the flow for the porous media, k is permeability, p is the pressure difference, L is the length of the sample across which there is an applied pressure difference of p, and u is the fluid viscosity. The distribution of pores in the cement paste can be obtained by CT images. The gray value 0 (black) and 255 (white) are set to represent the pore phase and the solid phase, respectively. Based on the finite difference method of the Navier–Stokes equation, the pores distribution information can be used to calculate the permeability coefficient of the tested object. The fluid in the pore is regarded as a steady incompressible viscous fluid, and its continuity equation is as follows:

5.5 Computed Permeability Based on Navier–Stokes Method

∂ρ + ∇ · (ρ v ) = 0 ∂t

173

(5.13)



where v is the local velocity of the fluid and ρ v is the mass flux. Because the density of the fluid ρ is constant, Eq. (5.13) can be simplified as:

∇·v =0

(5.14)

A derivative of pressure (p) to time is introduced into the continuity equation (Eq. (5.14)) by using the artificial compression algorithm, the entire set of governing equations can be transformed into a time-dependent and hyperbolic-parabolic-type equation. Under this circumstance, Eq. (5.14) can be expressed as: ∂p + c2 ∇ · v = 0 ∂t

(5.15)

where c2 is an arbitrary constant. On the other hand, the equation of motion of fluid can be written into substantial derivative with the Navier–Stokes equation as:

ρ

Dv = −∇ p + u∇ 2 v + f Dt

(5.16)



where f is the unit volume force, which can be neglected at the micro scale. For the viscous fluid, the Reynolds number is extremely small, so the acceleration term

ρ DDtv = 0. Therefore, Eq. (5.16) can be further simplified as a linear Stokes equation:

u∇ 2 v = ∇ p

(5.17)

To numerically solve the Stokes equations, a finite difference scheme in conjunction with the artificial compressibility relaxation algorithm is employed. Based on Eqs. (5.15) and (5.17), the velocity field can be solved, and then, the permeability coefficient can be obtained by substituting the velocity field into the Eq. (5.12). The discretized and non-dimensionalized approximation of Eqs. (5.15) and (5.17) can be written as: ( ) n+1 n+1 n+1 ∂v ∂v ∂v 1 n+1 y x − p n ) + c2 (p + + z =0 (5.18) t ∂x ∂y ∂z

174

5 Porosity Characterization and Permeability Prediction …

) ( 2 n ) ∂ pn 1 ( n+1 ∂ 2 vxn ∂ 2 vxn ∂ vx n 2 n = u∇ vx = u v − vx + + + t x ∂x ∂x2 ∂ y2 ∂ z2 ( ) 2 n 2 n ∂ vy ∂ vy ∂ 2 v ny ) ∂ pn 1 ( n+1 n 2 n v − vy + + + = u∇ v y = u t y ∂y ∂x2 ∂ y2 ∂ z2 ) ( 2 n ) ∂ pn ∂ 2 vzn ∂ 2 vzn ∂ vz 1 ( n+1 vz − vzn + + + = u∇ 2 vzn = u t ∂z ∂x2 ∂ y2 ∂ z2

(5.19)

where n is the time step, and the lattice spacing is considered as one. ∇ 2 is the Laplace operator which is discretized using a non-centered-difference-based method to ensure sufficient accuracy and to fulfill the boundary condition at the interface. The discretized forms in Eq. (5.19) have 6 forms depending on where the nearest solid voxels are located (Sect. 5.5.2). The velocity components at different locations will be determined when the following condition is met: ( max

) | 1 || n+1 vi − vin | < ε t

(5.20)

where ε is a sufficiently small value.

5.5.2 Six Conditions of Space Step Direction For three dimensions, the Laplace operator is divided into three parts: ∇2 =

∂2 ∂2 ∂2 + + ∂x2 ∂ y2 ∂ z2

(5.21)

One is the direction parallel to the fluid velocity and the other two are the directions perpendicular to the fluid velocity. For example, to solve the fluid velocity and the permeability in x-direction, the x-direction is assumed to be the parallel direction and then the y- and z-directions are assumed to be the perpendicular direction. For the Laplace operator in x-direction, it can be discretized using central difference method: ∂ 2 vx = (vx )−1 − 2(vx )0 + (vx )1 ∂x2

(5.22)

For the Laplace operator in y-direction (or z-direction), it can be divided into six conditions which are determined by the boundary condition at the interface between the pore phase and the solid phase. Because of the similarity, six conditions of the y-direction are discussed in this section (Figs. 5.25, 5.26, 5.27, 5.28, 5.29 and 5.30). (1) Solid phase step—computed step—solid phase step (as shown in Fig. 5.25). Conducting the Taylor series expansion on (vx )−1 and (vx )1 in the form of (vx )0 :

5.5 Computed Permeability Based on Navier–Stokes Method Fig. 5.25 Condition One of the discretization of space step for the Navier–Stokes equations

)

)

)

Fig. 5.26 Condition Two of the discretization of space step for the Navier–Stokes equations

Fig. 5.27 Condition Three of the discretization of space step for the Navier–Stokes equations

Fig. 5.28 Condition Four of the discretization of space step for the Navier–Stokes equations

Fig. 5.29 Condition Five of the discretization of space step for the Navier–Stokes equations

175

176

5 Porosity Characterization and Permeability Prediction …

Fig. 5.30 Condition Six of the discretization of space step for the Navier–Stokes equations

(vx )−1 = (vx )0 −

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 1 ∂(vx )0 · + · · − 2 ∂y 8 ∂ y2 48 ∂ y3

(vx )1 = (vx )0 +

1 ∂ 2 (vx )0 1 ∂(vx )0 1 ∂ 3 (vx )0 · − · · + 2 2 ∂y 8 ∂y 48 ∂ y3

(5.23)

(5.24)

The boundary condition in Condition One is: (vx )1 = (vx )−1 = 0

(5.25)

By combining Eqs. (5.23–5.25), the following equation can be obtained: ∂ 2 (vx )0 = −8(vx )0 ∂ y2

(5.26)

(2) Solid phase step—computed step—pore phase step—solid phase step (as shown in Fig. 5.26). Conducting the Taylor series expansion on (vx )−1 , (vx )1 , and (vx )2 in the form of (vx )0 : (vx )−1 = (vx )0 −

1 ∂ 2 (vx )0 1 ∂(vx )0 1 ∂ 3 (vx )0 · + · · − 2 ∂y 8 ∂ y2 48 ∂ y3

(5.27)

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 ∂(vx )0 + · · + ∂y 2 ∂ y2 6 ∂ y3

(5.28)

9 ∂ 2 (vx )0 9 ∂ 3 (vx )0 3 ∂(vx )0 · + · · + 2 ∂y 8 ∂ y2 16 ∂ y3

(5.29)

(vx )1 = (vx )0 + (vx )2 = (vx )0 +

The boundary condition in Condition Two is: (vx )2 = (vx )−1 = 0 By combining Eqs. (5.27–5.30), the following equation can be obtained:

(5.30)

5.5 Computed Permeability Based on Navier–Stokes Method

∂ 2 (vx )0 8 16 = (vx )1 − (vx )0 ∂ y2 3 3

177

(5.31)

(3) Solid phase step—computed step—pore phase step—pore phase step (as shown in Fig. 5.27). Conducting the Taylor series expansion on (vx )−1 , (vx )1 , and (vx )2 in the form of (vx )0 : (vx )−1 = (vx )0 −

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 1 ∂(vx )0 · + · · − 2 ∂y 8 ∂ y2 48 ∂ y3

(5.32)

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 ∂(vx )0 + · · + ∂y 2 ∂ y2 6 ∂ y3

(5.33)

(vx )1 = (vx )0 +

(vx )2 = (vx )0 + 2 ·

∂ 2 (vx )0 4 ∂ 3 (vx )0 ∂(vx )0 +2· · + ∂y ∂ y2 3 ∂ y3

(5.34)

The boundary condition in Condition Three is: (vx )−1 = 0

(5.35)

By combining Eqs. (5.32–5.35), the following equation can be obtained: ∂ 2 (vx )0 1 = − (vx )2 + 2(vx )1 − 5(vx )0 ∂ y2 5

(5.36)

(4) Solid phase step—pore phase step—computed step—solid phase step (as shown in Fig. 5.28). Conducting the Taylor series expansion on (vx )−2 , (vx )−1 , and (vx )1 in the form of (vx )0 : (vx )−2 = (vx )0 −

9 ∂ 2 (vx )0 9 ∂ 3 (vx )0 3 ∂(vx )0 · + · · − 2 ∂y 8 ∂ y2 16 ∂ y3

(5.37)

1 ∂ 2 (vx )0 ∂(vx )0 1 ∂ 3 (vx )0 + · · − ∂y 2 ∂ y2 6 ∂ y3

(5.38)

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 1 ∂(vx )0 · + · · + 2 ∂y 8 ∂ y2 48 ∂ y3

(5.39)

(vx )−1 = (vx )0 − (vx )1 = (vx )0 +

The boundary condition in Condition Four is: (vx )−2 = (vx )1 = 0

(5.40)

178

5 Porosity Characterization and Permeability Prediction …

By combining Eqs. (5.37–5.40), the following equation can be obtained: ∂ 2 (vx )0 8 16 = (vx )−1 − (vx )0 2 ∂y 3 3

(5.41)

(5) Pore phase step—pore phase step—computed step—solid phase step (as shown in Fig. 5.29). Conducting the Taylor series expansion on (vx )−2 , (vx )−1 , and (vx )1 in the form of (vx )0 : (vx )−2 = (vx )0 − 2 ·

(5.42)

1 ∂ 2 (vx )0 ∂(vx )0 1 ∂ 3 (vx )0 + · · − ∂y 2 ∂ y2 6 ∂ y3

(5.43)

1 ∂ 2 (vx )0 1 ∂ 3 (vx )0 1 ∂(vx )0 · + · · + 2 ∂y 8 ∂ y2 48 ∂ y3

(5.44)

(vx )−1 = (vx )0 − (vx )1 = (vx )0 +

4 ∂ 3 (vx )0 ∂(vx )0 ∂ 2 (vx )0 − +2· · ∂y ∂ y2 3 ∂ y3

The boundary condition in Condition Five is: (vx )1 = 0

(5.45)

By combining Eqs. (5.42–5.45), the following equation can be obtained: ∂ 2 (vx )0 1 = − (vx )−2 + 2(vx )−1 − 5(vx )0 ∂ y2 5

(5.46)

(6) Pore phase step—computed step—pore phase step (as shown in Fig. 5.30). Conducting the Taylor series expansion on (vx )−1 and (vx )1 in the form of (vx )0 : (vx )−1 = (vx )0 −

1 ∂ 2 (vx )0 ∂(vx )0 1 ∂ 3 (vx )0 + · · − ∂y 2 ∂ y2 6 ∂ y3

(5.47)

(vx )1 = (vx )0 +

1 ∂ 2 (vx )0 ∂(vx )0 1 ∂ 3 (vx )0 + · · + ∂y 2 ∂ y2 6 ∂ y3

(5.48)

By combining Eqs. (5.47–5.48), the following equation can be obtained: ∂ 2 (vx )0 = (vx )−1 − 2(vx )0 + (vx )1 ∂ y2

(5.49)

5.5 Computed Permeability Based on Navier–Stokes Method

179

After obtaining the discretized form of the second derivative of Eq. (5.19), the Eqs. (5.18) and (5.19) can be solved, and calculation termination condition can be set as Eq. (5.20). From the above equations, the velocity v can be obtained, and then, the permeability coefficient k can be calculated.

5.5.3 Process of Calculating Permeability Based on Navier–Stokes Method Based on the CT images of the cement paste, the three-dimensional microstructures of the paste can be reconstructed, and consequently, the permeability model can be established by referring the Navier–Stokes equation. The first step is to determine the threshold of the gray level of the pore phase. The method described in Sect. 4.7 is used to segment the phase in CT images. Only the division of the overall solid phase and the pore phase is conducted here, and it is not necessary to subdivide the solid phase into the clinker phase and the hydrates phase, because the type of the solid phase does not affect the permeability of the material (Monteiro, 1993). The second step is to construct the calculation model. The gray level of the pore phase is set to 0, and the gray level of the solid phase is set to 255. Then, the CT image is converted into a three-dimensional structure with two phases, which is taken as the calculation model for the permeability coefficient. The third step is to calculate the permeability coefficient. Based on the Stokes permeability solver (Bentz & Martys, 2007), the permeability coefficient can be solved by using the boundary conditions shown in Figs. 5.25, 5.26, 5.27, 5.28, 5.29 and 5.30.

5.5.4 Brief Introduction of Stokes Permeability Solver For many years, research in the Materials and Construction Research Division (formerly the Building Materials Division) at the National Institute of Standards and Technology (NIST) has focused on understanding the relationship between the microstructure and the properties of materials, particularly for cement-based materials. One of the key material properties is their resistance to fluid flow under a pressure gradient, e.g., permeability. The Stokes permeability solver was developed based on C and Fortran computer programs for computing the permeability of three-dimensional porous media under the incompressible Stokes flow conditions. The codes have two purposes: (1) a

180

5 Porosity Characterization and Permeability Prediction …

preprocessor written in C to convert a model or a real three-dimensional microstructure (represented by a three-dimensional set of voxels) into lists of coordinates representing the voxels where pressures, x-velocity, y-velocity, or z-velocity components will be present and (2) a set of Fortran programs to numerically solve the linear Stokes equations in three dimensions by using a finite difference scheme in conjunction with the artificial compressibility relaxation algorithm. In this program, a hydraulic pressure is applied as the initial input and the periodic boundary conditions are used. The flow condition in the pore system is considered to be incompressible and steady state based on the computational fluid mechanics. The finite difference method is applied on the image pixels combined with the artificial compressibility relaxation algorithm to solve the Stokes equation. Once the solutions of the average velocity at the four different depths converge to a close value, the permeability can be obtained from the Darcy equation based on that velocity. By making a comparison to the existing direct measurement of permeability made by other researchers (Bentz, 2008; Montes & Haselbach, 2006; Neithalath et al., 2006; Sisavath et al., 2001; Sumanasooriya et al., 2010), the accuracy of the code was validated with an error of less than 2%. The Stokes solver codes were originally developed in 1993, when computers were much slower and had much less memory than the current systems. Therefore, large efforts have been made to minimize the memory requirements of the codes and to develop codes that would be especially efficient on vector-based machines of that era. Rather than storing a complete representation of the three-dimensional microstructure, for example, simple lists of those nodes where pressures or x, y, or z components of the velocity vector are present were created and stored. The (x, y, z) coordinates were stored in and extracted from a single 4-byte integer using logical shift and logical operations. While the original version of the code was developed for (100 × 100 × 100) voxel microstructures and could therefore use an 8-bit representation (0–255) for each coordinate, the presented version of the code has a 9-bit size for each of these three coordinates. Thus, it could potentially be used on the threedimensional systems as large as 500 × 500 × 500. If execution for larger systems is desired, extensive modifications would have to be made to both the C and Fortran codes to take them to 10-bit (0–1023) versions. At present, the permeability codes have been used in a wide variety of applications, and the codes are being made available to the public for their utilization. The computer programs may be downloaded from the NIST anonymous ftp site at ftp://ftp.nist.gov/ pub/bfrl/bentz/permsolver.

5.5.5 Results of Permeability Prediction from the Navier–Stokes Method Based on the 3D Transmission X-ray Microscope (TXM) images (Fig. 5.14), Sun et al. (2014) used the Stokes permeability solver to predict the permeability of cement

5.5 Computed Permeability Based on Navier–Stokes Method

181

paste in the x-, y-, and z-directions, as shown in Table 5.6. The pore connectivity for each direction is defined as the number of pore pixels contained in the path that divides the total number of pore pixels. From Table 5.6, it is ready to notice that pore connectivity is increasing with porosity. Since the number of pores within the same volume increases with sample porosity, the pore connectivity is also enhanced. However, it should be noticed that the microstructure is not quite homogeneous, and one direction may have less percolated pores than other two directions. For example, the z-direction of sample 1 has less permeability than x- and y-direction. In Sun et al. (2014), the granulometry analysis was applied on each sample to obtain the critical pore size (L c ). Then, the permeability can be calculated by K-T equation (Eq. (5.1)) and listed in Table 5.6. For the samples with higher porosity like sample 1, the predicted permeability by the Navier–Stokes method is larger than that by the K-T equation. For the samples with lower porosity like sample 2, the predicted permeability by the Navier–Stokes method is smaller than the calculated by the K-T equation. The homogeneity of the computing volume is important to ensure that a representative permeability is obtained. To evaluate the effect of the heterogeneous nature of the cementitious materials on the predicted permeability by the Navier–Stokes method, Wei et al. (2020) selected five blocks for permeability prediction, and each block contains 300 voxels, as shown in Fig. 5.31a. The average predicted permeability at the five locations and their standard variance are plotted in Fig. 5.31b for the pastes of 5P-7d, 3P-1d, and 3P-7d. For each sample, the permeability predicted by these five blocks based on the Navier–Stokes method is basically the same, and the difference between the maximum and the minimum predicted permeability is about 10%. It is considered that the average permeability of the five blocks can represent the permeability of each sample. The permeability calculated by the average of these Table 5.6 Pore parameters and predicted permeability of cement paste (Sun et al., 2014) Samples

Porosity (%)

Orientations

Pore connectivity (%)

Predicted permeability by Navier–Stokes method (m2 )

Critical pore size (nm)

Calculated permeability by K-T equation (m2 )

Sample 1

23.17

x y z

96.50 96.90 95.60

1.21 × 10−17 2.30 × 10−17 7.49 × 10−18

85.7

7.24 × 10−18

Sample 2

12.19

x y z

83.20 84.40 85.70

1.24 × 10−19 2.42 × 10−19 1.79 × 10−19

62.9

1.79 × 10−18

Sample 3

22.91

x y z

94.14 94.75 94.81

1.00 × 10−17 1.79 × 10−17 8.94 × 10−18

74.6

5.30 × 10−18

Sample 4

12.51

x y z

85.14 88.56 86.25

6.37 × 10−19 1.91 × 10−19 6.33 × 10−19

64.9

2.01 × 10−18

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5 Porosity Characterization and Permeability Prediction …

five blocks of each sample based on the Navier–Stokes method is used to compare with permeability calculated by other methods (Fig. 5.32 and Table 5.2). It is seen that the magnitude of the calculated intrinsic permeability by the KatzThompson equation is about 1–2 orders of magnitude greater than that computed from the Navier–Stokes method, though the same set of pore parameters from X-ray CT measurement was used. The magnitude of the calculated permeability by the Katz-Thompson equation is × 10–14 –10–13 m2 , whereas it is × 10−16 –10−13 m2 by the Navier–Stokes method. Similar trend was also found in Das et al. (2015) that the Navier–Stokes method computed smaller permeability compared to the KatzThompson method, they found the intrinsic permeabilities are 5 × 10−14 and 5.2 × 10−16 m2 by using the Katz-Thompson equation and the Navier–Stokes method, respectively, for the fly ash-based geopolymer with porosity of 32%. In addition, the calculated intrinsic permeability by the Navier–Stokes method is consistent with the reported one in the literature (Das et al., 2015; Garboczi & Bentz, 2001). Garboczi and Bentz (2001) calculated the intrinsic permeability based on the 3D microstructure model reconstructed from the CT scanned images was × 10−15 –10−13 m2 for pastes with porosity of 20–30%.

Micro-CT Specimen

Calculation element left

up right down

Permeability (m2)

(b) 1.0E-12

(a)

1.0E-13 1.0E-14 1.0E-15

center 1.0E-16 5P-7d

3P-1d

3P-7d

Fig. 5.31 a Navier–Stokes method at five locations for sensitivity analysis of the predicted permeability, b the average predicted permeability of cement paste subject to freeze–thaw cycles by the Navier–Stokes method (Wei et al., 2020)

1.E-12 Intrinsic permeability (m2)

Fig. 5.32 Calculated intrinsic permeability by the Katz-Thompson and the Navier–Stokes methods based on X-ray CT data for control 3P and 5P pastes and the pastes subject to freeze–thaw cycling at the age of 1 and 7 days (Wei et al., 2020)

1.E-13

Katz-Thompson Navier-Stokes

1.E-14 1.E-15 1.E-16 3P-1d 3P-7d

3P

5P-1d 5P-7d

5P

5.6 Summary

183

The computed permeability by the Navier–Stokes method well reflects the effects of the freeze–thaw cycles and the w/c ratio on the specimen permeability. A clear increase in permeability was observed for the high w/c ratio paste (5P), and the increase of w/c ratio from 0.3 to 0.5 will increase the permeability by 2–3 orders of magnitude. The early-age freeze–thaw cycles increase the permeability by 2 orders of magnitude compared to the control sample. However, the computed permeability by the Katz-Thompson method does not reflect the effect of freeze–thaw cycles, and the permeability of the control paste and the paste subject to the early-age freeze–thaw cycles is almost the same. Therefore, it is reasonable to conclude that the Navier– Stokes method is more applicable to predict permeability of cement paste compared to the Katz-Thompson equation when pore parameters obtained from X-ray CT measurement are used.

5.6 Summary The purpose of this chapter is to provide an insight into the feasibility of different techniques and methods on measuring the pore parameters and predicting permeability of cementitious materials. To this end, the intrinsic permeability of cement paste is calculated by using the Katz-Thompson method, the general effective media (GEM) method, and the Navier–Stokes method based on the pore parameters measured from MIP, BSE, and X-ray CT scanning techniques. The major findings are. The pore parameters obtained from the MIP measurement is more applicable than that obtained from the BSE and the X-ray CT measurement, when KatzThompson equation is used for the permeability prediction. The Katz-Thompson equation performs well for pore size below 1 µm. The GEM method is a more suitable tool for cementitious materials with low porosity. Generally, the computed intrinsic permeability by the GEM method shows a similar trend to that by the Katz-Thompson equation for the same pore measuring technique, such as the MIP or the BSE measurements. Nevertheless, for the paste with a low w/c ratio, which is less porous than other paste, the predicted intrinsic permeability by the GEM method is expected to be closer to the real permeability. The X-ray CT scanning is a promising technique to detect the pore characteristics of cementitious materials non-destructively. The Navier–Stokes method is more applicable to predict the permeability of cement paste compared to the KatzThompson equation, when pore parameters obtained from the X-ray CT measurement are used. However, it should be aware that the computed permeability of cementitious materials is more suitable as a comparison tool for selecting mixture under the conditions of using the same technique for pore measurement and the same calculating method for permeability prediction.

184

5 Porosity Characterization and Permeability Prediction …

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Chapter 6

Testing and Analysis of Micro Elastic Properties

Abstract Elastic modulus is one of the key mechanical properties which determines the magnitude of the stress and strain of the structures. This chapter reviews the existing methods to measure the elastic modulus of cementitious materials at both macro and micro scales. The macroscale methods fail to measure the local mechanical properties which instead can be accurately measured by both quasi-static and dynamic indentation techniques. The CSM can assess whether the measured results reflect the homogeneous properties of cementitious materials. The MI test with a load larger than 2 N and a holding period with sufficient duration (generally large than 180 s) is necessary to obtain the homogeneous elastic properties of cementitious materials without the creep effect. This chapter provides various details on measuring the local elastic modulus of cementitious materials. Keywords Deconvolution techniques for statistical nanoindentation · Homogeneous properties · Loading rate · Storage modulus · Small-scale mechanical properties

6.1 Introduction Quantification of the mechanical properties of cementitious materials is one of the keys to evaluate the service condition of concrete structures. Among all the mechanical properties, elastic modulus, referring to the ratio of the stress in a body to the corresponding elastic strain, is one basic material parameter which determines the magnitude of the stress and the strain of the structures. It is well known that the elastic modulus of cementitious materials is influenced by many factors, including the mixture proportion, the curing age, the curing condition, etc. (Alexander & Milne, 1995; Dunant et al., 2020; Han & Kim, 2004; Li et al., 2018; Moosavi et al., 2013). Up to date, many experimental methods have been developed to measure the elastic modulus of cementitious materials, however, most of these methods can only provide an assessment of the macroscopic elastic properties of cementitious materials. Due to the heterogeneous features of cementitious materials, the macroscopic deterioration of concrete structures usually originates locally, therefore, it is very important to evaluate the local microscopic properties of cementitious materials. However, the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_6

189

190

6 Testing and Analysis of Micro Elastic Properties

conventional experimental methods mentioned above fail to give an assessment of the local mechanical properties of cementitious materials. Recently, nanoindentation (NI) has become a popular technique to evaluate the local mechanical properties of cementitious materials at the micro scale due to its simple, short testing time, and non-destructive characteristics (Vandamme & Ulm, 2009; Zhang et al., 2014). The small-scale mechanical properties including the indentation modulus and the contact hardness can be determined by analyzing the load (P)depth (h) curve obtained by the instrumented indentation, as discussed in Chapter 3. With the introduction of NI technique into cementitious materials, many researchers have successfully measured the elastic properties of cementitious materials at micro scale (Hu & Li, 2014; Mondal et al., 2007; Wei et al., 2016; Zhang et al., 2014). This chapter is dedicated to summarizing the current research efforts concerning the testing and analysis of the micro elastic properties of cementitious materials. First the existing methods to measure the elastic modulus of cementitious materials at macro scale are reviewed. Then the quasi-static method is introduced, followed by the dynamic method based on indentation technique. Finally, the measured results by the indentation technique are discussed. This chapter is expected to provide an indepth understanding of testing and analysis of micro elastic properties of cementitious materials.

6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale Several distinct methods have been proposed throughout the years that allow the assessment of the elastic properties of cementitious materials. Granja (2016) made a literature review on the various experimental methods that enable to estimate the elastic properties of cementitious materials, which include the cyclic loading methods, the wave propagation methods, the dielectric methods, the resonance-based methods, etc. The following section will provide a brief summary of these methods.

6.2.1 Cyclic Loading Method The cyclic loading method is the most commonly-used method to obtain the elastic properties of cementitious materials. The typical one is the uniaxial cyclic compression method, which has already been standardized by various entities, such as ISO 1920–10 ISO (2010), LNEC E397 (1993), RILEM CPC8 (1975), and ASTM C469 (2006). In this method, a unidirectional stress is applied to a specimen and the corresponding deformation in the same direction will be recorded. Figure 6.1 shows the scheme of the uniaxial cyclic compression method (Granja, 2016). The height/diameter ratio of the cylindrical concrete specimen is 2:1. To obtain the elastic

6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale Front view Load

Section A-A Pivot Metallic ring Setscrew

LVDT A

191

LVDT

Spacing

Specimen height

A

Setscrew Specimen Ф

Metallic ring

Specimen

Specimen

Fig. 6.1 Scheme of the uniaxial cyclic compression method to measure the macro elastic modulus of cementitious materials (Delsaute et al., 2016)

modulus, a series of uniaxial loading/unloading cycles are required to be performed between an initial stress of 0.5 MPa and one third of the average compressive strength (f cm /3) of the specimens at the age of testing. The strains should be measured by at least three displacement sensors attached to the specimen. Normally, the first cycle of the loading/unloading is disregarded and the elastic modulus can be obtained through the analysis of the remaining cycles, which can be expressed as: E=

Δσ Δε

(6.1)

where, E is the elastic modulus of material; Δσ and Δε are the stress difference and strain difference during the loading/unloading cycles, respectively. It should be noted that the uniaxial cyclic compression method is sensitive to the loading rate, the measured elastic modulus will vary with the loading rates. And it is suggested that the strain rate must be between approximately 10−2 and 10−7 s−1 (Bischoff & Perry, 1991). Besides, the classical uniaxial cyclic compression method requires the use of a demoulded specimen, which will prevent such tests from being performed at very early ages. To overcome this limitation, Boulay et al. (2010) developed a method called BTJASPE (acronym for BéTon au Jeune Age, Suivi de la Prise et du module d’Élasticité), which can provide the measurement of elastic modulus by application of compression cycles without demoulding the specimen. The model of BTJASPE test method is shown in Fig. 6.2. It should be noted that, the measured result obtained from the BTJASPE test will be affected by the lateral confinement created by the mould, since the test is conducted on the specimen inside the mould. It is necessary to separate the lateral confinement from the material modulus. The separation of these two parts is based on a relationship between the elastic modulus of the material inside the mould and the experimental stiffness K BT (defined as the slope of the load/displacement curve, which reflects the

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6 Testing and Analysis of Micro Elastic Properties

(a) Mould

(b) + float, 3 LVDT’s and double wall

(c) + Upper bearing

(d) + lower bearing and clamping devices

Fig. 6.2 Model of BTJASPE test method to measure the macro elastic modulus of cementitious materials (Boulay et al., 2013)

combined stiffness of the tested material and the mould) obtained through a finite element calculation performed during the design of the device (Boulay et al., 2010): 2 E = 1.03 × 10−8 K BT + 22.7 × K BT (K BT in N/m)

(6.2)

Another method that also allows evaluation of the elastic modulus by cyclic mechanical tests is the Temperature Stress Testing Machine (TSTM) (Schoppel et al., 1994; Staquet et al., 2012; Klausen et al., 2015; Delsaute et al., 2016). Similar to the BTJASPE method, TSTM can allow the implementation of short loading cycles during the curing process without the need to demould the sample. As shown in Fig. 6.3, TSTM usually is composed of a dog-bone shaped specimen fixed at one end by a steel grip and the other end is connected to a movable steel grip that is controlled by an actuator. In addition, the TSTM machine is capable of controlling the temperature of the specimen by the constant temperature oil bath (Yang et al., 2021). During the experiments, the distance between the two points in the specimen and the applied load are both recorded autonomously, which can be used to evaluate the elastic modulus of specimens. Fig. 6.3 General scheme of a TSTM to measure the macro elastic modulus of cementitious materials (Klausen et al., 2015)

Dog-bone shaped specimen

TSTM

6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale

193

6.2.2 Wave Propagation Method The wave propagation method is also called the acoustic method. The idea of using the measurement of the wave propagation velocity to determine the start and end of the setting period in cement-based materials was first described in the 1940s (Jones, 1949). Currently, the wave propagation methods can be further divided into the wave transmission method and the wave reflection method. The principle of the wave transmission method is based on that the velocity of a propagating wave through a medium depends on the elastic properties and density of the medium (Meyers & Chawla, 2008). Both compressional waves (P-waves) and shear waves (S-waves) can be used in the wave transmission method (Van Den Abeele et al., 2009). Several studies on the cementitious materials reveal that the velocity of the propagated waves is sensitive to the solid hydrate formation, and the variation of velocity observed over time corresponds to the cement hydration and the hardening process (Chotard et al., 2001; Smith et al., 2002). The application scheme of the ultrasound transmission method is shown in Fig. 6.4. In this method, an ultrasonic wave is generated at one side of the sample, which will be transmitted through the material and ultimately received at the opposite side of the specimen. Both the generated and the transmitted signals are measured to determine the propagation time of the waves through the material. In the wave transmission method, the velocity of wave propagation (V ) can be calculated based on the following equation: V =

Δl Δt

(6.3)

where, Δl is the distance between the probes; Δt is the wave propagation time. When the velocity of wave propagation is determined, one can assess the elastic modulus of the tested material by assuming that the tested material is a homogeneous and isotropic media: Fig. 6.4 Application scheme of ultrasound transmission method to measure the macro elastic modulus of cementitious materials (Granja, 2016)

Specimen Receiver Transmitter

Signal generator

Propagation time measurement Graphical interface

Signal recorder

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6 Testing and Analysis of Micro Elastic Properties

/ ⎧ ⎪ (1 − vd )E d ⎪ ⎪ ⎪ ⎨ V P = (1 + vd )(1 − 2vd )ρ / ⎪ ⎪ Ed ⎪ ⎪ ⎩ VS = 2(1 + vd )ρ

(6.4)

where, V P and V S are the compressional wave velocity and the shear wave velocity (m/s), respectively; ν d is the dynamic Poisson’s ratio, E d is the dynamic elastic modulus (Pa); ρ is the mass density of the tested material (kg/m3 ). It should be noted that concrete may not always be considered as a homogeneous and isotropic medium, particularly at early ages (Tuleubekov, 2012). The heterogeneity introduced by the presence of large aggregates in the mixture may lead to higher velocities than the expected (Granja, 2016). In addition, since the dynamic Poisson’s ratio of cementitious materials is unknown, in order to determine the dynamic elastic modulus, both the velocities of compressional waves and the shear waves need to be assessed (Boumiz et al., 1996), which will cause an increase in the complexity of these methods. It is well known that a wave propagating an interface between two different mediums is partly transmitted and partly reflected (Reinhardt & Grosse, 2004). The decrease in amplitude of the reflected wave depends on the reflection coefficient, which is a function of the acoustic properties of the materials at the interface. Thus, it can be used to determine the elastic modulus of materials (Chung et al., 2012; Suraneni et al., 2015; Voigt et al., 2006). The operation scheme of this experimental method applied to the study of cement-based materials is shown in Fig. 6.5. In this method, a plate made of a buffer material is placed in contact with the fresh cementbased material. Then an ultrasonic wave reflection probe is attached to the plate and a S-wave pulse is emitted at the buffer material surface. The pulse is almost completely reflected in the interface between the buffer-material and the tested material when the tested material is in the liquid state where S-waves do not propagate (t 0 ). Therefore, the reflection coefficient is close to one. As the hydration of cement progresses and a rigid skeleton begins forming, shear waves are able to propagate through the cementitious material. Thus, a portion of the shear wave can be transmitted through cementitious material, resulting in the loss of the reflection signals during the reflection process (t 1 ). Consequently, the reflection coefficient starts to decrease. This process evolves until the end of the hardening process (t 2 ). It has been reported that the accuracy of the reflection technique depends on the ability to detect changes in characteristics of the reflected ultrasonic wave, which varies with the acoustic impedance of the cementitious material (Wang et al., 2010). A decrease of the acoustic impedance of the material can improve the applicability of this method. A buffer material with low acoustic impedance such as poly-methyl methacrylate (PMMA) can be used (Subramaniam & Lee, 2003), which can make the changes in the shear modulus of the cementitious material be very sensitively detected through the setting process. The ultrasonic wave reflection method can be very useful especially in construction sites since it only needs one single surface of the

6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale

Buffer material UWR Probe Reflected wave

Cement-based material

Transmitted wave

Buffer material

Cement-based material

UWR Probe

Transmitted wave

Reflected wave

Cement-based material

UWR Probe

Transmitted wave

Reflected wave Curing time

t1

t0

Buffer material

195

t2

Fig. 6.5 Application of the ultrasonic wave reflection (UWR) method to measure the macro elastic modulus of cementitious materials (Granja, 2016)

material for testing. However, it is noted that the results obtained from the ultrasonic wave reflection method reflect the elastic modulus of surface materials rather than the internal properties of structures. In addition, this method is highly influenced by the presence of large aggregates in the vicinity of the interface. The applicability of this method to measure the properties of concrete is therefore arguable (Voigt, 2005).

6.2.3 Dielectric Method The dielectric properties of cementitious materials can provide valuable information about the material mechanical properties, which are related to the response of the material when subject to an electric field. Dielectric properties (εr ) are always presented relative to the dielectric constant of the vacuum (ε0 ), as expressed in Eq. (6.5), which can be determined by the electrical permittivity (ε ) and the electrical conductivity (σ ). As shown in Fig. 6.6a, the permittivity of the cementitious materials can be presented as a capacitor, while the conductivity can be represented by a resistor in parallel. A prototype of the dielectric measurement system to assess the mechanical properties of cementitious materials is shown in Fig. 6.6b. '

εr = ε − j

(

σ 2π f ε0

) (6.5)

where, ε0 = 8.854 × 10−12 F/m; f is the frequency of the alternating electrical field; j is the imaginary unit, and j2 = −1. Cementitious material usually has a certain amount of water in it, which will be consumed during the hydration process or escape to the drying environment. The change of the microstructures of cementitious materials (e.g. the water content and the volume fraction of solid phase) will result in the variation of the mechanical properties. It has been reported that the pore water has a high permittivity and a high conductivity, which does not occur with the cement particles (Van Beek & Milhorst,

196

6 Testing and Analysis of Micro Elastic Properties (a)

(b)

Conductivity Permittivity

Dielectric sensor 20MHz Resistor

Capacitor

Temperature probe

Electrodes

Fig. 6.6 a Dielectric properties of the concrete represented as a resistor and a capacitor, and b prototype of the dielectric measurement system to measure the macro mechanical property of cementitious materials (Van Beek & Milhorst, 1999)

1999). The difference of the dielectric properties of the water and the solid phases makes it possible to detect the mechanical property changes. The dielectric methods can assess the elastic properties of cementitious materials immediately after mixing, furthermore, it can provide continuous measurement of the elastic modulus without disturbing the samples. However, this method requires the experimental establishment of correlations between the dielectric and the mechanical properties for each mix.

6.2.4 Resonance-Based Method Based on the resonance principles, many resonance-based methods have been developed to measure the elastic modulus of cementitious materials, which can be classified into three main categories, i.e. the classical resonance method, the electromechanical impedance (EMI) method, and the ambient vibration method. The classical resonance method was first developed by Powers (1938). When the vibration of specimen with known dimensions is considered, its natural frequency of vibration is mostly related to its geometry, support conditions, density, and elastic modulus of the material. Therefore, the elastic modulus of a material can be determined from the measurement of the natural vibration frequency, provided that all other parameters are known. Currently, the classical resonance methods have been standardized by ASTM C215 (2002). The tests can be conducted in several alternative configurations, which are shown in Fig. 6.7. Among these configurations, the most common test setup is the longitudinal mode of vibration. Although the classical resonance method is relatively simple, it cannot allow the continuous measurement starting from the fresh state. The electromechanical impedance (EMI) method was first developed by Liang et al. (1996) for the assessment of damage in structures, which is based on the concept of mechanical impedance. The mechanical impedance of a point on a structure is the relationship between force applied at such point and the resulting velocity in the same

6.2 Elastic Modulus Measurement of Cementitious Materials at Macro Scale (a) Transverse Mode

197

(b) Longitudinal Mode

(c) Torsional Mode

Fig. 6.7 Position and direction of the impact and accelerometer for the various vibration configurations of the sample to measure the macro mechanical property of cementitious materials (ASTM C215, 2002)

point, which has a direct relationship with the physical parameters such as the elastic modulus and the density (Shin et al., 2008; Wang & Zhu, 2011). The mechanical impedance of the tested specimen can be assessed by using both the direct and the inverse piezoelectric effects of the piezoelectric lead zirconate titanate (PZT). When a fixed alternating electric field trigger the PZT attached to the specimen, a small deformation is produced in the PZT as well as in the specimen area where the PZT is attached. This excitation is then reflected as a mechanical vibration and is transferred back to the PZT which transforms in an electrical response (Park & Inman, 2005). Thus, any change in the mechanical properties of the material will cause a change in the measured response of PZT. The electromechanical model describing the above process is shown in Fig. 6.8a. One typical photo of the specimen used in an experiment with PZT is shown in Fig. 6.8b. The electromechanical impedance method can provide a continuous assessment of the elastic properties of cementitious materials. However, pre-calibrated correlations between the property and the electrical admittance are needed in order to estimate the mechanical properties of the tested material. The ambient vibration method is a variant of the classical resonance method, which is initially proposed by Azenha et al. (2010). In this method, the evolution of the flexural resonant frequency of the first mode of vibration can be obtained based on the continuous non-parametric modal identification of a composite beam (composed by the mould filled inside with the material to be tested) with known geometry and support conditions. Then the evolution of elastic modulus can be directly and quantitatively estimated without any kind of ambiguity of user dependency in the data processing (Granja, 2016). In the ambient vibration method, the specimen is usually excited by the environmental noise (e.g., wind, people walking, noises

198

6 Testing and Analysis of Micro Elastic Properties

(a)

(b)

PZT

m

Material c

k

Fig. 6.8 a Mechanical model of an assembly of a PZT (Granja, 2016), and b photo of the specimen used in an experiment with PZT made by Shin et al. (2008) to measure the macro mechanical property of cementitious materials

from construction site, etc.), which can conceptually be assumed to have an average behavior of white noise. Figure 6.9 shows a flowchart with a brief overall description of non-parametric data processing and modulus estimation. As shown in Fig. 6.9, the vibrations at the mid span of the beam are collected by the accelerometer in packages of 900 s. Then the collected data is converted from the time domain to the frequency domain through the Welch procedure (Welch, 1967), which can result in the normalized power spectrum density (NPSD) of each measured package of data. The NPSDs can be included side-by-side in a colored frequency vs. time surface. Finally, the resonance frequencies of the first vibration mode are identified through the highest peak in each amplitude spectrum. When the geometry and boundary condition is given, one can obtain the unique equation between the elastic modulus and the resonance frequency, by which the elastic modulus of the cementitious materials can be easily obtained based on the resonance frequency determined experimentally. In order to obtain a continuous evolution of the resonance frequency of the composite beam, this whole process can be repeated every 60 min.

6.3 Small-Scale Mechanical Properties from Indentation Technique Unlike the homogeneous materials, the measured mechanical properties may vary with the indented location due to the heterogeneous characteristics of the cementitious materials, which is more obvious for NI test with tiny probe and very small indentation force. Currently, to distinguish the mechanical properties of different phases in cementitious materials, several methods including the least-square estimation, the maximum likelihood estimation, the clustering analysis method, and the discrete measurement have been proposed to process the data acquired from NI test, which will be introduced as followings.

6.3 Small-Scale Mechanical Properties from Indentation Technique

199

Fig. 6.9 Non-parametric data processing and stiffness estimation flow chart based on the ambient vibration method to measure the macro mechanical property of cementitious materials (Granja, 2016)

6.3.1 Least-Square Estimation The deconvolution technique refers to the process of fitting a number of probability density functions (PDF) to the experimental frequency plot (normalized histogram) of the measured quantity (Constantinides et al., 2006). Vandamme and Ulm (2008) found that an automated process by the least-square estimation can be successfully utilized to extract mean property values, volume fractions, and standard deviations of the mechanical properties of individual phase in cementitious materials. The automated process is shown as following (Randall et al., 2009). First it is necessary to find the best choice of distribution function uniquely defined by its statistical moments (e.g. the mean value and the variance) for each peak in the frequency plot. If the measurements and the material are perfect, then for infinitely shallow indentations one would expect the peaks to be infinitely sharp with each peak being characterized by its first moment (mean value) only. However, there are

200

6 Testing and Analysis of Micro Elastic Properties

several reasons for which the peaks of the histogram are not infinitely sharp, therefore requiring the use of moments of higher order: i.

The measurements from NI tests exhibit some noise which is considered as random and creates a spread of peaks, but no asymmetry. ii. Each phase has its own intrinsic variability, causing a spread of the peaks. iii. The indentations are not infinitely shallow, even for an ideal material and ideal measurements, due to their finite depth, some indentations will mechanically solicit two (or more) phases simultaneously, resulting in a composite property. For the sake of simplicity, the distribution is chosen so that all the standardized central moments of order higher than the second are zero, i.e. the distribution function of the measured properties of each phase in cementitious materials follows the Gaussian distribution: ( ) −(X − μ j )2 2 p(X ; μ j , s j ) = / exp (6.6) 2s 2j 2π s 2 j

where, p(X ; μ j , s j ) is the value of the theoretical probability density function of the measured properties of the single phase; μj and sj are the mean and the standard deviation of X = (M, H) of the phase j = 1,…, n, which are still unknown and can be fitted by the experimental results. And the theoretical cumulative distribution function (CDF) of the measured properties of the single phase, i.e. D(X ; μ j , s j ), can be expressed as: D(X ; μ j , s j ) = /

2

∫X

2π s 2j −∞

(

) −(x − μ j )2 exp dx 2s 2j

(6.7)

Then the deconvolution with the aim to determine the mean and the standard deviation of X = (M, H) of each phase and the corresponding volume fraction begins with the generation of the experimental cumulative distribution function (CDF). By denoting the number of indentation tests performed on a specimen as N, {X i }i=1,…,N is used to represent the sorted values of the measured property which is to be deconvoluted, where X i can be the indentation modulus, M, the contact hardness, H, or any other property (creep properties, packing density, etc.) obtained from the i th indentation test. And the N points of the experimental CDF of X, denoted by DX , are obtained explicitly from: D X (X i ) =

1 i − ; for i ∈ [1, N ] N 2N

(6.8)

It is assumed that the heterogeneous material is composed of n material phases with sufficient contrast in mechanical phase properties. The j th phase occupies a volume fraction f j of the indented zone. The unknown quantity (f j , μj , sj ) in

6.3 Small-Scale Mechanical Properties from Indentation Technique

201

) ( the theoretical CDF of the j th Gaussian distributed phase (D X i ; μ j , s j ) can be determined by the least-square estimation: ⎞2 ⎛ N Σ Σ n Σ ) ( ⎝ min f j D X i ; μ j , s j − D X (X i )⎠ i=1

X

(6.9)

j=1

In order to ensure that the phases have sufficient contrast in properties, and to avoid the case where two neighboring Gaussians overlap, the optimization problem should be additionally constrained by the following equation: μ j +s j ≤ μ j+1 − s j+1

(6.10)

In addition, the volume fractions of the different phases should sum to one, which can be expressed as: n Σ

fj = 1

(6.11)

j=1

Sorelli et al. (2008) characterized the nanomechanical properties of the phases governing the UHPC microstructure by means of the statistical NI technique and the least-square estimation. In order to facilitate the deconvolution, a total population of 700 indentations were made on the test sample. As shown in the SEM images (Fig. 6.10a), the hardened UHPC microstructure can be considered to be composed of porosity, LD CSH, HD CSH, quartz powder, quartz sand, cement clinker, and fiber. By minimizing Eq. (6.9) for the above phases, one can obtain the mechanical properties and the corresponding volume fraction of each phase. The fitted frequency plots for indentation modulus (M) and hardness (H) are displayed in Fig. 6.10 b and c, respectively. It is shown that the mean values of the indentation modulus of the LD and HD CSH phases are in good agreement with the characteristic values found for different cement pastes. The intrinsic elastic properties differ up to 15%, which is a fairly good agreement considering the experimental measurements’ errors arising from material variability and instrumented calibration. In summary, the idea of the least-square estimation to obtain the mechanical properties and the volume fraction of each phase in cementitious materials is clear and straightforward. However, the accuracy of this method depends on the selection of the initial values of the model parameters and the estimation of phase numbers. It has been shown that the PDF plot construction mainly depended on the selection of the bin size when the least-square estimation was used to fit the experimental PDF of the NI test (Ulm et al., 2007). In addition, as reported by Lura et al. (2011), numerical instability may occur when fitting experimental PDF of the indentation modulus by using the multiple Gaussian functions.

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6 Testing and Analysis of Micro Elastic Properties

Fig. 6.10 a SEM images of the UHPC microstructure, and the overall PDF evaluated from the deconvolution technique for b the indentation modulus and c the contact hardness (Sorelli et al., 2008)

6.3.2 Maximum Likelihood Estimation The maximum likelihood estimation (MLE) has also been employed to process the measured results by indentation tests (Davydov et al., 2011; Gao & Wei, 2014; Luo et al., 2020). However, unlike the least-square estimation, MLE gives the mechanical property values and the fraction of each phase based on the Expectation–Maximization (EM) algorithm. In MLE, it is assumed that the mechanical properties of each phase follow the Gaussian distribution, which is the same as the least-square estimation. However, the use of MLE doesn’t require choosing of bin size and the starting parameters which can significantly affect the fitting results. One of the methods to reduce the influence of the starting parameters is to seek the global optimization by conducting an iteration procedure with multiple random starting values (Gautham & Sasmal, 2019). In the MLE, the probability density function (PDF) of the multi-phases is the sum of distributions of all phase, which can be expressed as:

6.3 Small-Scale Mechanical Properties from Indentation Technique

P D F(X i ) =

n Σ j=1

203

] [ )T −1 ( ) 1 1( (6.12) fj | |1 2 exp − X i − μ j Σ j X i − μ j 2 (2π ) D/ 2 |Σ j | /

where, D is the dimension of the normally distributed quantity X, for example, D = 2 for X = (M, H); Σ j is the covariance matrix of phase j, which describes the correlation between various dimensional variables, it is noted that diagonal matrix stands for no correlations between variables in different dimensions, while full matrix indicates existing correlations within them; μk and f k are the mean value and the volume fraction of the k th Gaussian distribution, respectively. The model parameters can then be taken by maximizing the function Q as the new probable value, which is given by (Paalanen et al., 2006): Q(θ ; θi ) = E Y [ ln ϕ(X, Y ; θ )|X ; θi ]

(6.13)

where, θ = {μ1 , Σ 1 , f 1 , ..., μn , Σ n , f n } is the parameters of Gaussian models; θi is the i th iteration result of parameters. Y is the unknown feature that refers to the category of all data Π N points in this case. E Y is the expect value of Y determined by θi . ϕ(X, Y ; θ ) = i=1 p(xi , yi ; θ ) is the likelihood function of the test data, and the iterative formula of θ is expressed as: θi+1 = arg max Q(θ ; θi ) θ

(6.14)

For a given initial value of θ0 , after several iterations, the probable value would be achieved as the optimal solution when it changes rarely from the result of the last iteration. By the MLE method, each experimental data will be classed into the group that it belongs to with the maximum posterior probability, the normal distribution parameters (i.e. the mean value and the covariance matrix) of the mechanical properties, and the corresponding volume fraction of each phase can be determined after conducting the maximum likelihood estimation. Gao and Wei (2014) employed the maximum likelihood estimation to assess the microscale mechanical properties of cement pastes blended with slag (replacing pure cement at 0%, 50%, and 70% by weight, denoted as 3O, 3S50, and 3S70, respectively) from the grid NI tests. The mechanical properties including the indentation modulus and the contact hardness, the packing density, and the volume fraction of each phase can be calculated based on the measured NI results of 225 indents, as shown in Fig. 6.11. Recently, Chen et al. (2021) compared the maximum likelihood estimation results with those of the least-square estimation and found that the elastic modulus and the contact hardness of LD CSH and HD CSH obtained by these two methods are well agree with each other, which verifies the feasibility of these two methods to obtain the micromechanical properties of each phase in cementitious materials from a large number of NI data.

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6 Testing and Analysis of Micro Elastic Properties

Fig. 6.11 Probability density functions of indentation modulus, contact hardness, packing density, and the mixture Gaussian distribution calculated by maximum likelihood estimation (Gao & Wei, 2014)

6.3.3 Clustering Analysis Method Both the least-square estimation and the maximum likelihood estimation should assume that the distribution of the test data for various phases follows the combination of Gaussian distribution. However, the Gaussian distribution assumption is not always necessarily the real case. For example, Liu et al. (2018) found that both the indentation modulus and the contact hardness of hydrated cement paste follow a lognormal distribution instead of the Gaussian distribution. Recently, Chen et al. (2021) employed the clustering analysis method to process the indentation data where there is no need to presuppose any distribution laws followed by the mechanical properties of each phase. The clustering process is to partition a data set into several clusters so that the data within a cluster are more similar to each other than data in

6.3 Small-Scale Mechanical Properties from Indentation Technique

205

different clusters. Since the mechanical properties of individual phase in cementitious materials are distinguished from each other, it is possible to employ the clustering analysis method to assign the indented points to the corresponding phases. The clustering method offers an alternative mathematical tool to group the mixed data by a similarity criterion. K-mean clustering, a classical partitional clustering algorithm, has been applied widely (Jain, 2010), in which the Euclidean distance is adopted to measure the similarity. For a data set X consisting of n points with m features and K clusters, the Euclidean distance between data points and the adjacent cluster center can be expressed as: (

dis X i , C j

)

[ | m |Σ =] (X it − C jt )2

(6.15)

t=1

where, X i is the i th data point, i = 1–n; C j is the j th cluster center, j = 1–K; X it denotes the tth feature of X i , while C jt represents the tth feature of C j . For a given initial value C 0 , all data points are assigned to the nearest cluster center to get K clusters {S 1 ,S 2 ,…,S K }, then the cluster center for each cluster will be recalculated based on the data points belonging to the corresponding cluster until a stable grouping with cluster centers is obtained after enough numbers of iterations. After each iteration, the cluster center coordinate can be updated as the mean value of the incorporated data points in each cluster, which can be expressed as: Σ Ck =

X i ∈Sk

|Sk |

Xi

(6.16)

where, |Sk | is the number of data points in cluster k. In order to carry out the iterative operation of clustering analysis, the initial value of C 0 must be given in advance. It has been reported that K-means clustering would be sensitive to the abnormal data (Dhamecha, 2021), the low amount of data, and the highly dispersed data, which may lead to large deviations of the cluster center. To solve this problem, K-medoids clustering method (Park & Jun, 2009) can be used, which takes the real data points as the cluster centers. Figure 6.12 show the clustering results of indentation data of cement sample with w/c ratio = 0.2 at 3 days age based on the unnormalized data and the normalized data. It is necessary to normalize the data to balance the weight of the elastic modulus and the hardness in the distance metric function. ⎧ E i − E min ⎪ ⎪ Ei' = ⎨ E max − E min (6.17) Hi − Hmin ⎪ ⎪ ⎩ H i' = Hmax − Hmin

6 Testing and Analysis of Micro Elastic Properties

Elastic modulus (GPa)

120

(a) Unnormalized

100 80 60 40 20

120 Elastic modulus (GPa)

206

80 60 40 20 0

0 0

2 4 6 8 10 Indentation hardness (GPa)

(b) Normalized

100

0

2 4 6 8 10 Indentation hardness (GPa)

Fig. 6.12 Clustering results of indentation data of cement sample with w/c ratio = 0.2 at 3 days age based on the unnormalized data and the normalized data (Chen et al., 2021)

where, E i and H i are the elastic modulus and the contact hardness of the ith indent; E min and H min are the minimum elastic modulus and contact hardness of all indents; E max and H max are the maximum elastic modulus and contact hardness of all indents. By comparing Fig. 6.12a and b, it can be seen that the clustering results can be refined by the normalization treatment (Eq. (6.17)), especially in high values data region. The reason lies in that the elastic modulus is generally about ten times larger than the indentation hardness in values, direct use of real values in Euclidean distance will lead to unreasonable results.

6.3.4 Discrete Measurement In grid NI test, the test area is chosen randomly which may increase the uncertainty of the measured results due to the fact that the indent may be located at the interface between different phases. In order to reduce the experimental uncertainty of the grid indentation method and to avoid the deconvolution process of the NI results, several authors have used discrete measurement to process the NI results (Mondal et al., 2007; Davydov et al., 2011; Liang & Wei et al. 2020c). In the discrete measurement, the position of indentation points is chosen in advance such that NI would probe a particular phase and thus increase the accuracy of the obtained results. The discrete measurement on the HD CSH around the clinker for alite sample with w/c ratio = 0.4 by Davydov et al. (2011) will be illustrated. As shown in Fig. 6.13, HD CSH can be identified as a rim around the unhydrated clinker on a BSE image, then the rim regions around the unhydrated clinker were detected in the optical microscope equipped in the NI machine (e.g. Hysitron Tribo Indenter). A precise navigation of the NI points was done in an atomic force microscope. However, it is noted that the accuracy of the indenter on such areas is not sufficient to hit exactly the prescribed points. As shown in Fig. 6.13, several indented points were located outside of the HD CSH rim (e.g. the indents in the circle symbols). Therefore, the

6.3 Small-Scale Mechanical Properties from Indentation Technique

207

area should be scanned again after each test and only indents which really went into HD CSH were used in the analysis. Figure 6.14 clearly shows three distinct peaks in the histogram, from left to right the peak is attributed to the HD CSH with big capillary pores nearby (around 13 GPa), the HD CSH (around 30 GPa), and a mixture of HD CSH and CH (around 40 GPa), respectively. The experimental results reveal the complexity of the properties of the HD CSH rim around the unhydrated grain, confirming the hypothesis of the mixture of HD CSH and CH.

HD CSH

HD CSH

HD CSH

Fig. 6.13 BSE (bottom; before indentation) and AFM (top; after indentation) images showing the regions selected for HD CSH in alite paste (Davydov et al., 2011)

Fig. 6.14 Indentation modulus histogram on HD CSH rim around the clinker for alite sample with w/c ratio = 0.4, hmax = 230 ± 49 nm by discrete indentation (Davydov et al., 2011)

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6 Testing and Analysis of Micro Elastic Properties

6.4 Dynamic Method Based on Indentation Technique 6.4.1 Pixel Spacing Effect on Storage Modulus An interaction may exist between the indentation stress fields if the distance between the two adjacent indents is too close (Constantinides & Ulm, 2007). The close pixel spacing in scanning probe microscope (SPM) modulus mapping may also affect the measured mechanical properties of individual phase, since it is influenced by the deformation of the adjacent pixels. Therefore, this section will discuss the influence of the pixel spacing on the measured storage modulus. Figure 6.15a shows the SPM images of a target area which includes a slag grain, a dicalcium silicate (C2 S) clinker, and the hydration products in paste samples. Two small areas denoted as Area I and Area II as shown in Fig. 6.15a at the edges of the slag grain and the C2 S clinker were selected to assess the influence of pixel spacing on the measured storage modulus. Herein, the pixel spacing is determined by dividing the image size by the amount of pixels in one dimension, i.e., 256. Therefore, the pixel spacing in Fig. 6.15a is 35 µm/256 = 137 nm and it is 7 µm/256 = 27 nm in Fig. 6.15b and c. The variations of the storage modulus with the distance along the same scanning lines measured by the different pixel spacing are shown in Fig. 6.15d and e. It is seen from Fig. 6.15d and e that the measured storage modulus with the pixel spacing of 137 nm and 27 nm are almost the same, which indicates that the pixel spacing has minor effect on the storage modulus. Based on the Hertzian contact theory, / the theoretical influencing radius of the

indenter tip can be expressed as a = 3FR (where, F is the constant force; R is the 2K s tip radius of indenter; K s is the stiffness of the contact which can be found in Fig. 6.11 in Sect. 3.3.2). Since the storage modulus of pastes shown in Fig. 6.15 is 15–90 GPa, the influencing radius of the indenter tip is calculated as 24–43 nm when the contact force is 4 µN. It can be seen from Fig. 6.15b and c that the pixel spacing is about 27 nm, which is less than the influencing radius of the indentation tip. However, the close pixel spacing doesn’t influence the measured storage modulus, since the storage modulus obtained in Fig. 6.15b and c are almost the same as that obtained in Fig. 6.15a. This indicates that the sample deformation under the indenter tip is elastic, therefore, the elastic deformation will totally recover without residual local stress or permanent deformation when the applied force is removed. Therefore, the adjacent pixels do not affect the measured storage modulus in the SPM measurement.

6.4.2 Effect of Surface Roughness on Storage Modulus It has been reported that the surface roughness will affect the measure results by NI test, a surface roughness criterion for measuring the small-scale mechanical properties of cementitious materials by NI has been developed by Miller et al. (2008). In

6.4 Dynamic Method Based on Indentation Technique

209

35μm

Area II

7μm

7μm

35μm

C2S

Area I

Area II (c)

7μm Scanning line

Area I (b)

5μm

7μm

(a)

Slag

Data obtained in (b) with pixel spacing of 137nm Data obtained in (c) with pixel spacing of 27nm (e) 80 Storage modulus (GPa)

Storage modulus (GPa)

(d) 80 60 40 20 0

60 40 20 0

0

1 2 3 4 5 6 7 Distance (μm)

0

2

8 4 6 Distance (μm)

10

Fig. 6.15 Effect of the pixel spacing on the variability of storage modulus: a modulus image of target area with pixel spacing of 35 µm/256 = 137 nm; b and c modulus images of area I and area II in a with pixel spacing of 7 µm/256 = 27 nm; d and e storage modulus along the same scanning lines for large and small pixel spacing (Gao et al., 2018)

this criterion, the ratio of the root-mean-squared average of the surface roughness Rq to the maximum indentation depth (hmax ) should be less than 0.2 in an area with the edge size of 200hmax . It is expected that a surface roughness of less than 50 nm in the randomly selected area with size of 50 × 50 µm is required to obtain the valid data in NI test. However, it is noted that the roughness would vary with the location even on the same sample due to its inherently heterogeneous features. Both the surface elevation of the target area and the storage modulus of the same area were measured simultaneously by the SPM technique, which makes it possible to quantify the influence of surface roughness on the measured storage modulus in exactly the same positions. Figure 6.16a and b show the modulus image and the topography of an area including C3 S clinkers and hydration products, which were meshed into 16 × 16 elements with size of about 2.2 × 2.2 µm. The meshing process can be seen in the study of Trtik et al. (2012), which will provide the data matrix that is required in the calculation of Rq (Miller et al., 2008). Figure 6.16c shows the storage modulus and the surface roughness of the meshed elements. The surface roughness of the outer product (OP) is 10–200 nm and exhibits the most significant variation. The reason lies in that OP will form in the position where water will occupy initially, however, OP won’t occupy fully the space during the hydration and thus pores and voids will

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be formed. As a result, the surface roughness of the OP phase is large due to the existence of pores and voids. However, the surface roughness has an insignificant influence on the variation of the storage modulus of both OP and IP. The storage modulus of OP and IP is 17–31 GPa and 31–37 GPa, respectively, which indicates that the surface roughness of less than 200 nm for the 2.2 × 2.2 µm element is sufficiently low to conduct the SPM mapping test. Figure 6.16d shows the average values of Rq of the different phases in the entire image in Fig. 6.16b, which ranges from 30 to 50 nm. Although the surface roughness of the clinker is smaller than that of OP and IP, the standard variation of the storage modulus of the clinker is remarkably large, which suggests that the surface roughness is not the main reason for the large variation of storage modulus of the clinker. The reason for the insignificant influence of surface roughness on the stability of the measured storage modulus is illustrated in Fig. 6.17. It is seen from Fig. 6.17b that the surface elevation along the scanning line is 100–300 nm and the variation of (a)

(b)

nm

IP

OP

150 100

IP

C3S

50 0 -50

OP C3

(c)

5μm

5μm

-100 -150

(d)

Fig. 6.16 Influence of the surface roughness on the variability of storage modulus of individual phases in OPC paste: a and b meshing the modulus image and topography into 16 × 16 elements; c relationship between the surface roughness and the storage modulus of individual phases quantified in every element; d mean value and standard variation of storage modulus and surface roughness of individual phases (Gao et al., 2018)

6.4 Dynamic Method Based on Indentation Technique

C3S HP Pore HP

C3S

Scanning line

(b) 200

Elevation Amplitude

0 2 0

7

14

21

28

Pore

HP

35

-200 C3S -400

(c) Indenter tip

Indenter tip displacement = 2nm

HP

C3S

Distance (μm) (d)

Surface elevation = 10nm

4 Displacement amplitude (nm)

5μm

Surface elevation (nm)

(a)

211

0

Indenter tip Surface elevation = 200nm

Indenter tip displacement = 2nm

Fig. 6.17 a A topography of target area in OPC paste; b surface elevation and displacement amplitude of the indenter tip along the selected line in a; and schematic representations of the small displacement amplitude of the indenter tip on the rough surface with c small and d large elevations (Gao et al., 2018)

the surface elevation is mainly related to the pores and the interfaces between the C3 S clinker and the hydration products. On the other hand, the displacement amplitude of the indenter tip is only a few nanometers. The above results indicates that the differences of the elevation of the sample surface is two orders of magnitude more than the displacement amplitude of the indenter tip. The schematic representations of the small displacement amplitude of the indenter tip on a rough surface with small and large elevation are illustrated in Fig. 6.17c and d, respectively. When the variation of surface elevation ranges from dozens of nanometers to hundreds of nanometers, the influence of the surface elevation on the indentation displacement is the same due to the very small displacement amplitude of the indenter tip. Thus, the surface roughness (10–200 nm in this section) has minor effect on the measured storage modulus.

6.4.3 Effect of Magnitude of Applied Load: Quasi-Static and Dynamic Loads To select a proper load to accurately measure the mechanical properties of individual phase, the influence of the magnitude of applied load on the displacement amplitudes of the indenter tip and the storage modulus will be discussed by using four sets of

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6 Testing and Analysis of Micro Elastic Properties

loads including 2 ± 1.5, 4 ± 1.5, 4 ± 3.5 and 8 ± 3.5 µN (quasi-static ± dynamic). A target area covering a slag grain and the hydration products in the slag-blended paste shown in Fig. 6.18a was used to examine the influence of different loads magnitudes on the measured results. A vertical line in the target area was selected to extract the line data. The sharp contrast of both the displacement amplitude of the indenter tip and the measured storage modulus between the slag grain and the hydration products will be used to quantify the effect of the applied load. (a)

20

HP

10

Slag

5 5μm

HP

0

Displacement amplitude (nm)

3

Desired load of 4 3.5 μN which achieves the largest displacment amplitude 2.3

Slag HP

0.3nm

2.5 2 1.5 0.4nm 1.5

0.2nm

1.5 0.9

0.2nm

1 1.1

0.5 0.6

0 2

0.3nm 0.4

1.5

4

0.3nm 0.7

0.1nm

1.5

4

Standard variation of storage modulus (GPa)

2 4 4 8

25

Slag

20 15

(c)

3.5

8

Slag HP

100

63 23GPa

Desired load of 4 3.5 μN which achieves the smallest 72 modulus variation 27GPa 70 16GPa

80 60

85 22GPa

40 20

0.1nm

3.5

20 40 60 80 100 Storage modulus (GPa)

120

0

Magnitude of applied load (quasistatic±dynamic μN) (d) 30

0

1 2 3 Displacement amplitude (nm)

Storage modulus (GPa)

0

(b)

Distance (μm)

15

25

2

8GPa 28

1.5

4

9GPa 27

5GPa

1.5

3.5

4

33

6GPa

8

3.5

Magnitude of applied load (quasistatic±dynamic μN)

1.5μN 1.5μN 3.5μN 3.5μN

Desired load

10 5

HP

0 0

1

2

3

Mean value of displacement amplitude (nm)

Fig. 6.18 a Variation of the displacement amplitudes and the storage modulus along the selected line under the load of 4 ± 3.5 µN; effect of magnitudes of loads on b the displacement amplitude, and c the storage modulus of slag grain and hydration product (denoted HP); d decease of the standard variations of storage modulus with the mean value of displacement amplitude of slag grain and HP (Gao et al., 2018)

6.4 Dynamic Method Based on Indentation Technique

213

Effect on Displacement Amplitudes Figures 6.18b and c show the displacement amplitudes in the same position induced by SPM mapping test conducted on the area shown in Fig. 6.18a under different loads of 2 ± 1.5, 4 ± 1.5, 4 ± 3.5 and 8 ± 3.5 µN (quasi-static ± dynamic). It can be seen from Fig. 6.18b that the displacement amplitude of the indenter tip is influenced more by the dynamic force than the quasi-static force. For example, under the same quasi-static force of 4 µN, the average displacement amplitude on the slag grain and the hydration products induced by the dynamic load of 3.5 µN is about 1.1 nm and 2.3 nm, respectively. However, it is only 0.4 and 0.9 nm on the slag grain and the hydration products when the sample is subjected to a dynamic load of 1.5 µN. A sharp contrast of displacement amplitudes in different phases can be achieved by using a large dynamic load of 3.5 µN and a small quasi-static load of 4 µN. Therefore, SPM test with a load of 4 ± 3.5 µN can be used to identify different phases in cementitious materials. Effect on Storage Modulus Storage modulus is an important parameter in SPM mapping that can be used to differentiate phases in cementitious materials. It can be seen from Fig. 6.18c that the measured storage modulus of both the slag grain and the hydration products increase slightly with increasing quasi-static load. And the storage modulus of the slag grain is obviously different from that of the hydration products, which can be used to identify each other under all four sets of loads. The standard variation of the storage modulus of individual phases is found to be influenced significantly by the dynamic load. For example, the standard variation of the storage modulus measured by using a fixed quasi-static load of 4 µN decreases from 27 to 16 GPa for the slag grain when the dynamic load increases from 1.5 to 3.5 µN, and it decreases from 9 to 5 GPa for the hydration products. The comparison of the standard variation of storage modulus of different phases in cementitious materials is shown in Fig. 6.18d. As can be seen in Fig. 6.18d, the variability of storage modulus of the slag grain is significantly large. Moreover, it can be observed that the influence of different applied loads on the standard variation of storage modulus of individual phases is opposite to that on the displacement amplitudes, which indicates that a large displacement amplitude contribute to decreasing the variability of storage modulus.

6.4.4 Continuous Stiffness Measurement (CSM) of Homogeneous Mechanical Properties at Small-Scale As a dynamic indentation technique, the CSM is capable of measuring continuously the elastic modulus and the contact hardness of the tested materials at different indentation depth (or indentation force) during the loading stage. Therefore, it is possible to assess whether the measured results can reflect the homogeneous properties of cementitious materials by the CSM, which will be discussed in this section.

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6 Testing and Analysis of Micro Elastic Properties

The CSM results obtained from Liang et al. (2017) will be introduced in this section. Agilent G200 Nano Indenter equipped with a Berkovich indenter was used to conduct the CSM tests. The load precision and the depth precision were 1 mN and 0.01 nm, respectively. During the CSM test, a maximum indentation depth of 30 µm was set and the applied strain rate was 0.05 s−1 . The frequency of the harmonic motion was set as 45 Hz and the sinusoidal depth amplitude was set as 2 nm. There were 6 indents made on each cement paste sample, and all the CSM tests were performed under the temperature of 20 °C. Prior to the CSM tests, the cement paste samples were kept in the equipment for several hours to minimize the thermal drift effect, which will result in a low thermal drift rate of less than 0.05 nm/s. Therefore, the maximum depth induced by the thermal drift was lower than 75 nm during the CSM test within the testing duration of less than 1500 s, which is expected to have little effect on the indentation depth caused by the indentation load. Two typical types of indents can be normally observed after the CSM tests on each sample, taking the cement paste with w/c ratio of 0.3 shown in Fig. 6.19 as an example, one indent is initially located on the hydration product, while the other one is initially located on the unhydrated clinkers. It can be seen from Fig. 6.19 that the microstructures of both the two types of indents are made on the composite of the pore, the hydrate, and the clinker. When the applied maximum indentation depth was set as 30 µm, the characteristic length of the residual indent was about 170 µm. Due to the fact that the characteristic length of the individual phase in cement paste is smaller than that of the indented area, it is expected that the homogeneous mechanical properties of the cement paste can be evaluated by the CSM tests. Figures 6.20 and 6.21 show the variation of the elastic modulus and contact hardness with the indentation depth or indentation load. Since the measured properties by the CSM test reflect the mixed properties of materials within the indented volume, the elastic modulus obtained at very shallow indentation depth reflects the elastic property of the hydration products when the indenter tip is initially located on the hydration products. As the indentation depth (force) increases, the elastic modulus increases slightly and finally reaches a constant value due to the growing involvement of clinkers. When the initial contact is located on the clinker, the measured (a)

(b) Pore

Pore Hydrate

Clinker

Hydrate

169μm 63μm

118μm

200μm

Clinker 167μm 200μm

Fig. 6.19 Typical BSE images of indented zone in hardened cement paste with w/c ratio = 0.3: a indenter tip located on hydration product, and b indenter tip located on clinker (Liang et al., 2017)

6.4 Dynamic Method Based on Indentation Technique

215

elastic modulus reflects the property of the unhydrated clinker at the very beginning of the CSM test. Then the elastic modulus decreases significantly with increasing indentation depth (force) until reaching a nearly constant value, which is due to the fact that other phases (pore and hydrate) with low elastic modulus involve within the indented zone. Similar experimental results were also observed for the cement pastes with w/c ratios of 0.4 and 0.5. And the stable elastic modulus obtained from the CSM test reflects the homogenous property of cement paste, which will be further illustrated as follows. The variation of the contact hardness (H) with h and P for the cement paste with w/c ratio of 0.3 is shown in Fig. 6.21. Similar to the elastic modulus, the H–h curves and the H-P curves can be divided into two categories according to the initial tip location. The contact hardness will also reach a constant value when h (P) is large enough. However, the indentation depth (force) beyond which the contact hardness is constant is larger than that for the elastic modulus, which may be related to the size effect in the CSM test. It has been widely accepted that the measured property will become stable when the indentation volume size is comparable to that of the (a) 120

strain rate=0.05 s-1

(b) 120

E (GPa)

Indenter tip on clinker

80

E (GPa)

strain rate=0.05 s-1

100

100

Ehomo

60

hhomo=7.9 μm

40

Indenter tip on clinker

80 60

Ehomo

40

Phomo=1.5 N

20

20 Indenter tip on hydration product

Indenter tip on hydration product 0

0 0

2

4

6

8 10 h (μm)

12

14

0

16

1

2

3

4

5

P (N)

Fig. 6.20 Relation between elastic modulus and a indentation depth, and b indentation force for the cement paste with w/c ratio = 0.3 (Liang et al., 2017) (a) 7

(b) 7 strain rate=0.05 s-1

H (GPa)

5 Indenter tip on clinker

4

Hhomo

5

Hhomo hhomo=10.1 μm

3 Indenter tip on hydration product

2

strain rate=0.05 s-1

6 H (GPa)

6

Indenter tip on clinker

4 3

Indenter tip on hydration product

2

1

Phomo=1.9 N

1

0 0

2

4

6

8 10 h (μm)

12

14

16

0 0

1

2

3

4

5

P (N)

Fig. 6.21 Variation of contact hardness with a indentation depth, and b indentation force for cement paste with w/c ratio = 0.3 (Liang et al., 2017)

6 Testing and Analysis of Micro Elastic Properties

hhomo (μm)

12

13.2μm

7

(b) 24

6

20

5 9

7.2μm 8.6μm

4

6

3

3.6N

3

1.4N

2

1.9N

0 0.2

0.3

0.4 0.5 w/c ratio

hhomo (μm)

E

Phomo (N)

(a) 15

12

H

16 12

4

0

0

0.6

8

12.9μm

6

9.9μm

8

1

10

17.9μm

4

5.1N 2N

2.7N

Phomo (N)

216

2 0

0.2

0.3

0.4 w/c ratio

0.5

0.6

Fig. 6.22 Influence of w/c ratio on hhomo and Phomo to achieve homogeneous mechanical properties of cement paste for a elastic modulus, and b contact hardness (Liang et al., 2017)

representative elementary volume. It has been reported by Larsson et al. (1996) that a larger indentation force is required to obtain the bulk plastic material properties by indentation test. Since the contact hardness is influenced by the plastic property of the material, a larger indentation depth (force) is observed to measure the constant contact hardness. In this book, the minimum depth (force) beyond which the mechanical properties (E and H) remain almost constant is denoted as hhomo (Phomo ), which means that the measured results reflect the homogenous property of the tested materials underneath the indenter tip when the depth (force) is no less than hhomo (Phomo ). Figures 6.22a and b show the relation between hhomo and Phomo with w/c ratio of cement paste. It can be seen from Fig. 6.22 that the indentation depth (force) to obtain the homogeneous mechanical properties of cement paste is influenced by the w/c ratio of the cement paste. A high w/c ratio of cement paste will result in a greater hhomo and Phomo . The reason lies in that the mechanical properties of materials depends greatly on their microstructures that is controlled by the w/c ratio. Cement paste with a higher w/c ratio will enhance the heterogeneity. As a result, a greater hhomo or Phomo is required to obtain the homogeneous properties of cement paste with a higher w/c ratio.

6.5 Analysis of Small-Scale Mechanical Properties 6.5.1 Influence of Indentation Load The influence of magnitude of the indentation load on the measured elastic modulus of cementitious materials was evaluated through conducting MI tests on cement paste sample. The w/c ratio of the tested cement paste was 0.3, and the applied indentation forces were 2 N, 4 N, and 8 N, respectively (Liang & Wei, 2020a). Table 6.1 summarizes the experimental programs of the MI tests. In addition, the CSM, as

6.5 Analysis of Small-Scale Mechanical Properties

217

an alternative technique, was conducted on the paste sample to verify the measured results by MI. A maximum indentation depth of 20 µm was used in the CSM test. The strain rate used in the CSM test was 0.05 s−1 , the excitation frequency was 45 Hz and the amplitude of the sinusoidal depth was 2 nm. To illustrate whether CSM test can determine the homogeneous properties of cement paste, the measured elastic modulus by the CSM test will be compared to those measured by the MI test with different indentation forces of 2 N, 4 N, and 8 N, which is shown in Fig. 6.23. It can be seen from Fig. 6.23 that the elastic modulus measured by MI test with the maximum indentation force of 2 N, 4 N, and 8 N is 31.5 ± 3.8 GPa, 30.7 ± 2.2 GPa, and 30.2 ± 1.4 GPa, respectively. A slight decrease in the elastic modulus can be observed when the maximum indentation force is increased, which may be related to the minor damage condition of cement paste induced by the larger indentation force. The measured elastic moduli by the MI tests with different maximum indentation forces are in good agreement with that measured by the CSM test (30.2 ± 2.3 GPa). This suggests that the MI tests with a maximum indentation force of greater than 2 N can measure the homogenous elastic properties of cement paste, which also validates that the CSM test is feasible to assess the homogeneous properties of cementitious materials. Davydov et al. (2011) investigated the influence of the maximum indentation force on the indentation modulus and the contact hardness of individual phase in cementitious materials by NI test. The maximum indentation force used in the NI test was 1–5 mN and the corresponding indentation depth was 150–600 nm. The relation between the indentation modulus and the contact hardness with the maximum indentation force are shown in Fig. 6.24a and b, respectively. It can be seen from Fig. 6.24 that the maximum indentation force has an effect on the measured elastic modulus and contact hardness. It is concluded by Davydov et al. (2011) that the experimental discrepancies can be probably accounted for by the fact that the influence of the porosity near the indenter is more significant as the indentation force is increased, which will lead to a smaller elastic modulus and contact hardness. Another possible reason may be related to the damage effect. As the damage is more significant when a larger indentation load is used, a greater reduction in the elastic modulus and contact hardness will be observed.

6.5.2 Influence of Loading Rate/Loading Time To investigate the influence of loading rate/loading time on the measured elastic modulus of cementitious materials, Liang and Wei (2020a) conducted MI tests with the maximum indentation force of 4 N on cement paste sample with w/c ratio of 0.3 using different loading times of 5 s, 10 s, 20 s, and 40 s, respectively. The holding time of all the MI tests was set as 180 s and the unloading time was set as 10 s. Figure 6.23 shows the measured elastic modulus of cement paste by the MI test with different loading times. As shown in Fig. 6.23, the loading time has little effect on the measured elastic modulus, which is nearly the same as that measured by the

5

0.8

8

10

0.4

0.8

0.8

14

15

16

10

10

10

10

10

0.4

0.4

12

40

0.1

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13

10

20

0.4

0.2

9

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0.2

0.2

10

6

0.2

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7

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3

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Holding force (N)

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Holding stage

Loading stage

2

Test No

180

0

300

300

0

180

180

180

180

300

40

0

180

180

180

180

Holding time (s)

Table 6.1 Experimental programs of MI tests (Liang & Wei, 2020a)

0.8

0.8

0.8

0.4

0.4

0.4

0.4

0.4

0.4

0.2

0.2

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0.05

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Unloading rate (N/s)

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10

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Unloading time (s)

Unloading stage

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20

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Reholding force (mN)

625

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625

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625

625

625

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625

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625

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625

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625

625

Reholding time (s)

Reholding stage

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

No. of indents

218 6 Testing and Analysis of Micro Elastic Properties

6.5 Analysis of Small-Scale Mechanical Properties 60

loading time=10 s holding time=180 s unloading time=10 s

Elastic modulus (GPa)

50 40

indentation force=4 N holding time=180 s unloading time=10 s

ECSM=30.2

219 indentation force=4 N indentation force=4 N loading time=10 s loading time=10 s unloading time=10 s holding time=180 s

2.3 GPa

30 20 10 0

2N 4N 8N Indentation force

5s

10s 20s 40s Loading time

0s 180s 300s Holding time

5s 10s Unloading time

Fig. 6.23 Comparison of the measured elastic modulus of cement paste by MI tests and CSM tests (Liang & Wei, 2020a)

(b) 10

90

Hardness (GPa)

Indentation modulus (GPa)

(a) 110

70 50 30 10

1000 2000

4000 5000

Maximum load (μN)

8 1 2 3 4

6 4 2 0

1000 2000

4000 5000

Maximum load (μN)

Fig. 6.24 Variation of a indentation modulus and b contact hardness with the maximum indentation load in NI test (Davydov et al., 2011)

CSM test. This suggests that the MI test can measure the homogeneous properties of cement pastes as long as the maximum indentation force is large enough despite the duration of the loading stage.

6.5.3 Influence of Holding Time To investigate the influence of holding time on the measured elastic modulus of cementitious materials, Liang and Wei (2020a) performed MI tests on cement paste sample with w/c ratio of 0.3 by different holding times of 0, 180, and 300 s. The maximum indentation force was set as 4 N, and the loading time was set as 10 s, which is the same as the unloading time. Figure 6.23 shows the measured elastic modulus of cement paste under different holding times. As shown in Fig. 6.23, the elastic modulus measured by the MI test with no holding stage is 37.2 ± 1.1 GPa, which is about 15–24% larger than those measured by the MI test with holding times of 180 s and 300 s. The possible reason is related to the nose effect (Ngan et al.,

220

6 Testing and Analysis of Micro Elastic Properties

Indentation force (N)

10

Ac0 Ac180

3O-7 days

8

holding time

180 s 0s

6 Ac0 < Ac180 S0 > S180

4

1

nose effect

S180

S0 1

creep is significant at the unloading segment of P-h curve of MI tests with no holding stage slope of unloading segment of P-h curve ↑ contact stiffness ↑

2 0 0

10000 20000 30000 Indentation depth (nm)

40000

Fig. 6.25 Illustration of the influence of nose effect on the measured contact stiffness by MI tests with and without a holding stage

2005; Oliveira et al., 2014), i.e. the contact stiffness obtained from the MI test with no holding stage will increase due to the creep property of cement paste, which will result in a greater elastic modulus. It is illustrated in Fig. 6.25 that the contact area in the MI test with no holding stage at the onset of unloading is smaller than that with a holding stage of 180 s since the indentation depth is less in the MI test with no holding stage. In addition, the creep effect is still significant at the onset of unloading in the MI test with no holding stage, which will overestimate the contact stiffness. Based on Eqs. (3.8) and (3.10), the MI test with no holding stage would give a higher estimation of the elastic modulus. However, the MI test with a holding stage of 180 s gives an almost the same elastic modulus as that with a holding stage of 300 s. And the measured elastic moduli by the MI tests with a holding stage of both 180 s and 300 s are in agreement with that measured by the CSM test, which suggests that a holding stage is required in the MI test to measure accurately the elastic modulus of cementitious materials normally with a creep property. It has also been reported that the incorporation of a holding stage with suitable holding duration into the indentation test can improve the measuring accuracy of the elastic modulus of cementitious materials by eliminating the creep effect (Nˇemeˇcek et al. 2009; Oliveira et al., 2014).

6.5.4 Influence of Unloading Time To investigate the influence of unloading time on the measured elastic modulus of cementitious materials, Liang and Wei (2020a) conducted MI tests with different unloading times of 5 and 10 s on the cement paste with w/c ratio of 0.3. The maximum indentation force was set as 4 N. The loading and holding time were set as 10 s and 180 s, respectively. The elastic modulus of cement paste measured by the MI test

6.5 Analysis of Small-Scale Mechanical Properties

221

with different unloading times is shown in Fig. 6.23. It is found that the measured elastic modulus by the MI test with unloading time of 5 s is comparable to that with unloading time of 10 s, and both of them are in agreement with that measured by the CSM test. This is because the contact stiffness measured from the unloading stage depends mainly on the elastic property of cement paste as long as most of the creep deformation of cement paste have developed during the holding stage of 180 s. Therefore, the unloading time has little influence on the measured elastic modulus of cement paste by the MI test with a suitable holding stage.

6.5.5 Influence of Mixture Proportion of Cementitious Materials Several studies have revealed that the mixture proportions of cementitious materials have a great influence on their mechanical properties (Haecker et al., 2005; Helmuth & Turk, 1966). To evaluate the influence of water-to-cement ratio of cement paste on the measured elastic modulus and the contact hardness by indentation test, a series of MI tests have been performed by Liang and Wei (2020b) on cement pastes with different w/c ratios of 0.3, 0.4, and 0.5. Figure 6.26 shows the elastic moduli of cement paste with different w/c ratios measured by MI test. A higher w/c ratio will result in a lower elastic modulus of cement paste, which is in agreement with those measured by the macroscopic method (Haecker et al., 2005; Helmuth & Turk, 1966). In addition, it can be seen from Fig. 6.26 that the measured elastic modulus of cement paste by the MI test with a holding time of 180 s is smaller than that measured by the MI test with no holding stage, which is related to the nose effect described in Sect. 6.5.3 (shown in Fig. 6.25). Figure 6.27 shows the relation between the measured contact hardness and the w/c ratio of cement paste. As shown in Fig. 6.27, a larger w/c ratio of cement paste will result in a lower contact hardness, which is similar to the elastic modulus. Besides, the measured contact hardness of cement paste by the MI test with a holding time of 180 s is smaller than that with no holding time, which is also related to the creep 30

Elastic modulus (GPa)

Fig. 6.26 Measured elastic modulus of cement paste with w/c ratios of 0.3, 0.4, and 0.5 at the age of 7 days (Liang & Wei, 2020b)

180 s holding time

25

0s

20 15 10 5 0 0.3

0.4

w/c ratio

0.5

Fig. 6.27 Measured contact hardness of cement paste with w/c ratios of 0.3, 0.4, and 0.5 at the age of 7 days (Liang & Wei, 2020b)

6 Testing and Analysis of Micro Elastic Properties 0.5

Contact hardness (GPa)

222

holding time

0.4

180 s 0s

0.3 0.2 0.1 0 0.3

0.4

0.5

w/c ratio

effect of cement paste. By comparing Figs. 6.26 and 6.27, it can be observed that the creep effect is more significant on the elastic modulus than that on the contact hardness. For example, the elastic modulus will increase by 27.7–38.9% due to the lack of holding stage in the MI test, while the increment is only 12.2–18.5% for the contact hardness. This can be explained by the fact that both the contact area and the contact stiffness affect the elastic modulus (see Eqs. (3.8) and (3.10)), while only the contact area affects the contact hardness (H = Pmax /Ac , H is the contact hardness, and Pmax is the maximum indentation depth).

6.5.6 Influence of Curing Age of Cementitious Materials To evaluate the influence of curing age, Liang and Wei (2020b) conducted MI tests on the cement paste with w/c ratio of 0.3. The cement paste used for the MI tests were cured for 1 day, 7 days, and 3 years, respectively. As shown in Fig. 6.28, no matter whether the holding stage is set in the MI test, a longer curing age will result in a greater elastic modulus, which is in agreement with that measured by the macroscopic method (Haecker et al., 2005; Helmuth & Turk, 1966). Moreover, the elastic modulus measured by the MI test with no holding is 27.4–28.0% greater than that measured by the MI test with a holding time of 180 s. The reason can refer to Sect. 6.5.3. Figure 6.29 shows the contact hardness of the cement pastes with w/c ratio of 0.3 measured at the age of 1 day, 7 days, and 3 years by the MI test. Similar to the elastic modulus, a longer curing time will result in a greater contact hardness. Moreover, the contact hardness measured by the MI test with no holding is 12.2–12.7% greater than that with a holding time of 180 s, which can also be explained by the creep effect of cement paste. However, the curing age seems to have little effect on the creep effect of cement paste on the measured contact hardness, which is seen from the minor difference between the increasing rates of the contact hardness under different curing ages.

6.5 Analysis of Small-Scale Mechanical Properties 40

Elastic modulus (GPa)

Fig. 6.28 Measured elastic modulus of cement paste with w/c ratio of 0.3 at the age of 1 day, 7 days, and 3 years (Liang & Wei, 2020b)

223

holding time 30

180 s 0s

20 10 0

1 day

7 days

3 years

Curing age

1.2

Contact hardness (GPa)

Fig. 6.29 Measured contact hardness of cement paste with w/c ratio of 0.3 at the age of 1 day, 7 days, and 3 years (Liang & Wei, 2020b)

1.0

holding time

180 s 0s

0.8 0.6 0.4 0.2 0.0

1 day

7 days

3 years

Curing age

6.5.7 Influence of Loading Strain Rate It has been reported that the strain rate would affect the macro mechanical properties such as strength and elastic modulus of cementitious materials (Wu et al., 2012). To investigate the influence of strain rate on the mechanical properties of cementitious materials at micro scale, Liang et al. (2017) performed CSM tests on the cement paste with w/c ratios of 0.3, 0.4, and 0.5, which were cured for 1 year, 4 years, and 4 years, 1 = dh based on respectively. In the CSM test, the strain rate is defined as ε˙ = dε dt h dt / the concept of true strain (ε = dh h) (Nieh et al., 2002). Four strain rates of 0.01, 0.05, 0.1, and 0.5 s−1 were set in the CSM test. A maximum indentation depth of 30 µm was set in all the CSM tests. The excitation frequency of the harmonic motion added in the loading stage was 45 Hz, and the amplitude of the sinusoidal indentation depth was 2 nm. The number of the indents was 6 for each strain rate level on each cement paste. Figure 6.30a shows the variation of the measured elastic modulus with the strain rate for different cement pastes. When the strain rate lies in between 0.01 and 0.5 s−1 , the measured elastic modulus is almost unchanged for each cement paste, which is different from that measured at macro scale. For example, it has been shown by Wu et al. (2012) that the elastic modulus of concrete increases by about 20% when the

224

6 Testing and Analysis of Micro Elastic Properties

strain rate is increased from 10−6 to 10−3 s−1 . The difference between the micro and macro experimental results may be attributed to the fact that the applied strain rate range is quite narrow (from 0.01 to 0.5 s−1 ) and only the cement paste was measured in the CSM test. The aggregate in concrete may also affect the strain rate effect, which cannot be assessed by the CSM test. The variation of the contact hardness of cement paste measured by the CSM test with the strain rate is shown in Fig. 6.30b. Obviously, a higher strain rate will result in a greater contact hardness. Moreover, the relation between the contact hardness and the strain rate can be well captured by Eq. (6.18). The fitted parameters are listed in Table 6.2 and the fitted results are also plotted in Fig. 6.30b for comparison. It can be seen from Fig. 6.30b that the fitted results match well the measured ones, which suggests that the empirical power-law equation shown in Eq. (6.18) can characterize the variation of the contact hardness with the strain rate of cement paste at micro scale. ( )n H (6.18) ε˙ = B H0 where, B (in s−1 ) and n are constant parameters, which can be determined based on the measured contact hardness at different strain rate. Besides, n is defined as the strain-rate sensitivity at micro scale (Nieh et al., 2002), H 0 = 1 GPa is a reference hardness, which can ensure that n is a constant with a dimensionless unit. Figure 6.31 shows the fitted parameters of the cement paste with different w/c ratios of 0.3, 0.4, and 0.5. It can be seen that both B and n decrease linearly with 30

(a)

1.0

15 10 5 0 0.01

(b)

0.8

20 H (GPa)

E (GPa)

25

w/c ratio

0.3 Eave=24.3 GPa 0.4 Eave=21.4 GPa 0.5 Eave=16.7 GPa 0.1 strain rate (s-1)

0.6 0.4

1

0.0 0.01

0.3 0.4 0.5

w/c ratio

0.2

0.1 strain rate (s-1)

1

Fig. 6.30 Influence of strain rate on a elastic modulus, and b contact hardness of cement pastes with different w/c ratios (Liang et al., 2017)

Table 6.2 Fitted power-law parameters in Eq. (6.18)

w/c ratio

0.3

0.4

0.5

B (s−1 )

0.889

0.729

0.551

n

0.096

0.068

0.036

6.5 Analysis of Small-Scale Mechanical Properties

225

0.3

1

Fig. 6.31 Influence of w/c ratio on the power-law parameters in Eq. (6.18) (Liang et al., 2017)

( /

) 0.2

0.6

n

B (s-1)

0.8

0.4

0.1

0.2 0

0 0.2

0.3

0.4 w/c ratio

0.5

0.6

increasing w/c ratio despite that the curing age of the three cement pastes is different, i.e., the curing age of the cement paste with w/c ratios of 0.3, 0.4, and 0.5 was 1 year, 4 years, and 4 years, respectively. When the w/c ratio increases from 0.3 to 0.5, B will decrease by 38.0% and n will decrease by 62.5%. The relation between the fitted parameters (B and n) and the w/c ratio shown in Fig. 6.31 can be used to determine the power-law parameters for pastes with other w/c ratios.

6.5.8 Comparison of Modulus Measured by Quasi-Static and Dynamic Methods It has been reported that the storage modulus measured from the dynamic method reflects the elastic property of the tested materials, which can be considered to be practically equivalent to the indentation modulus (Balooch et al., 2004). To investigate the relation between the modulus of cement pastes measured by the quasi-static and the dynamic methods at micro scale, Wei et al. (2018) performed the in-situ quasi-static NI and the dynamic SPM mapping on the OPC and the slag-blended pastes. The results are shown in Fig. 6.32. The measured modulus was located on the same position in order to achieve a reasonable comparison. Since damage would be induced on the sample surface by the large static NI force, modulus mapping without causing damage was always conducted before the NI test, then NI test was conducted on the same testing location, as shown in Fig. 6.32a. In the NI test, indents were selected to be made on the desired phases to measure their indentation modulus (Fig. 6.32a). Then, the position lines were drawn intentionally on the modulus mapping image to trace back to the indent locations, which can ensure that an overlapping image is obtained with the position lines on the same coordinate. A 0.5 × 0.5 µm small area covering the indent by the NI test was identified on the mapping image, the average storage modulus within this area will be used to compare with the indentation modulus.

226

6 Testing and Analysis of Micro Elastic Properties

Fig. 6.32 Relationship between the indentation modulus and the storage modulus for hydration products and the unreacted grains in OPC and the slag-blended pastes (Wei et al., 2018)

Figure 6.32b shows the comparison of the indentation modulus and the storage modulus for both the OPC and the slag-blended pastes. As shown in Fig. 6.32b, the storage modulus measured by modulus mapping is generally in good agreement with the indentation modulus measured by NI, which is consistent with findings by Li et al. (2015). A good linear relation is observed for the indentation modulus and the storage modulus. The slightly greater value of the indentation modulus may be related to the stronger interaction between the multiple phases in the indented area, since the indent depth of NI is much larger than that in SPM modulus mapping test. In fact, the linear relation has also been reported for glassy materials with storage modulus of E’≈1GPa, however, no good linear relation between the indentation modulus and the storage modulus exits for materials with lower storage modulus (i.e., E’≈1 MPa) (Odegard et al., 2005; White et al., 2004). And the above good relation between the storage modulus and the indentation modulus indicates that the modulus mapping can be used to characterize the mechanical property and distinguish different phases in cementitious materials as the static NI technique does.

6.5.9 Comparison of Elastic Modulus Measured by NI and Macroscopic Methods To investigate the influence of porosity on the elastic modulus of pure constituents of Portland cement clinker, Velez et al. (2001) performed NI test at micro scale and resonance frequency measurements at macro scale on four Portland cement clinkers (C3 S C2 S, C3 A and C4 AF). The samples used for the measurements were synthesized from reagent grade oxide and calcium carbonate. A load-time curve shown in Fig. 6.33a was used in the NI test. The maximum indentation depth was set as 500 nm in the NI test. The indenter was loaded to the maximum load and then unloaded three time at a constant loading rate in order

6.5 Analysis of Small-Scale Mechanical Properties 40

(a)

30

Load (mN)

Load (mN)

40

20 10 0

0

227

50

100

150

200

(b)

30 20 10 0

0

100

Time (s)

200

300

500

400

600

Depth (nm)

Fig. 6.33 a Indentation load-time curve and b indentation load–depth curve measured during the indentation test to evaluate the elastic modulus of cementitious materials (Velez et al., 2001)

to eliminate the non-elastic property of the cement clinker on the measured elastic modulus. The indentation force at the end of each unloading cycle is 10% of the maximum one. One typical indentation load-indentation depth curve is shown in Fig. 6.33b. The macro elastic modulus of the four Portland cement clinkers was measured by the resonance frequencies technique using flexural vibration of the specimens with size of 10 × 1 × 2 cm. In the resonance frequency test, the specimen vibrates as a whole in one of its natural frequency mode. The vibration frequency of the specimen is influenced by many factors including the geometry of the tested materials, mass density of the materials, pores distribution within the specimen, and the elastic properties of the tested materials. More details about the resonance frequencies technique can be found in the reference (Brunarski, 1969). The measured elastic modulus of the four Portland cement clinkers by NI test are shown in Table 6.3. The measured elastic modulus of the four Portland cement clinkers lies in between 125 and 145 GPa, which can be considered as the elastic modulus of the solid phase in clinker without any porosity. Figure 6.34 shows the measured dynamic modulus of C3 S, C2 S, and C3 A with different porosities by the resonance frequencies technique. As the porosity of the clinker increases, the measured elastic modulus will decrease, which can be captured by the equation E = E 0 (1-p)n (E 0 is the modulus of clinker with no porosity, p is the porosity, n is a fitted parameter). The comparison of the measured elastic moduli of the clinkers including C3 S, C2 S, and C3 A by the NI test to those with zero porosity extrapolated from the measured Table 6.3 Measured elastic modulus (E) of calcium silicates, calcium aluminate, and calcium aluminoferrite present in Portland cement clinker by indentation technique (Velez et al., 2001) C3 S

C2 S

C3 A

C4 AF

Alite

Belite

E (GPa)

135

130

145

125

125

127

S.D

7

20

10

25

7

10

228

6 Testing and Analysis of Micro Elastic Properties

(b) 150

Experimental results Fit E=146.89e-3.4P R2=0.99

Experimental results Fit E=140.2(1-P)4.65 R2=0.92

100 E (GPa)

E (GPa)

(a) 150

50 0

0

0.1 0.2 0.3 Relative porosity (%)

(c)

50 0

0

0.05

0.1 0.15 0.2 Relative porosity (%)

0.25

Experimental results Fit E=164.2(1-P)3.36 R2=0.97

150 E (GPa)

0.4

100

100

50

0

0.1 0.2 Relative porosity (%)

0.3

Fig. 6.34 Macroscopic elastic modulus as a function of the relative porosity of the phases of a C3 S, b C2 S, and c C3 A (Velez et al., 2001)

elastic modulus by resonance frequency technique is shown in Table 6.4. The extrapolated elastic modulus of the four clinkers with 0% porosity is comparable to those measured by NI test, which suggests that the porosity of the particles with volume fraction of a few µm3 indented by the NI is close to zero, and the measured indentation modulus by NI test is close to that of cement clinker without porosity extrapolated from the measured results by resonance frequency technique. Table 6.4 Comparison between the elastic modulus obtained by NI and by resonance frequency technique for C3 S, C2 S, and C3 A (Velez et al., 2001) Phases

E measured by resonance frequency (GPa)/S.D

E measured by NI/S.D

C3 S

147/5

135/7

C2 S

140/10

130/20

C3 A

160/10

145/10

References

229

6.6 Summary The existing research findings are reviewed in this chapter concerning the testing and analysis of micro and macro scale elastic properties of cementitious materials. To measure the elastic properties of cementitious materials at macro scale, the cyclic loading method, the wave propagation method, the dielectric method, and the resonance-based method are commonly used. While, to measure the elastic properties of cementitious materials at micro scale, nano- or micro-indentation test and the dynamic method based on indentation test are recommended. To distinguish the mechanical properties of different phases in cementitious materials at micro scale, the least-square estimation, the maximum likelihood estimation, the clustering analysis method, and the discrete measurement can be used to process the data acquired from NI test. For the test in which the small-scale elastic properties of cementitious materials are measured by the SPM, the effect of the surface elevation on the indentation displacement is minor, no matter whether the variation of surface elevation is dozens or hundreds of nanometers. A large dynamic load of 3.5 µN and a small quasistatic load of 4 µN achieve a sharp contrast of displacement amplitudes in different phases by the SPM, which can be used to distinguish different phases. The measured storage modulus by the SPM correlates well with the indentation modulus, which indicates that the modulus mapping is capable of characterizing the mechanical property and distinguishing phase as the static NI technique does. The CSM can provide a continuous measurement of the elastic modulus and the contact hardness at different indentation depth (or indentation force) during the loading stage. Thus, it is possible to assess whether the measured results can reflect the homogeneous properties of cementitious materials by the continuous stiffness measurement. Strain rate in CSM has a negligible effect on the homogeneous elastic modulus of cement paste at micro level. However, contact hardness will increase with increasing strain rate. To test the elastic properties at meso scale, the MI test with load larger than 2 N can be used to obtain the homogeneous elastic properties of cementitious materials. Meanwhile, although the loading duration and the unloading duration have minor effect on the measured elastic properties, a holding period with sufficient duration (generally large than 180 s) in MI test is necessary to reduce the influence of creep effect on the measurement of elastic modulus.

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Chapter 7

Testing and Analysis of Micro Fracture Properties

Abstract The fracture properties of cementitious materials have been mainly investigated at macro scale in the last several decades. In view of this, the existing methods for the fracture toughness measurement of the film materials and the rock materials at the micro scales are first summarized in this chapter, including the Lawn–Evans– Marshall (LEM) method and the energy-based method. As a different technique from the LEM method and the energy-based method, the nanoscratch method measures and calculates the fracture properties of individual phases based on the linear elastic fracture mechanics assumption, which is verified to perform well when characterizing the fracture properties of cementitious materials. The fracture toughness can be quantified for individual phases, including the ITZ phase between the unreacted grain and the surrounding hydrate. A linear relationship exists between the macro fracture properties of the paste and the micro fracture properties of the ITZ and the hydrates phases. The nanomaterials such as carbon nanotubes and nanosilica can improve the fracture toughness of both the hydrates and the ITZ phase. The content of this chapter provides guidance on evaluating the fracture behavior of cementitious materials at micro scale. Keywords Energy-based method by indentation · Freeze–thaw cycles · Lawn–Evans–Marshall (LEM) method · Micro fracture properties · Strengthening mechanism by nanomaterials

7.1 Introduction Understanding the fracture behaviors of (quasi-) brittle materials and their relations to the material microstructures is always a fundamental basis but a challenging task for fracture resistance enhancement. The cement-based materials usually have inherent flaws (or cracks) emerging within the material matrices and/or the interfacial areas between different phases, which could further grow and propagate under loads, potentially causing structure failure. The fracture behaviors of the cementbased materials remained being mainly viewed and studied in macro scales in the last several decades. The indentation technique has been extensively used to estimate the material properties and to probe the structures of cement-based materials © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_7

235

236

7 Testing and Analysis of Micro Fracture Properties

at both the nano and the micro scales. As to the assessment of material fracture properties by indentation, the Lawn–Evans–Marshall (LEM) (Lawn et al., 1980) method may be the most commonly used one by calculating the fracture toughness based on the directly measured crack length. In addition, the energy-based model has been frequently used to estimate the fracture properties of the quasi-brittle materials (Chen & Bull, 2007; Cheng et al., 1998; Mencik & Swain, 1994), such as the fracture toughness of shale rocks (Liu, 2015; Zeng et al., 2019). As compared to the NI technique, the nanoscratch can perform continuous measurement of the mechanical properties. Hoover and Ulm (2015) derived the formula based on the linear fracture mechanics theory for calculating the fracture toughness from the scratched results, which greatly promotes the application of the scratching technique to obtain the fracture properties of materials at the micro scale. This technique has been applied in few types of materials including cement, shales, and the amorphous polymers (Akono & Ulm, 2014, 2017; Kabir et al., 2017). In this chapter, the LEM method, the energy-based method, and the nanoscratch method are discussed along with their applications for characterizing the film materials, the rock materials, and the cementitious materials. Meanwhile, this chapter investigates the development of fracture toughness of the pure cement paste and the nanomaterials-modified cement pastes at the micro scales.

7.2 Lawn–Evans–Marshall (LEM) Method by Indentation LEM method can be dated back to Hertz (1881). Hertz utilized the spherical indenter to indent the surface of glass, the cone crack appeared, and the indent force at this time was recorded as the critical force Pc . It was found that Pc is proportional to the radius of the indenter probe (r) when the fracture occurred on the materials surface (Auerbach, 1981), and thus, Pc is proportional to the contact area between the indention probe and the materials surface. Roesler (1956) later found that the stored energy induced by indentation is proportional to the contact area and the fractured area of the cone crack. The fractured area of the cone crack is difficult to be measured directly; however, based on the geometry of the cone crack, it can be calculated from the crack length. In 1967, Lawn proposed the mechanisms of generation and propagation of cracks based on the fracture mechanics, and the experimental results of the indentation test, such as the crack length (marked as c in Sect. 7.2.2) is utilized to evaluate the release of the stored energy (Lawn et al., 1980), which is known as the LEM method. This method has been proved suitable for determining the fracture properties of brittle materials (glass, ceramics, coatings, alloys, and rock), as the cracks can be generated on the material surface easily by indentation and measured precisely by microscope (Chen, 2012; Liu, 2015).

7.2 Lawn–Evans–Marshall (LEM) Method by Indentation

237

7.2.1 Typical Crack Patterns When using the LEM method to calculate the fracture toughness (K C ) of materials, the key is to identify the crack form and measure the crack length. Earlier studies have found that there are five major forms of cracks depending on the load, the material, the environment conditions, and the indenter (Cook & Pharr, 1990), which are illustrated in Figs. 7.1, 7.2, 7.3, 7.4 and 7.5. Almost all of these crack forms can be found in glass (Argon et al., 1960; Bush, 1999), ceramics (Cook & Pharr, 1990), coatings (Cook & Pharr, 1990), crystal (Knight et al., 1977), and other materials. Only a few crack forms, such as the radial crack, can be found in rocks and cementbased materials (Liu, 2015). The crack length and its measuring method depend on the form of the crack, and the measured crack length can be used to calculate the fracture toughness (K C ), which will be illustrated in Sect. 7.2.2. (a)

(b) Cone crack

Cone crack

Fig. 7.1 a Schematic of cone crack and its propagation (brown zone) which can be formed by spherical indenter in glass, fused silica, and coatings materials (Cook & Pharr, 1990); b top view of the cone cracks formed by the spherical indenter on the surface of glass under the force of 1 N applied within a duration of around 30–60 s (Argon et al., 1960)

(a)

Radial crack

(b)

Radial crack Radial crack Indentation

Fig. 7.2 a Schematic of radial crack and its propagation (brown zone) which can be formed by Vickers, Berkovich or spherical indenters in glass, crystalline, and rock materials under a large force (Cook & Pharr, 1990); b top view of the radial crack formed by the Vickers indenter probe on the surface of sapphire under the force of 40 N (Cook & Pharr, 1990)

238

7 Testing and Analysis of Micro Fracture Properties (b)

(a)

Median crack Median crack

Fig. 7.3 a Schematic of median crack and its propagation (brown zone) which can be formed by Vickers or Berkovich indenter in glass materials (Cook & Pharr, 1990); b side view of median crack in soda–lime glass formed by Vickers indenter under the force of 250 N (Lawn & Swain, 1975)

(a)

Half-penny crack

(b)

Half-penny crack

Fig. 7.4 a Schematic of half-penny crack and its propagation (brown zone) which can be formed by Vickers or Berkovich indenter in glass materials (Cook & Pharr, 1990); b side view of half-penny crack in soda–lime glass formed by Vickers indenter under the unloading stage when the maximum force is 250 N (Lawn & Swain, 1975)

(b) Radial crack

(a)

Indentation

Lateral crack

Lateral crack

Fig. 7.5 a Schematic of lateral crack and its propagation (brown zone) which can be formed by Vickers or Berkovich indenters in ceramics, glass, and thin coatings materials during unloading, and formed in crystalline materials during loading (Cook & Pharr, 1990); b top view of the lateral crack in soda–lime glass formed by Vickers indenter when unloading from 40 to 0.8 N (Cook & Pharr, 1990)

7.2 Lawn–Evans–Marshall (LEM) Method by Indentation

239

7.2.2 KC for Different Crack Patterns The differences of the indenter geometry and the material properties lead to different crack patterns; therefore, the calculation method of the fracture toughness (K C ) should be adjusted according to different crack patterns. For the above five introduced crack patterns, the calculation methods of fracture toughness are summarized in this section. Cone Crack Cone cracks have been formed by using a spherical indenter on materials of silicate glass, single crystal ceramics (particularly those with diamond structure) and some hard, fine grain polycrystalline ceramics. One major advantage of using a spherical indenter is that it enables one to follow the entire evolution of the damage modes from the initial elasticity to the full plasticity. Cone crack begins as a surface ring crack and propagates downwards and flares outwards into a truncated cone configuration when the load exceeds the critical load (Lawn, 1998), as shown in Fig. 7.6. For a well-developed cone crack, the fracture toughness is given by (Lawn, 1993, 1998): K C = χ P/C 3/2

(7.1)

where P is the applied load, χ is a crack geometry coefficient. C is the extended crack length and is given by C =c+

R0 cos α0

(7.2)

where c is the crack length, R0 is the surface ring radius, and α 0 is the cone base angle. Radial Crack, Median Crack, and Half-penny Crack In a well-developed radial crack (shown in Fig. 7.7), or a median crack, or a halfpenny crack caused by indentation, the toughness K C of the indented materials was found to be proportional to the applied load P divided by crack length, c3/2 . (b) (a)

C α0

c R0 Cone crack

Fig. 7.6 Side view of a cone crack in soda–lime glass (Roesler, 1956), and b schematic showing critical geometrical parameters (Lawn, 1998)

240

7 Testing and Analysis of Micro Fracture Properties

Fig. 7.7 SEM image of radial crack in soda–lime glass caused by indentation with a Berkovich indenter (Cuadrado et al., 2015)

Radial crack

c

K C = χ P/c3/2 = ξvR (E/H )m P/c3/2

(7.3)

where ξvR is a calibration coefficient which depends on the crack pattern and the indenter geometry. For the well-developed radial cracking produced by a Vickers or a Berkovich indenter, the range of ξvR is 0.012–0.020 (Anstis et al., 1981). E and H are the elastic modulus and the contact hardness of the indented material, respectively. The empirical exponent m was found to be between 2/5 and 2/3 (Laugier, 1987; Lawn, 1993; Niihara et al., 1982). The commonly used value m = 1/2 was based on the expanding spherical cavity model and the dimensional analysis. Lateral Cracking Lateral cracking is responsible for the abrasive wear and the erosion of ceramics and ceramic coatings. Based on a similar expanding cavity model (Lawn et al., 1980) and a simple plate theory, Marshall et al. (1982) presented a model to correlate the fracture toughness of the brittle bulk materials to the critical load for the lateral cracks to occur during unloading: K C4 =

( 4) H P0 A2 (cot ψ)2/3 δ0 E

(7.4)

where P0 is the threshold load for the lateral crack, δ 0 is the dimensionless constant and equals to 1200. ψ is the angle between the pyramidal edges of the indenter and equals to 74° for Vickers indenter (Marshall et al., 1982). A is a dimensionless, geometrical constant, and it can be calculated by: A = 3 × (1 − ν 2 )/4π (lateral crack > radial crack, Fig. 7.8a)

(7.5a)

7.2 Lawn–Evans–Marshall (LEM) Method by Indentation

241

Lateral crack

Fig. 7.8 Relationship between lateral crack and radial crack during the formation of radial crack (Marshall et al., 1982)

(a)

A = 3/4

Radial crack

(b)

(7.5b)

(radial crack > lateral crack, Fig. 7.8b). where ν is the Poisson’s ratio. For the tensile stress-induced delamination during unloading, this method can be extended to assess the interfacial failure of a coated system which has been discussed in (Chen & Bull, 2007, 2009).

7.2.3 Typical Studies of LEM Method Schiffmann (2011) used Cube–corner indenter to perform the NI tests on a number of different bulk (single crystal silicon, fused silica, and single crystal sapphire) and thin film materials (Si-DLC, aC, SnO2 , ZnO:Al, and In2 O3 :Sn) to characterize their fracture toughness. The indentation experiments were load controlled and consisted of 5 s loading to the maximum load of 32 mN, 10 s holding, and 5 s unloading. Radial crack appeared in all of the above materials, and their fracture toughness can be calculated by Eq. (7.3). The crack length (c) was measured by the AFM, as the one in single crystal silicon shown in Fig. 7.9. The calculated fracture toughness, the measured crack length, and other mechanical parameters are summarized in Table 7.1. The LEM method is normally used to measure the fracture toughness of glass, film, and crystal materials, and barely used for testing rock materials. Liu (2015) utilized the Berkovich indenter to measure the fracture toughness of shale by NI. To produce obvious cracks on the sample surface for measurement, the maximum indentation depth of 4000 nm, 6000 nm, and 8000 nm was applied for NI test, and the radial crack was observed for all the three measurements, as shown in Fig. 7.10. The mean crack lengths of the samples under the indentation depth of 4000 nm, 6000 nm, and 8000 nm were 9.09 µm, 15.07 µm, and 28.11 µm, respectively. And the mean fracture toughness calculated by the LEM method was 0.069 MPa·m½ , and the standard deviation was 0.013 MPa·m½ . A number of different models based on the crack morphology have been proposed to evaluate the fracture toughness of materials. However, the morphology of the cracks is difficult to obtain due to the small scale involved. Cuadrado et al. (2015)

242

7 Testing and Analysis of Micro Fracture Properties

c

Fig. 7.9 AFM image of single crystal silicon with radial crack created by the Cube–corner indenter (Schiffmann, 2011) Table 7.1 Measured elastic modulus (E), contact hardness (H), crack length (c), and fracture toughness (K C ) of typical materials by LEM method (Schiffmann, 2011) Materials

H (GPa)

E (GPa)

c (µm)

K C (MPa·m½ )

Single crystal silicon

12

170

3.69 ± 0.16

0.96 ± 0.07

Fused silica

9

72

3.70 ± 0.26

0.68 ± 0.08

Single crystal sapphire Al2 O3

31

420

0.83 ± 0.03

8.94 ± 0.50

Si-DLC (diamond-like carbon)

15.5

150

1.23 ± 0.07

4.13 ± 0.38

aC (amorphous, H-free carbon)

35.6

350

1.12 ± 0.04

4.80 ± 0.29

ZnO: Al

14

131

1.71 ± 0.03

2.37 ± 0.06

SnO2 (not tempered)

10

110

1.91 ± 0.11

2.57 ± 0.28

SnO2 (tempered)

10

110

1.79 ± 0.12

2.32 ± 0.23

In2 O3 : Sn (ITO)

12

133

1.51 ± 0.15

3.54 ± 0.62

40μm

40μm

40μm

Fig. 7.10 Shale with radial crack created by Berkovich indenter under the indentation displacement of 4000 nm, 6000 nm, and 8000 nm (Liu, 2015)

7.3 Energy-Based Method by Indentation

Front view

Top view Lateral view

243

Front view

5 mN

Top view Lateral view Lateral crack

50 mN 10 mN Crack length

100 mN

20 mN

Half-penny crack

Fig. 7.11 Crack morphology system for a Cube–corner indentation at 5 mN, 10 mN, 20 mN, 50 mN, and 100 mN in soda–lime glass using FIB tomography (Cuadrado et al., 2015)

utilized the focused ion beam (FIB) tomography to obtain the crack morphology to quantify the fracture toughness of different materials. The soda–lime glass and the three single crystals (SiC, Si, ZrO2 ) indented at different loads have been selected for FIB reconstruction. Figure 7.11 presents the FIB tomography reconstruction of the soda–lime glass for indentations performed at 5, 10, 20, 50, and 100 mN. The lateral cracks appeared at higher load during unloading, which is consistent with the observations by Cook and Pharr (1990). The half-penny cracks and the lateral cracks were also formed in the single crystals. Soler et al. (2018) used the NI technique to study the effect of the deposition temperature on the fracture behavior of the sputtered Mo2 BC/Si systems by measuring their fracture toughness. Four Mo2 BC/Si systems were formed at the temperature of 380 °C, 480 °C, 580 °C, and 630 °C. The fracture toughness measurements were conducted by using a Keysight G200 Nanoindenter equipped with a Berkovich indenter tip. The fracture toughness, K C , was calculated from the indentation experiments with 20 indents per system at a constant strain rate of 0.05 s−1 and a maximum indentation load of 500 mN. The maximum indentation depth was approximately 35% of the film thickness, and the cracks were typical half-penny radial cracks with the length determined by the high-resolution SEM analysis (as shown in Fig. 7.12). The fracture toughness can be calculated by Eq. (7.3), which is summarized in Table 7.2 along with the crack length. The higher the temperature, the longer the crack length, and the lower the fracture toughness. The results in Ref. (Soler et al., 2018) show that the LEM method can provide a new way to characterize the fracture toughness of materials in various environments.

7.3 Energy-Based Method by Indentation The LEM method requires the accurate measurement of the crack length. However, the measuring error is inevitable due to the precision of the microscope, and the crack lengths along different directions might be different due to the anisotropy nature of

244 (a)

7 Testing and Analysis of Micro Fracture Properties (c)

(b)

(d)

Fig. 7.12 NI fractographs at an indentation maximum load of 500 mN for Mo2 BC/Si systems with different deposition temperatures (Soler et al., 2018)

Table 7.2 Crack length and the calculated fracture toughness of Mo2 BC/Si systems under different temperatures (mean value ± standard deviation) (Soler et al., 2018) 380 °C

480 °C

580 °C

630 °C

Crack length (µm)

7.92 ± 0.46

8.60 ± 0.40

14.28 ± 1.02

40.79 ± 10.25

Fracture toughness (MPa·m½ )

1.20 ± 0.12

1.05 ± 0.09

0.55 ± 0.06

0.11 ± 0.04

some materials, such as shale (Chen, 2012; Liu, 2015). In addition, the LEM method has an inherent defect for evaluating the fracture toughness of quasi brittle (such as cement-based materials) and plastic materials, of which the radial cracks or the halfpenny cracks are quite difficult to be measured. In fact, literature surveying indicates that almost no radial cracks have been found in cement-based materials after either micro or nanoindentation (Jennings et al., 2007; Liang et al., 2017; Nguyen et al., 2014; Vandamme & Ulm, 2009; Wei et al., 2017b). Since the end of the twentieth century, within the frame of indentation tests, energy-based models have been frequently used to estimate the fracture properties of materials (Chen & Bull, 2007; Cheng et al., 1998; Mencik & Swain, 1994). The energy-based method calculates the fracture toughness based on the load–displacement curve obtained from indentation test. Depending on the type of the indented materials and the loading process, three energy-based methods were proposed: the energy-based method with an excursion in the load–displacement curve (Li & Bhushan, 1998; Li et al., 1997), the energy-based method with no excursion in the load–displacement curve (Korsunsky et al., 1998), and the improved energy-based method with no excursion in the load–displacement curve (Xu et al., 2018), which will be discussed as follows. The “excursion” is a leap that occurs in the load– displacement curve (as the AD points shown in Fig. 7.15), due to the fracture of the materials.

7.3 Energy-Based Method by Indentation

245

7.3.1 Energy-Based Method with an Excursion in Load–Displacement Curve This method is normally used during the testing and calculation of the coating and the film materials where the delamination and bulking occurs between the stiff coating and the stiff substrate. Three stages involve in this situation, as shown in Fig. 7.13. The stage I refers to the stage when the cone crack or the radial crack occur in coating/substrate during the indentation, the stage II is the widening of the crack with the increasing indentation load and consequently the bulking damage and the delamination. The stage III is the spalling of the coating when the crack propagates through the coating. A typical example (SiC coating on Si) for this is depicted in Fig. 7.14. The typical load–displacement (P–h) curve with an excursion is illustrated in Fig. 7.15. It is thought that the material fractures at point A and propagates to point D. Point C is constructed by extending line OA and sharing the same x value with point D. The area of ACD is considered as the dissipated energy (U fra ) during the fracture of materials. This method is also termed as the ld–dp method, and the fracture toughness (K C ) for this type of materials can be expressed as (Li & Bhushan, 1998; Li et al., 1997):

Stage

P

First through-thickness crack

P Stage

Stage

P

Delamination and bucking

Secondary through-thickness crack

Partial spalling 2CR

Fig. 7.13 Schematic of various stages in indentation fracture for the film/substrate system (Chen, 2012)

Radial crack Delamination (a) Stage

(b) Stage

Partial spalling (c) Stage

Fig. 7.14 Indentations on SiC/Si system by Berkovich indenter at a 100 mN, b 250 mN, and c 500 mN (Chen, 2012)

246

7 Testing and Analysis of Micro Fracture Properties

Fig. 7.15 Typical load–displacement (P–h) curve with an excursion created by indentation in film or coating materials (Chen, 2012)

Load

C

A

D

O Displacement

[( )( | ) | E Ufra ] ) ) KC = t 1 − ν 2 2π C R

(7.6)

where E is the elastic modulus, ν is Poisson’s ratio, 2C R is the crack length shown in Fig. 7.13, t is the thickness of film materials. However, this method ignores the elastic–plastic behavior of the coated system when fracture occurs. Toonder et al. (2002) also argued that the area ACD was not the actual energy dissipated by the fracture. Therefore, a method based on the total work–displacement curve was proposed by Chen (2006), Chen and Bull (2006), as shown in Fig. 7.16. In addition to line AC, line BD is constructed following the same procedure as that for line AC, and the energy dissipated (U fra ) due to the material fracture is expressed as: Ufra = UCD − UAB

(7.7)

where U CD is the length of CD, and U AB is the length of AB in Fig. 7.16. And thus, the fracture toughness of materials can be calculated by substituting Eq. (7.7) into Eq. (7.6), this method is also termed as W t –dp method.

7.3.2 Energy-Based Method with No Excursion in Load–Displacement Curve For the bulk materials or the stiff coating on a soft substrate, there is normally no excursion in the load–displacement (P–h) curve. In this situation, the critical energy release rate (GC ) can be determined by the ratio of the fracture energy to the crack area. And the fracture toughness (K C ) can be determined by the fracture energy release rate:

7.3 Energy-Based Method by Indentation

C

E

After fracture

Total work

Fig. 7.16 Typical total work–displacement (W –h) curve with an excursion created by indentation in film or coating materials (Chen, 2012)

247

A

Before fracture

D

B

O Displacement

GC = / KC =

Wfra Afra /

GC E ) )= 1 − v2

(7.8) Wfra E ) ) 1 − v 2 Afra

(7.9)

where W fra is the fracture energy, Afra is the total cracking area, E and v is the elastic modulus and Poisson’s ratio. The key way to obtain the fracture toughness is to determine the values of the fracture energy and the total crack area. For an indentation test, the total work done by the indenter may involve all the possible mechanical regimes, including elastic, plastic, and fracture processes, if the possible thermal exhaustion and dissipation during the test are neglected (Chen, 2012). Therefore, the fracture energy, W fra , may be expressed as: Wfra = WT − W E − W P

(7.10)

where W T is the total energy made by the indenter on the materials, W E is the elastic energy, and W P is the plastic energy. The parameters W T and W E can be calculated from Fig. 7.17. W T is the area of OAC, and W E is the area of BAC. Both pure plastic and fracture energy represent the irreversible work produced in the system. For a material showing regularly smooth indentation load–depth curves, it is easy to obtain the total and the elastic energies by reading the load–depth curves directly, because the unloading curve represents exactly the response to the elastic recovery of the material (Cheng et al., 1998). However, the only unknown term, the plastic energy, W P , cannot be obtained directly from the load–depth curves as the plastic energy is implicitly involved in the loading curve which needs to be assessed by a theoretical model, which is shown as follows:

248

7 Testing and Analysis of Micro Fracture Properties

Fig. 7.17 Schematic illustration of indentation P–h curves in original energy-based method

Load

A

hm

hf

B

O

C

Displacement

Combining the experimental and the finite element modeling results, Cheng et al. (2002) suggested that the pure-plastic to total energy ratio may be determined by the function that contains the characteristics of load–depth curves and the maximum and the residual depths: WP =1− WT

1−3

(

hf hm

( )3 h + 2 h mf ( )2

)2

1−

(7.11)

hf hm

where hf is the residual depth, hm is the maximum depth, and these two parameters can be seen in Fig. 7.17. As a matter of fact, Eq. (7.11) can be derived theoretically the quadratic functions by assuming that the loading and the unloading curves follow ( ) 2 2 2 of indentation depth, i.e., P ∝ h for loading and P ∝ h − h f for unloading. By combining Eq. (7.9)–(7.11), one may estimate the fracture energy without knowing the crack geometries of a material under the indentation tests. Indeed, this method has been applied by many scholars to estimate the fracture toughness of quasi-brittle materials, such as shale and cement paste (Liu, 2015; Soliman et al., 2017; Taha et al., 2010). Based on this method, the holding stage is absent from the indentation process, which is termed as the original energy-based method (Xu et al., 2018, 2020; Zeng et al., 2017).

7.3.3 Improved Energy-Based Method with No Excursion in Load–Displacement Curve The classical energy-based models, however, may yield biased results, as it was found that these models did not consider an important segment of indentation: the

7.3 Energy-Based Method by Indentation

249

holding segment at the peak load. This load-holding step at the peak load seems to be mandatory to obtain the stable unloading data for calculating the contact stiffness and becomes a standard procedure for indentation test nowadays. Furthermore, when an indenter is held at the maximum load, plastic deformation including creep deformations for cement-based materials (Liang et al., 2017; Mallick et al., 2019; Nguyen et al., 2014; Vandamme & Ulm, 2009, 2013; Wei et al., 2017a) will continuously develop, which could remarkably change the distributions of different energy terms, and consequentially affect the fracture data. Indeed, recent studies suggested that the correction of the holding segment at the peak load can significantly modify the estimated fracture energy and toughness (Xu et al., 2018; Zeng et al., 2019). As compared to the original energy-based method, the energy method for quantifying a NI test with the holding stage is termed as the improved energy-based method (Xu et al., 2018, 2020; Zeng et al., 2017). A load–displacement (P–h) curve with a holding stage is shown in Fig. 7.18. The area under the loading curve, W 2 , represents the energy during the loading process. The area under the holding curve, W 5 , represents the energy during the holding process, and the total energy is written as W T = W 2 + W 5 . The area under the unloading curve, W 4 , represents the elastic energy. Following the concepts proposed by Attaf (2003), the energies associated with the straight linear loading and unloading paths are the referenced loading energy (W 1 + W 2 ) and the referenced unloading energy (W 3 + W 4 ), respectively. To characterize the loading and the unloading curves, we employed the loading and the unloading energy constants that are defined as the ratio of the referenced energy to the actual energy under loading just before the holding step, V t , and the one under unloading, V e . They can be defined as: Vt =

W1 + W2 W2

(7.12)

Ve =

W3 + W4 W4

(7.13)

Here, V t and V e can be identical to the total and the elastic energy constants (Sneddon, 1965). Indeed, V t and V e remove the influences of the maximum load, and represent how the curve shape plays a part (Jha et al., 2015). For modeling purpose, the loading and the unloading curves are often represented by the mathematical functions, such as, the simple power functions, the quadratic functions, and the polynomials (Attaf, 2003; Oliver & Pharr, 1992; Sneddon, 1965). Here, we employed the power functions that have been extensively used to represent the loading and the unloading curves of indentation (Oliver & Pharr, 1992). Considering the holding step, the entire indentation load–depth curves can be described by:

7 Testing and Analysis of Micro Fracture Properties

Load

Load

250

1 2

3

5 4

hf Displacement

hl

hm

Displacement

Fig. 7.18 Schematic illustration of total energy (W 2 + W 5 ), elastic energy (W 4 ), referenced loading energy (W 1 + W 2 ) and referenced unloading energy (W 3 + W 4 ) proposed by Attaf (2003) in an indentation loading-holding-unloading cycle

P = Pm P = Pm P = Pm

( )2Vt −1

0 ≤ h < hl

h hl

(

h−h f h m −h f

)2Ve −1

Loading

h l ≤ h ≤ h m Holding

(7.14)

h f ≤ h < h m Unloading

From the load–depth curves provided by Eq. (7.14), the elastic recovery energy, W E , and the total energy, W T , of indentation can be characterized by: ∫hl W T = W2 + W5 =

∫h m Pdh +

0

( ) 2Vt − 1 h l Pdh = Pm h m 1 − 2Vt h m

(7.15)

hl

(

∫h m W E = W4 =

Pm hl

h −hf hm − h f

)2Ve −1

) ) Pm h m − h f dh = 2Ve

(7.16)

There is no direct way to calculate the plastic energy and the fracture energy based on the curve in Fig. 7.18 due to the simultaneous occurrence of the plastic deformation and the fracture during the loading and the holding stages. To separate the plastic energy and the fracture energy from the total energy, the following assumptions have to be made: 1. The fracture process disperses averagely in the loading and holding steps. 2. The fracture energy is stored in the total energy. 3. The plastic process follows a behavior of pure plastic materials. These three assumptions allow us to estimate the plastic energy by using the relationship for the ideal elasto-plastic materials. In this way, the plastic energy then can be directly estimated as:

7.3 Energy-Based Method by Indentation

1+ WP = WT

(

251

( ) − 1 hhml − 2V1 e 1 − ( ) 1 + 2V1 t − 1 hhml

1 2Vt

hf hm

) (7.17)

The fracture energy of the quasi-brittle materials under indentation tests is then estimated by substituting Eq. (7.17) into Eq. (7.10):

Wfra = WT − W E − W P =

( Vt 1 −

hf hm

)

[ ] WT − W E Ve 2Vt − (2Vt − 1) hhml

(7.18)

Combining Eqs. (7.8), (7.9) and (7.18), the fracture properties of the quasi-brittle materials can be estimated without knowing the specific cracking patterns and the geometries.

7.3.4 Typical Studies of Energy-Based Method A NI test was conducted on a 400-nm-thick amorphous carbon film deposited on silicon by a Cube–corner indenter with a maximum load of 200 mN (Li & Bhushan, 1998). The load–displacement curve with an excursion and the SEM image of the indent are shown in Fig. 7.19. The fracture toughness of the amorphous carbon film is calculated by using the ld–dp method discussed in Sect. 7.3.1, and the results are summarized in Table 7.3. Even if the center of the indentation deviates from that of the delamination, the measuring method of 2C R was provided by Li and Bhushan (1998). 2C R is the longest distance from the tip of the radial crack to the boundary of the delamination zone. Chen and Bull (2006) utilized Berkovich indenter for NI test on the 400-nmthick coating materials, such as TiOx Ny , ITO and SnO2 , the coating materials were Excursion

(b) Radial crack Delamination

Load (mN)

(a)

2CR Partial spalling

Displacement (nm) Fig. 7.19 a P–h curves with an excursion under the peak load of 200 mN, and b SEM image of indentation on amorphous carbon/silicon system (Li & Bhushan, 1998)

252

7 Testing and Analysis of Micro Fracture Properties

Table 7.3 Parameters and results of amorphous carbon/silicon by the ld–dp method (Li & Bhushan, 1998) Material

t (µm)

E (GPa)

ν

U fra (nN·m)

C R (µm)

K C (MPa·m½ )

Amorphous carbon/silicon

0.4

300

0.25

7.1

7.0

10.9

sputtered on the soda–lime glass, and the indentation results are shown in Fig. 7.20. This system belongs to the stiff coating and the stiff substrate, the radial cracks, the bulking damage, and the delamination will occur on the material surface. According to the ld–dp method discussed in Sect. 7.3.1, the elasto-plasticity behavior is ignored, which is not able to calculate the accurate fracture toughness. Therefore, Chen and Bull (2006) utilized the W t –dp method to calculate the fracture toughness of the above three coating materials, and compared with the LEM method, the results are shown in Table 7.4. The fracture toughness of the three materials obtained by the LEM method is similar to that of the soda–lime glass substrate (about 0.9 MPa•m½ ). This is because that the substrate material is softer than the coating material, and the significant deformation of the substrate cannot be eliminated in the LEM method. However, the W t –dp method can eliminate the influence from the substrate to provide the actual fracture toughness, particularly for a very thin coating (< 500 nm). Delamination (b)

(a)

Radial crack

Fig. 7.20 SEM images of indentation on a TiOx Ny coating, and b ITO coating (Chen & Bull, 2006)

Table 7.4 Parameters and results of coating materials by W t –dp method, and the comparison of fracture toughness calculated by LEM method and W t –dp method (Chen & Bull, 2006) Materials

E (GPa)

ν

t (nm)

U fra /Afra (J/m2 )

K C calculated by W t –dp method (MPa·m½ )

K C calculated by LEM method (MPa·m½ )

TiOx Ny

122

0.25

400

24.1

1.8

1.0

ITO

141

0.25

400

32.7

2.1

0.7

SnO2

139

0.25

400

29.3

1.9

1.3

7.3 Energy-Based Method by Indentation

253

Liu (2015) investigated and compared the fracture toughness of the quasi brittle material opalinus clay shale by utilizing the LEM method and the improved energybased method. The NI test was conducted on this type of material and a holding stage was set. The load–displacement curve did not show the excursion, and thus, Eqs. (7.9) and (7.18) in the improved energy-based method were used to calculate the fracture toughness. The calculated results are shown in Table 7.5. It was found that the fracture toughness determined by the improved energy-based method is greater than that of the LEM method. A radial crack (as shown in Fig. 7.7) normally seen in metal or the brittle materials such as shale is not observed in the quasi brittle material of opalinus clay shale. Instead, many nanocracks and spalling would develop in the fracture process zone, as shown in Fig. 7.21. In this case, ignoring the nanocracks in fracture zone but only considering the length of radial crack may greatly underestimate the fracture toughness of the material. Therefore, the improved energy-based method is more suitable than other methods for calculating the fracture toughness of the quasi-brittle materials. Guo et al. (2018) testified the applicability of the original energy-based method by nanoindentating five types of bulk metallic glasses materials and calculated their fracture toughness. The load was applied at a rate of 0.05 mN/s up to the maximum load of 24 mN and then unloaded. It is seen that no excursion occurs in the load– displacement curve (i.e., Fe65 Mo14 C15 B6 , as shown in Fig. 7.22), and the fracture toughness of materials can be calculated by Eqs. (7.8)–(7.11). The results are shown in Table 7.6. The results are compared with those obtained from the crack tip opening displacement (CTOD) method based on the three-point bending test (Guo et al., 2017; Jia et al., 2009; Kawashima et al., 2005; Lewandowski et al., 2008; Xi et al., 2005; Table 7.5 Fracture toughness of opalinus clay shale calculated by LEM method and improved energy-based method (Liu, 2015)

Methods

Avg. K C (MPa·m½ )

Std. K C (MPa·m½ )

LEM method

0.41

0.12

Improved energy-based method

1.23

0.29

Fig. 7.21 SEM images of indentation on opalinus clay shale (Liu, 2015)

Nano cracks and spalling

254

7 Testing and Analysis of Micro Fracture Properties

Zhu et al., 2011); the minor difference between these two methods indicates the good applicability of the original energy-based method on evaluating the fracture toughness of brittle materials. Liu (2015) and Xu et al. (2020) compared the original energy-based method (with non-holding stage in test) with the improved energy-based method (with holding stage in test) on calculating the fracture toughness of shale rocks; they found that the calculated fracture toughness by the original energy-based method is about 5–10 times greater than that calculated by the improved energy-based method, as shown in Fig. 7.23. Zeng et al. (2019) also compared the fracture toughness calculated by these two methods for shale; the results are shown in Fig. 7.24. It was found that the fracture toughness of shale is overestimated by 3–6 times by the original energy-based method with non-holding stage compared to the improved energy-based method with holding stage. Xu et al. (2020) evaluated the influence of the holding stage on the measured fracture toughness by the energy-based method, as shown in Fig. 7.25. It is seen that neglecting the holding stage by using the original energy-based method would lead to the overestimations of fracture toughness of the unreacted clinker and hydration matrix by 25% and 80%, respectively.

Load (mN)

Fig. 7.22 Load–displacement curve of Fe65 Mo14 C15 B6 (Guo et al., 2018)

Displacement (nm)

Table 7.6 Fracture toughness of five bulk metallic glasses materials calculated by CTOD method and original energy-based method (Guo et al., 2018) Metallic glasses materials Original energy-based method (K C–E /MPa·m½ )

CTOD method (K C–C /MPa·m½ )

Difference (|K C–E −K C–C |/K C–C ) (%)

Fe65 Mo14 C15 B6

41 ± 2

38 ± 10

Cu49 Hf42 Al9

62 ± 3

65 ± 10

5

Ni40 Cu5 Zr28.5 Ti16.5 Al10

42 ± 2

38 ± 7

10

Zr55 Al10 Ni5 Cu30

50 ± 4

52 ± 10

Ce60 Al20 Ni10 Cu10

11 ± 1

10

8

4 10

Fig. 7.23 Comparison of the fracture toughness results of shale rocks estimated by the original energy-based method with non-holding stage and the improved energy-based method with holding stage (Xu et al., 2020)

255

Fracture toughness by energy-based method (MPa.m0.5)

7.3 Energy-Based Method by Indentation

Fracture toughness by LEM method (MPa.m0.5) Fig. 7.24 Comparison of the fracture toughness of Antrim shale obtained by the original energy-based method with non-holding stage and the improved energy-based method with holding stage (Zeng et al., 2019)

Fig. 7.25 Differences of fracture toughness between the original energy-based method with non-holding stage and the improved energy-based method with holding stage (Xu et al., 2020)

Improved energy-based method Original energy-based method

256

7 Testing and Analysis of Micro Fracture Properties

(a) Fracture energy Gc (J/m2)

Gc Kc

Fracture toughness Kc (MPa•m0.5)

Based on the improved energy-based method, Xu et al. (2020) discussed the influence of the loading rate during NI on the measured fracture toughness and the fracture energy of cementitious materials, as shown in Fig. 7.26a. It is seen that greater fracture energy is obtained at a very low loading rate, the reason is attributed to the greater deformation of the hydration products at a very low loading rate which increases the fractured surface and consequently the fracture energy (Liang et al., 2017; Mallick et al., 2019). From Fig. 7.26b and c, the length of the indent edge decreases from 41.1 to 38.3 µm, when the loading rate increases from 5 to 12.5 mN/s. Moreover, the radial crack is not observed in the SEM images in Fig. 7.26, indicating the LEM method might not apply to cementitious materials.

Loading speed (mN/s) (b)

(c)

Fig. 7.26 a Changes of fracture toughness K C and fracture energy GC with loading speed, and SEM images of indentation under the loading speed of b 5 mN/s and c 12.5 mN/s for cement paste (Xu et al., 2020)

7.4 Scratch Method

257

7.4 Scratch Method Both the LEM method and the energy-based method were proposed based on the MI or NI test. All these expressions (shown in Sects. 7.2 and 7.3) were derived from a combination of dimensional analysis and empirical observations, but they are not supported by a closed-form analytical model (Akono et al., 2011). Moreover, the LEM method requires considerable efforts to measure the average length of the cracks that expand from the corners of the probe. Despite recent advances in optical imaging devices, considerable uncertainties can arise because of the observer’s skill and subjectivity or because of possible spalling around the indent. A novel technique has been proposed to extract the fracture toughness from the measured scratching forces and the depth during a scratch test. The scratch test is modeled by using the linear elastic fracture mechanics, and an expression of the fracture toughness is derived. The details of testing fracture properties by using the scratch method can be found in Sect. 3.5. Hoover and Ulm (2015) measured the fracture toughness and the fracture energy by scratch test for the cement paste with w/c ratio of 0.4 at the age of 7 hours–28 days to quantify their relationship with the degree of hydration (ξ ). The scratching speed was 6 mm/min, and the maximum vertical force was 30 N which was loaded with a constant loading rate of 60 N/min. The scratch grooves in sample at the age of 14 days and 1 day are shown in Fig. 7.27a. The indent groove at the age of 1 day was wider than that at the age of 14 days due to the less degree of hydration of cement paste. It is also observed from Fig. 7.27b that there is a logarithmic relationship between the fracture properties and the degree of hydration, which is consistent with the findings by Plassard et al. (2005) and Del Gado et al. (2014). The natural logarithmic form also shows a smooth evolution of the mechanical properties as a function of ξ and that the rate of the evolution of these properties depends on 1/ξ. Kabir et al. (2017) evaluated the micro fracture behavior of gray shale considering the effect of the scratching speed and the loading rate. Figure 7.28 shows the scratch toughness K C for 270 scratch tests carried out on the gray shale at different scratch speeds and loading rates. Kabir et al. (2017) found that the fracture toughness increases with the increasing scratch speed. As the scratching speed increases by nearly two orders of magnitude, the scratch toughness doubles its original value. A convergence of the scratch toughness is found at the high scratch speeds, which is toward a constant value and invariant with respect to the loading rate. The similar results was found by Akono and Ulm (2017) that the fracture toughness of shale increases with the scratching speed and maintains stable at the scratching speed greater than 8 mm/min. Akono and Ulm (2017) investigated the influence of scratching speed and vertical loading rate on the fracture toughness of polyvinylchloride (amorphous polymer) and polymethylene oxide (semicrystalline polymer) through microscratch test. The scratching speeds ranged 0.2–11 mm/min, and the vertical loading rates were 0, 45, 60 and 90 N/min. The results (Fig. 7.29) show that for polymer, the scratch toughness increases with the scratching speed at a given loading rate and converges

258

7 Testing and Analysis of Micro Fracture Properties

(a) 14 days

1 day

(b)

Fig. 7.28 Fracture toughness of gray shale specimen under different scratching speeds and loading rates (Kabir et al., 2017)

Kc (MPa•m0.5)

Fig. 7.27 a Scratch groove in samples cured for 14 days and 1 day, and evolution of b fracture toughness and fracture energy with hydration degree (ξ ) (Hoover & Ulm, 2015)

60 N/min 90 N/min V (mm/min)

toward a horizontal asymptote for large speeds. The value of this asymptote seems to be invariant with the loading rate, which is about 2.39 MPa·m½ for polyvinylchloride and 2.17 MPa·m½ for polymethylene oxide. The results are consistent with that tested by Kuo (1999) and Saenz et al. (2011). On the contrary, the scratch toughness decreases with increasing loading rate for a given scratching speed. The rate dependence of the scratch toughness points toward viscous processes which in the case of polymers usually involves a distribution of relaxation times and consists of the inelastic mechanisms such as the molecular entanglements, the rotation of the

(a)

FT/(2pA)0.5 (MPa•m0.5)

FT/(2pA)0.5 (MPa m0.5)

7.5 Fracture Properties Across Scale for Cementitious Materials

V (mm/min)

259

(b) 0 N/min 45 N/min 60 N/min 90 N/min

V (mm/min)

Fig. 7.29 Fracture toughness of a polyvinylchloride and b polymethylene oxide specimen under different scratching speeds and loading rates (Akono & Ulm, 2017)

molecular segments or the relative sliding of the molecular segments between the cross-linking points (Gittus, 1975).

7.5 Fracture Properties Across Scale for Cementitious Materials In this section, the development of the fracture properties of cement paste at both macro and micro scales is investigated by using the macro three-point bending test method and the nanoscratch technique, respectively. The samples subjected to the freeze–thaw (FT) cycles at the age of 7, 3 and 1 day are noted as “7d-FT,” “3dFT,” and “1d-FT,” respectively, and those not subjected to FT cycles are noted as “unfrozen.” A total of 28 FT cycles were conducted within 7 days, and each cycle is finished within 6 h with the temperature ranging between −20 and 20 °C. The samples are then sealed cured at the room temperature until the age of 28 days for the fracture property tests. For the unfrozen and frozen samples, the micro fracture properties of the clinker, the hydrates, and the ITZ phases are measured by nanoscratch technique, which are then compared to the macro fracture properties of the cement paste beam tested by the three-point bending test. The findings are expected to be beneficial for understanding the mechanism of the fracture properties development and the frost damage to cementitious materials.

260

7 Testing and Analysis of Micro Fracture Properties

7.5.1 Development of Macro Fracture Toughness and Fracture Energy The three-point bending tests were conducted on the cement paste beam samples with size of 40 mm × 40 mm × 160 mm to measure the macro fracture properties. A notch with the wide of 2 mm and the height of 20 mm was made at the bottom of the middle span to induce the crack propagation during the three-point bending test based on the recommendation by (Draft RILEM, 1985; Guinea et al., 1992; Hengst & Tressler, 1983). In the three-point bending test, the notched beam samples were loaded by the MTS810 under a loading speed of 0.05 mm/min. The average fracture toughness and fracture energy of each mixture were calculated from the results of six samples by Eqs. (7.19)–(7.22) (Draft RILEM, 1985). ∫ δmax

P(δ)dδ + mgδmax (7.19) B(D − a0 ) Pmax S ( a0 ) (7.20) KC = f B D 3/2 D ( a )1/2 (a ) ( a )3/2 ( a )5/2 ( a )7/2 ( a )9/2 0 0 0 0 0 0 = 2.9 f − 4.6 + 21.8 − 37.6 + 38.7 D D D D D D (7.21) W0 + mgδmax = ζC = B(D − a0 )

0

where ζ C and K C is the macro fracture energy and fracture toughness of cement paste beam, respectively; W 0 is the area under the load–deflection (P–δ) curve as shown in Fig. 7.30a and can be calculated from Eq. (7.22) with Δδ of 6 × 10–4 mm which is the displacement resolution of the MTS810; m is sample weight; δ max and Pmax are the maximum deflection of the sample and the maximum applied load, respectively; B, D, and S are the width, the height, and the span of the beam, respectively; a0 is the depth of the notch. W0 =

n ∫ 1

Pδn · (δn+1 − δn ) =

n ∫

Pδn · Δδ

(7.22)

1

where n is the number of data points recorded by the MTS during the three-point bending test, Pδn is the load when the deflection reaches δn . It should be noted that the end of the P–δ curve is very flat (Fig. 7.30a), resulting in a very long time period for the load to drop to zero, but having little effect on the fracture energy calculation. Therefore, each test was terminated after the deflection reaching 0.35 mm, which means δ max = 0.35 mm (Figs. 7.30b–e). The age of the FT cycles start has significant effects on the fracture properties of cement paste. The P–δ curves of the unfrozen, 7d-FT, 3d-FT, and 1d-FT cement paste beam samples are shown in Figs. 7.30b–e, respectively. It is seen that both the

7.5 Fracture Properties Across Scale for Cementitious Materials

261

400 300 200 100 0

Load (N)

700

(b) 600 P = 542.1N max 500 δ = 0.173mm 400

Pδn W0 δmax

δn δn+1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mid-span deflection (mm) 700 (c) Unfrozen 600 Load (N)

Load (N)

600 (a) Pmax 500

300 200 100

500 400 200 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mid-span deflection (mm)

(d) Pmax = 461.5N δ = 0.141mm

3d-FT

Load (N)

Load (N)

500

δ = 0.162mm

300

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mid-span deflection (mm) 600

Pmax = 500.4N

100

0

700

7d-FT

400 300 200 100 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mid-span deflection (mm)

700 600 500 400 300 200 100 0

(e)

1d-FT

Pmax = 413.5N δ = 0.132mm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mid-span deflection (mm)

Fig. 7.30 a Typical load–deflection curve of notched beam, and the measured load–deflection curves with crack pictures at the age of 28 days under the curing conditions of b unfrozen, c 7d-FT, d 3d-FT, and e 1d-FT (Kong et al., 2021)

maximum applied load and the corresponding deflection of the beams are reduced due to the early-age FT cycles. Compared to the unfrozen sample, the applied maximum load and the corresponding deflection of the frozen samples (7d-FT, 3d-FT and 1d-FT) reduce by 7.7%–23.7% and 6.4%–23.6%, respectively. The average macro fracture roughness and fracture energy calculated by Eqs. (7.19)–(7.22) of the four types of beam samples are shown in Table 7.7. It is found that the FT cycles significantly reduce the macro fracture toughness and the fracture energy, and more reduction occurs in the earlier frozen samples. Compared to the

262

7 Testing and Analysis of Micro Fracture Properties

unfrozen sample, the reduction of the macro fracture toughness of the 7d-FT, 3dFT and 1d-FT beam samples is 7.38%, 14.86%, and 23.73%, respectively, and the reduction of macro fracture energy is 13.98, 30.26, and 38.98%. Figure 7.31 shows the development of the macro fracture toughness (Fig. 7.31a) and the fracture energy (Fig. 7.31b) of cement paste beams subject to different curing conditions. It is seen that the development of fracture properties of the unfrozen specimen follows a logarithmic trend, which is consistent with the results from (Hoover & Ulm, 2015). Moreover, the development of fracture properties of the frozen samples (7d-FT, 3d-FT, and 1d-FT) during the original sealed curing period has the similar trend with that of the unfrozen specimen, indicating the reproducibility of the threepoint bending test results. Although the fracture properties of the frozen samples are reduced compared to that of the unfrozen samples after the FT cycles, the development of the fracture properties of the frozen samples is still following the logarithmic trend during the re-curing period. However, the reduced fracture properties of the frozen samples cannot be recovered completely by the later curing, as shown in Fig. 7.31.

7.5.2 Development of Micro Fracture Properties of Unreacted Clinker To calculate the micro fracture energy of the ITZ, the hydrates, and the clinker phase by Eq. (3.51), the indentation modulus of each phase should be measured first by NI technique. In NI tests, the maximum load of 4 mN with loading speed of 0.4 mN/s was used for clinker and hydrates, and the maximum load of 1 mN with loading speed of 0.1 mN/s is used for ITZ phase. In all NI tests, the holding time and the unloading time were set as 5 s and 10 s, respectively. For the nanoscratch test, the transverse scratching speed, the maximum vertical force, and the scratching path length were set as 1 µm/s, 10 mN, and 100 µm, respectively. The fracture toughness (K C-clinker ), the indentation modulus (M clinker ), and the fracture energy of the unreacted clinker (ζ C-clinker ) of the unfrozen, 7d-FT, 3d-FT, and 1d-FT samples are shown in Figs. 7.32a–c, respectively, in which each measured data is the averaged results of six scratchings or indentations. It is seen that neither the frozen conditions nor the curing age affects the three mechanical properties of the unreacted clinker, the measured fracture toughness ranges between 1.46– 1.80 MPa·m½ with an average of 1.67 MPa·m½ , the indentation modulus ranges between 100.74–118.87 GPa with an average of 108.94 GPa, and the fracture energy ranges between 26.81–31.63 J/m2 with an average of 29.32 J/m2 . The micro fracture toughness, the indentation modulus, and the fracture energy of clinker phase at the age of 28 days from the previous studies are summarized in Table 7.8 (Gautham & Sasmal, 2019; Gao et al., 2017; Kong et al., 2021; Nemecek & Hrbek, 2016; Nemecek et al., 2016). The fracture properties in Kong et al. (2021) were measured by nanoscratch technique, and that in other studies were measured

500.4

461.5

413.5

7d-FT

1d-FT

36.3

50.7

49.9

42.8

542.1

Unfrozen

3d-FT

Std. Pmax (N)

Avg. Pmax (N)

0.344

0.384

0.417

0.451

Avg. K C-paste (MPa·m½ )

0.030

0.042

0.041

0.035

Std. K C-paste (MPa·m½ )

27.47

31.58

38.95

45.28

Avg. ζ C-paste (J/m2 )

2.26

3.01

2.84

2.58

Std. ζ C-paste (J/m2 )

Table 7.7 Macro fracture toughness and fracture energy of paste at the age of 28 days under the condition of sealed curing (unfrozen sample) and subject to freeze–thaw cycles at the age of 7 (7d-FT), 3 (3d-FT), and 1 (1d-FT) days (Kong et al., 2021)

7.5 Fracture Properties Across Scale for Cementitious Materials 263

264

7 Testing and Analysis of Micro Fracture Properties (a)

50

y = 0.0814ln(x) + 0.172

0.3 y = 0.1364ln(x) - 0.0322

0.2 0.1

(b)

y = 10.608ln(x) + 9.6006

40

0.4

y = 0.1519ln(x) - 0.1136 y = 0.1666ln(x) - 0.2035 y = 0.0596ln(x) + 0.1875

0 0

5

10

Unfrozen 3d-FT

15 20 Age (d)

ζC-paste (J/m2)

KC-paste (MPa•m½)

0.5

7d-FT 1d-FT

25

30 y = 10.787ln(x) + 3.1331 y = 12.766ln(x) - 9.0161 y = 12.7ln(x) - 12.22 Unfrozen 7d-FT y = 8.4326ln(x) + 11.766 3d-FT 1d-FT

20 10 0 0

30

5

10

15 20 Age (d)

25

30

Fig. 7.31 Measured development of a fracture toughness, and b fracture energy of cement paste beams under the condition of sealed curing (unfrozen sample) and subject to freeze–thaw cycles at the age of 1 (1d-FT), 3 (3d-FT), and 7 (7d-FT) days (Kong et al., 2021) 2

(a)

KC-clinker (MPa•m½)

1.80

1.8

(b)

118.87

Mclinker (GPa)

120

1.6

110

1.4 Unfrozen 3d-FT

1.2

1.49

7d-FT 1d-FT

100 Unfrozen 3d-FT

90

1

100.74

7d-FT 1d-FT

80 0

33 ζC-clinker (J/m2)

130

5

10

15 20 Age (d)

25

(c)

30

0

5

10

15 20 Age (d)

25

30

31.63

31 29 27 Unfrozen 3d-FT

25

26.81

7d-FT 1d-FT

23 0

5

10

15 20 Age (d)

25

30

Fig. 7.32 Development of a micro fracture toughness, b indentation modulus, and c micro fracture energy of unreacted clinker in paste under the condition of sealed curing (unfrozen sample) and subject to freeze–thaw cycles at the age of 1 (1d-FT), 3 (3d-FT), and 7 (7d-FT) days (Kong et al., 2021)

by the NI technique and calculated by the improved energy-based method described in Sect. 7.3.3. It is seen that these three properties remain stable for different curing conditions, indicating the little effect of the frost attack and the age on the fracture properties of the clinker phase. Consequently, it is concluded that the frost deterioration of the macro fracture properties of cement paste beams is not associated with the unreacted clinker. Meanwhile, the fracture properties of clinker phase measured

7.5 Fracture Properties Across Scale for Cementitious Materials

265

Table 7.8 Summary of the average micro fracture toughness, indentation modulus, and fracture energy of clinker phase References

Method

Curing K C-clinker (MPa·m½ ) M clinker (GPa) ζ C-clinker (J/m2 ) condition

Nˇemeˇcek NI et al. (2016)

Unfrozen 1.69 ± 0.55

–

42.8 ± 22.6

Nˇemeˇcek and Hrbek (2016)

NI

Unfrozen 1.25 ± 0.33

–

23.4 ± 10.1

Gautham NI and Sasmal (2019)

Unfrozen 1.54 ± 0.40

–

–

Gao et al. (2017)

NI

Unfrozen –

101.6 ± 24.8

–

Kong et al. (2021)

Nanoscratch for fracture properties; NI for indentation modulus

Unfrozen 1.69 ± 0.10

109.5 ± 16.2

29.1 ± 1.5

7d-FT

1.64 ± 0.14

111.0 ± 19.3

28.7 ± 1.6

3d-FT

1.62 ± 0.16

107.8 ± 20.2

29.6 ± 2.0

1d-FT

1.68 ± 0.12

107.2 ± 18.8

29.8 ± 1.6

by nanoscratch technique are similar to that measured by NI technique, indicating the reliability of the emerging nanoscratch technique in characterizing the fracture properties of cement-based materials.

7.5.3 Development of Micro Fracture Properties of Hydrates The development of the micro fracture toughness (K C-hydrates ), the indentation modulus (M hydrates ), and the fracture energy (ζ C-hydrates ) of the hydrates phase of the unfrozen, 7d-FT, 3d-FT and 1d-FT samples is shown in Fig. 7.33. It is seen that the logarithmic trend also occurs in the development of these three mechanical properties of hydrates phase, which is similar to that of the macro fracture properties measured by the three-point bending test (Fig. 7.31). This is because the macro properties depend much on the hydrates phase which is the major phase in cementitious materials. Moreover, the development of the micro fracture properties of hydrates phase in frozen samples during the original sealed curing period is similar to that in the unfrozen samples, indicating the reproducibility of the nanoscratch test results. On the other hand, the fracture properties of the hydrates phase in the frozen samples are always less than that in the unfrozen samples during the entire re-curing period after the early-age FT cycles, and more reduction of the fracture properties can be found in the samples subjected to earlier FT cycles. Compared to the unfrozen samples, the reduction of the micro fracture toughness of the hydrates phase at the age of the 28 days for 7d-FT, 3d-FT and 1d-FT is 15.8%, 25.5%, and 38.5%,

266

7 Testing and Analysis of Micro Fracture Properties 50

(a)

1

y = 0.2409ln(x) + 0.1212

0.8 0.6 0.4 0.2 0 0 24

10

7d-FT

3d-FT

1d-FT

15 20 Age (d)

25

30

40

(b) y = 11.658ln(x) + 4.7725

30 20 10 0 0

5

10

Unfrozen

7d-FT

3d-FT

1d-FT

15 20 Age (d)

25

30

(c)

20

ζC-hydrates (J/m2)

5

Unfrozen Control

Mhydrates (GPa)

KC-hydrates (MPa•m½)

1.2

y = 4.8809ln(x) + 3.1157

16 12 8 4 0 0

5

10

Unfrozen

7d-FT

3d-FT

1d-FT

15 20 Age (d)

25

30

Fig. 7.33 Development of a micro fracture toughness, b indentation modulus, and c micro fracture energy of hydrates phase in paste under the condition of sealed curing (unfrozen sample) and subject to freeze–thaw cycles at the age of 1 (1d-FT), 3 (3d-FT), and 7 (7d-FT) days

respectively. For indentation modulus, they are 19.8%, 30.7%, and 42.9%. For micro fracture energy, they are 11.6%, 21.6%, and 33.8%. Figure 7.34 shows the development of the micro fracture toughness and the fracture energy of hydrates phase during the period of the 7 days’ FT cycles. It is seen that the samples show less development of the fracture properties of hydrates during the period of FT cycles. Moreover, the early-age FT cycles do not completely terminate the development of the fracture properties, and the fracture toughness and the fracture energy of the hydrates phase in 1d-FT samples still increase by 4.05% and 0.35% during the freeze–thaw cycling period, respectively. Table 7.9 summaries the micro fracture toughness, the indentation modulus, and the fracture energy of hydrates phase in unfrozen and frozen samples at the age of 28 days from the previous studies (Gautham & Sasmal, 2019; Kong et al., 2021; Nemecek & Hrbek, 2016; Nemecek et al., 2016; Wei et al., 2018). It is seen that these measured results agree well with each other.

7.5 Fracture Properties Across Scale for Cementitious Materials

0.8

Before F-T cycles After F-T cycles

9.28%

0.6

20

(b)

12

7.41%

0.4 4.05% 0.2

Before F-T cycles After F-T cycles

9.82%

16

ζC-hydrates (J/m2)

KC-hydrates (MPa•m½)

1 (a)

267

6.67%

8 0.35%

4 0

0 7d-FT

3d-FT

1d-FT

7d-FT

3d-FT

1d-FT

Fig. 7.34 Development of a micro fracture toughness and b micro fracture energy of hydrates phase during the period of freeze–thaw cycles (Kong et al., 2021)

Table 7.9 Summary of the average micro fracture toughness, indentation modulus, and fracture energy of hydrates phase References Method

Curing K C-hydrates (MPa·m½ ) M hydrates (GPa) ζ C-hydrates (J/m2 ) condition

Nˇemeˇcek et al. (2016)

NI

Unfrozen 0.70 ± 0.25

–

25.2 ± 10.9

Nˇemeˇcek and Hrbek (2016)

NI

Unfrozen 0.76 ± 0.13

–

14.93 ± 5.1

Gautham NI and Sasmal (2019)

Unfrozen 0.98 ± 0.32

–

–

Wei et al. (2018)

NI

Unfrozen –

39.3 ± 5.1

–

Kong et al. (2021)

Nanoscratch for fracture properties; NI for indentation modulus

Unfrozen 0.89 ± 0.13

42.3 ± 4.5

18.91 ± 2.1

7d-FT

0.75 ± 0.10

33.9 ± 2.8

16.7 ± 1.7

3d-FT

0.67 ± 0.09

29.3 ± 2.9

14.8 ± 1.9

1d-FT

0.55 ± 0.11

24.2 ± 2.1

12.5 ± 2.0

7.5.4 Development of Micro Fracture Properties of ITZ Figure 7.35 shows the development of the fracture toughness, the indentation modulus, and the fracture energy of the ITZ phase in unfrozen and frozen samples, and Table 7.10 summaries these three mechanical properties of ITZ in unfrozen and frozen samples at the age of 28 days. It is seen that the logarithmic trend of the development of these three mechanical properties is similar to that of the macro properties of the paste beam and the microproperties of the hydrates phase, indicating that the ITZ phases is also an important phase affecting the fracture properties of cement

268

7 Testing and Analysis of Micro Fracture Properties

paste. Meanwhile, earlier start of the FT cycles results in the lower fracture toughness and fracture energy of the ITZ phase, which is similar to that of the hydrates phase and the cement beam. From Fig. 7.35, the reduction of the micro fracture toughness of ITZ phase in 7dFT, 3d-FT and 1d-FT at the age of 28 days is 18.4%, 31.9% and 44.7%, respectively, and they are 15.4%, 30.1% and 41.8% for micro fracture energy. By comparing the data in Fig. 7.35 with those in Fig. 7.33, it is found that the frost damage in the ITZ phase is more severe than that in the hydrates phase, indicating that the effect of the frost damage on the vulnerable phase like ITZ is more pronounced. 1

40

(a)

0.8

y = 9.171ln(x) + 4.6462

30

MITZ (GPa)

KC-ITZ (MPa•m½)

y = 0.1949ln(x) + 0.0929

(b)

0.6

20

0.4 0.2

Unfrozen 3d-FT

0 0 18

5

10

15 20 Age (d)

10

7d-FT 1d-FT

25

Unfrozen 3d-FT

0

30

5

10

15 20 Age (d)

7d-FT 1d-FT

25

30

(c) y = 4.152ln(x) + 1.5849

15 ζC-ITZ (J/m2)

12 9 6 Unfrozen 3d-FT

3 0 0

5

10

15 20 Age (d)

7d-FT 1d-FT

25

30

Fig. 7.35 Development of a micro fracture toughness, b indentation modulus, and c micro fracture energy of ITZ phase between clinker and hydrates in paste under the condition of sealed curing (unfrozen sample) and subject to freeze–thaw cycles at the age of 1 (1d-FT), 3 (3d-FT), and 7 (7d-FT) days (Kong et al., 2021)

Table 7.10 Summary of the average micro fracture toughness, indentation modulus, and fracture energy of ITZ between clinker and hydrates K C-ITZ (MPa·m½ )

ζ C-ITZ (J/m2 )

Method

Curing condition

Nanoscratch for fracture properties; NI for indentation modulus

Unfrozen

0.69 ± 0.06

33.4 ± 2.9

14.6 ± 1.3

7d-FT

0.57 ± 0.07

24.5 ± 2.3

12.4 ± 1.9

M ITZ (GPa)

3d-FT

0.48 ± 0.04

20.8 ± 2.1

10.2 ± 0.8

1d-FT

0.39 ± 0.04

17.6 ± 1.7

8.5 ± 0.7

7.5 Fracture Properties Across Scale for Cementitious Materials (a) 9.02%

Before F-T cycles After F-T cycles

0.5 0.4

3.06%

0.3 -10.29%

0.2

12

Before F-T cycles After F-T cycles

(b) 7.60%

10

ζC-ITZ (J/m2)

KC-ITZ (MPa•m½)

0.6

269

3.57%

8 6 4

-16.68%

2

0.1

0

0 7d-FT

3d-FT

7d-FT

1d-FT

3d-FT

1d-FT

Fig. 7.36 Development of a micro fracture toughness and b micro fracture energy of ITZ phase during the period of freeze–thaw cycles (Kong et al., 2021)

Figure 7.36 shows the development of the micro fracture toughness and the fracture energy of ITZ phase during the period of the 7 days’ FT cycles. Compared to Fig. 7.34, the development of the fracture properties of ITZ phase during FT cycles is less than that of the hydrates phase, indicating that frost attack may lead to a more severe damage in the vulnerable ITZ phase, especially for the 1d-FT samples whose development of the fracture properties is negative during FT cycles.

7.5.5 Recovery of Fracture Toughness During Later Curing As seen in Figs. 7.31, 7.33 and 7.35, the macro and micro fracture properties can be recovered partially by the later sealed curing. To quantify this recovery, the ratio of the fracture properties between the frozen (7d-FT, 3d-FT and 1d-FT) and the unfrozen samples is plotted in Figs. 7.37, 7.38 and 7.39, respectively, for the paste, the hydrates phase, and the ITZ phase. It is seen that the development of the ratio of the micro fracture properties of ITZ phase and hydrates phase is similar to that of the macro fracture properties of cement 1.2

1.008 1.010

0.925 0.820

1..021

0.851

0.615 0.763

0.6 0.4

7d-FT 3d-FT 1d-FT

0.412

0.2

(b) 1.025 0.986

1 Ratio of ζC-paste

Ratio of KC-paste

1 0.8

1.2

(a)

0.8

0.720

0.586

0.6

0.611 7d-FT 3d-FT 1d-FT

0.4 0.354

0.2

0

0.860

0.791

0.989

0 0

5

10

15 20 Age (d)

25

30

0

5

10

15 20 Age (d)

25

30

Fig. 7.37 Ratio of a the macro fracture toughness and b the macro fracture energy between frozen/unfrozen paste samples (Kong et al., 2021)

270

7 Testing and Analysis of Micro Fracture Properties

1.2

1.009 1.013

1.026

0.842

0.776

0.8

0.745

0.538

0.6

0.615

0.4

7d-FT 3d-FT 1d-FT

0.2 0.230

0 0

5

10

15 20 Age (d)

25

(b)

0.978 1.011

1 Ratio of ζC-hydrates

1 Ratio of KC-hydrates

1.2

(a)

0.884

0.834

1.005

0.8

0.784 0.570

0.6

0.662

0.4

7d-FT 3d-FT 1d-FT

0.2 0.194

0 0

30

5

10

15 20 Age (d)

25

30

Fig. 7.38 Ratio of a the micro fracture toughness and b the micro fracture energy of hydrates phase between frozen/unfrozen samples (Kong et al., 2021) 1.2

0.989 0.981

1.030

0.816

0.723

0.6

0.681

0.471

0.553 7d-FT 3d-FT 1d-FT

0.4 0.2 0.183

0 0

5

10

15 20 Age (d)

25

30

(b)

0.997 0.991

1 Ratio of ζC-ITZ

Ratio of KC-ITZ

0.8

1.2

(a)

1

0.846

0.943

0.728

0.8

0.699 0.503

0.6

0.582

0.4

7d-FT 3d-FT 1d-FT

0.2 0.168

0 0

5

10

15 20 Age (d)

25

30

Fig. 7.39 Ratio of a the micro fracture toughness and b the micro fracture energy of ITZ phase between frozen/unfrozen samples (Kong et al., 2021)

beam. The ratios decrease first due to the frost attack and then increase during the later sealed curing. The frost damage to fracture properties of 1d-FT samples is the most severe which is revealed by the smallest ratio when the FT cycles are terminated. The ratios then increase after the FT cycles, which means that the frost damage to the fracture properties can be gradually recovered in the later curing period. However, none of the ratios equal to 1 when the later curing age reaches 28 days, indicating that the later curing can only recover partially the fracture properties at both micro and macro scales. At the age of 28 days, the recovery of the fracture properties is about 60%–90%, 50%–80%, and 60%–90% for the paste, the ITZ phase, and the hydrates phase, respectively. The recovery of the ITZ phase is less than that of the hydrates phase, indicating that the frost damage to the vulnerable ITZ phase is more severe. By comparing the ratio of the fracture properties between the frozen (7d-FT, 3dFT and 1d-FT) and the unfrozen samples, it is found that the earlier start of the FT cycles may result in the more recovery magnitude of the fracture properties. This is because the fracture properties of the paste and each phase are not yet well developed

7.5 Fracture Properties Across Scale for Cementitious Materials

271

due to the shorter original curing duration (such as in 1d-FT sample) and still have greater potentials for further development after FT cycles.

7.5.6 Effect of Frost Attack on Fracture Properties of CSH and CH Figures 7.40a–d show the probability distribution function (PDF) curves of the fracture toughness of the hydrates phases in the unfrozen sample and the frozen samples (7d-FT, 3d-FT and 1d-FT), in which three peaks can be found in the PDF curves. Accordingly, these three peaks can be recognized as the LD CSH, the HD CSH, and the CH phase. When the start age of the freeze–thaw cycles changes from 7 days to 1 day, all three peaks move toward the left, indicating the gradual decrease in the fracture toughness of the three phases. Based on the PDF curves shown in Fig. 7.40, the average fracture toughness of the LD CSH, the HD CSH, and the CH phases in the unfrozen and the frozen samples is summarized in Table 7.11. It is seen that more pronounced frost damage occurs in the three phases of samples subjected to the earlier age FT cycles, which is revealed by the lower ratio of the fracture toughness between the frozen and the unfrozen (a)

PDF

0.15

0.2

Unfrozen

HD CSH

0.1

(b)

7d-FT

LD CSH

0.15

LD CSH

0.05

PDF

0.2

HD CSH

0.1 0.05

CH

CH

0

0 KC (MPa•m½)

0.2

(c)

3d-FT

0.15

LD CSH HD CSH

PDF

PDF

0.15 0.1

0.05 0

KC (MPa•m½) 0.2

0.1

(d) LD CSH

1d-FT

HD CSH

0.05 CH

KC (MPa•m½)

CH 0 KC (MPa•m½)

Fig. 7.40 Distribution of fracture toughness of LD CSH, HD CSH and CH phases at the age of 28 days for a unfrozen sample, b 7d-FT, c 3d-FT, and d 1d-FT (Kong et al., 2021)

272

7 Testing and Analysis of Micro Fracture Properties

samples. The ratios of these phases at the age of 28 days are about 86%–89%, 76%– 78%, and 65%–70% for the 7d-FT sample, the 3d-FT sample, and the 1d-FT sample, respectively. Meanwhile, the effect of frost attack on the fracture toughness of the LD CSH phase is the highest, followed by the HD CSH phase and CH phase, indicating that the effect of the frost damage on the vulnerable phase is more pronounced. Table 7.11 also summaries the content of the LD CSH, the HD CSH, and the CH in hydrates phase. It is seen that the early-age freeze–thaw cycles will increase the content of the LD CSH but reduce the content of the HD CSH and the CH phases.

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials With the increasing interest and application of nanomaterials to cementitious materials, the mechanical properties of the nanomaterials-modified cementitious materials under the severe environmental conditions are critical for their successful application. The fracture toughness of the nanomaterials-strengthened cement paste by nanosilica (NS) particles and the carbon nanotubes (CNT) was investigated at the micro and the macro scales by using the nanoscratch technique and the conventional threepoint bending test, respectively. The aim is to better understand the strengthening mechanism of these two nanomaterials on the fracture properties of the cementitious materials subject to the early-age frost attack. The micro fracture toughness of the unreacted clinker, the hydrates, and the interfacial transition zone (ITZ) between the unreacted clinker and the surrounding hydrates was measured along with the thickness of ITZ. The effects of nanosilica and carbon nanotubes on the fracture properties are discussed and compared, and the different strengthening mechanisms of these two nanomaterials are recognized. The results are beneficial for designing mixtures to achieve better early-age frost resistance that satisfies the engineering applications. The nanosilica particles and the carbon nanotubes were added by substituting 2% and 0.08% of the cement by weight, respectively, which is recommended by Jalal et al. (2012), Metaxa et al. (2012), Stefanidou and Papayianni (2012), and Wang et al. (2013). The properties of nanosilica and carbon nanotubes are listed in Table 7.12. The dispersion method of the nanoparticles can be found in Sect. 2.2.1.

7.6.1 Effect of Nano Materials on Macro Fracture Properties of Paste The macro fracture toughness of each mixture was measured by using the conventional three-point bending test method. The measured load–deflection curves are shown in Fig. 7.41 for both the nanomaterials-modified paste and the OPC paste

7d-FT

3d-FT

1d-FT

0.836

0.981

1.060

LD CSH

HD CSH

CH

6.6

38.9

54.5

0.928

0.844

0.741

87.5

86.0

88.6

2.7

39.3

58.0

0.826

0.756

0.635

77.9

77.0

75.9

1.8

34.9

63.3

0.740

0.665

0.544

69.8

67.8

65.1

1.7

30.5

67.8

K C (MPa·m½ ) Content K C (MPa•m½ ) K C ratio of Content K C (MPa·m½ ) K C ratio of Content K C (MPa·m½ ) K C ratio of Content (%) frozen/unfrozen (%) frozen/unfrozen (%) frozen/unfrozen (%) (%) (%) (%)

Phase Unfrozen

Table 7.11 Micro fracture toughness and content of LD CSH, HD CSH, and CH phases at the age of 28 days obtained from PDF curves in Fig. 7.40 (Kong et al., 2021)

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials 273

274

7 Testing and Analysis of Micro Fracture Properties

Table 7.12 Properties of nanosilica and carbon nanotubes (Wei et al., 2021) Nanomaterials

Diameter

Length

Specific surface area (m2 /g)

Purity (%)

Bulk density (g/cm3 )

PH value

Electric conductivity

Nanosilica

15 nm

–

250

99.9

0.1–0.15

5–7

–

Carbon nanotubes

10–20 nm

15–30 µm

> 450

> 95

0.14

–

> 150 S/cm

subject to different curing conditions. Due to the fact that the tail of the load–deflection curve is very flat and takes a long time for the load to drop to zero, but has minor effect on the calculated macro K C , the test is terminated when the measured deflection reaches 0.35 mm, 0.40 mm, and 0.25 mm for the OPC paste beams (3P), the NS paste beams, and the CNT paste beams, respectively. It is seen clearly from Fig. 7.41 that the highest load is reached in the CNT beam followed by the NS beam, suggesting that the significantly improved bearing capacity is obtained in the carbon nanotubes-strengthened paste. Moreover, the CNT beam shows similar or overlapping load–deflection curves for all the curing conditions of either subject to early-age freeze–thaw cycles at the age of 1, 3, and 7 days or the

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Mid-span deflection (mm)

NS-Control NS-7d-F NS-3d-F NS-1d-F

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Mid-span deflection (mm)

CNT-Control CNT-7d-F CNT-3d-F CNT-1d-F

Load (N)

(c) 900 800 700 600 500 400 300 200 100 0

(b) 900 800 700 600 500 400 300 200 100 0

Load (N)

3P-Control 3P-7d-F 3P-3d-F 3P-1d-F

Load (N)

(a) 900 800 700 600 500 400 300 200 100 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Mid-span deflection (mm)

Fig. 7.41 Load–deflection curves measured by three-point bending test for a 3P sample, b NS sample, and c CNT sample at the age of 28 days (Wei et al., 2021)

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials

120

NS

100 KC increase (%)

Fig. 7.42 Increase of the macro fracture toughness (K c ) of nanomaterial-modified paste (NS and CNT) compared to OPC (3P) paste measured at the age of 28 days after subject to different curing conditions (Wei et al., 2021)

96.80

CNT 75.52

80 60

275

64.75 52.11

40 20

15.52

14.63

13.80

12.21

0 Control

7d-F

3d-F

1d-F

unfrozen conditions, which indicates the strong early-age frost damage resistance of the cementitious materials strengthened by the carbon nanotubes. Similar trend is found for the macro fracture toughness (K c ) calculated from the measured load–deflection curve. As shown in Fig. 7.42, the increase of the fracture toughness of the nanomaterials-strengthened paste compared to the OPC paste is plotted for different curing conditions. The fracture toughness was measured at the age of 28 days. It is seen that both the nanosilica particles and the carbon nanotubesstrengthened paste samples show improved macro fracture toughness compared to that of the control samples, and the strengthening effect from the carbon nanotubes is the best. The average increase of the fracture toughness of the CNT beam compared to the OPC beam ranges from 52.1–96.8%, and it is 12.2–15.5% for the NS beam. Moreover, the earlier the frozen age, the better the strengthening effect from the carbon nanotubes to improve the fracture toughness. The highest increase by 96.8% of fracture toughness of the CNT beam is reached under the severe frozen condition when subject to freeze–thaw cycles at the age of 1 day. This indicates the overwhelming advantage of using carbon nanotubes to improve the fracture properties of cementitious materials both under the unfrozen and the frozen conditions. Therefore, it is recommended to utilize carbon nanotubes to improve the fracture properties for important structures susceptible to subject to very early-age frost attack.

7.6.2 Effect of Nanomaterials on Micro Fracture Properties of Clinker Figure 7.43 shows the development of the fracture toughness of the unreacted clinker phase in the OPC paste (3P), the nanosilica-strengthened paste (NS), and the carbon nanotubes-strengthened paste (CNT) under the curing condition of both unfrozen and subject to freeze–thaw cycles at the age of 1, 3 and 7 days. It is found that the nanomaterials do not affect the fracture toughness of the unreacted clinker, and the

276

1.9 1.8 KC-clinker (MPa•m½)

Fig. 7.43 Development of fracture toughness of clinker in 3P, NS, and CNT samples subject to unfrozen (control) (green solid) and frozen conditions at the ages of 7 days (purples), 3 days (red), and 1 day (orange) (Wei et al., 2021)

7 Testing and Analysis of Micro Fracture Properties

1.7 1.6 1.5

3P NS CNT

1.4 1.3

0

5

10

15 Age (d)

20

25

30

early-age frost attack does not affect the fracture toughness of the unreacted clinker as well. This demonstrates that the clinker phase in cementitious materials is not the cause of deterioration of paste under the frost attack. The average measured fracture toughness of the unreacted clinker in 3P, NS, and CNT samples under all the curing conditions is 1.62 MPa·m½ , 1.59 MPa·m½ and 1.61 MPa·m½ , respectively.

7.6.3 Effect of Nanomaterials on Micro Fracture Properties of Hydrates The development of the micro fracture toughness of the hydrates phase was measured by the nanoscratching test during the 28 days period and plotted in Fig. 7.44 for both the nanosilica and the carbon nanotubes-strengthened paste. It is first seen that under the same early-age frozen conditions (7d-F, 3d-F, and 1d-F), the pastes strengthened either by nanosilica or by carbon nanotubes show improved fracture toughness during the entire 28 days testing period when compared to the OPC paste. More improvement is seen in CNT pastes than that in NS pastes. The highest fracture toughness of hydrates phase is always reached in CNT paste. At the age of 28 days, the hydrates fracture toughness of the NS sample has been increased by 32.8%, 46.8%, and 61.1% compared to the OPC sample for each curing conditions of 7d-F, 3d-F, and 1d-F, respectively. It is 104.2%, 139.7% and 181.1% for the CNT pastes. The early-age frozen hinders the fracture toughness development of the hydrates phase in both the OPC paste and the NS paste. The earlier the frozen, the more reduced fracture toughness at the age of 28 days. For nanosilica-strengthened paste (NS), the reduction of the fracture toughness at the age of 28 days is 5.30%, 11.54%, and 16.18% for sample frozen at the age of 7, 3, and 1 day, respectively, when compared to that of the unfrozen (NS-Control) sample. It is seen that the early-age freeze–thaw cycles do not affect or reduce the fracture toughness of the hydrates phase when the carbon nanotubes are added. And the age

2.1

3P-7d-F

2.1

3P-3d-F

1.8

NS-7d-F

1.8

NS-3d-F

NS-Control

1.5

1.057

1.2

39.50%

0.9

1.001 0.754

29.81%

0.6

9.28%

0.3

KC-hydrates (MPa•m½)

KC-hydrates (MPa•m½)

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials

0

KC-hydrates (MPa•m½)

1

3

7

2.1

3P-1d-F

1.8

NS-1d-F

8 10 Age (d)

14

21

1.057

1.2 104.20%

0.9

0.935 0.637

59.12%

0.6 0.3

7.41%

1

3

7

8 10 Age (d)

14

21

28

NS-Control

1.5

1.057

1.2 0.9 0.6

NS-Control

1.5

0

28

277

0.886 0.550

411.22% 214.15%

0.3

4.05%

0 1

3

7

8 10 Age (d)

14

21

28

(a) Development of micro fracture toughness of hydrates phase in NS samples 2.1

3P-7d-F CNT-7d-F CNT-Control 17.03%

1.8 1.5 1.2

1.548 1.540

14.80%

0.9

0.754

9.28%

0.6 0.3

3P-3d-F CNT-3d-F CNT-Control

1.8 1.5

1.548

36.81%

1.2

1.527

33.13%

0.9 0.6

0.637

7.41%

0.3 0

0 1

3

2.1

KC-hydrates (MPa•m½)

KC-hydrates (MPa•m½)

KC-hydrates (MPa•m½)

2.1

7

8 10 Age (d)

14

21

3P-1d-F CNT-1d-F CNT-Control

1.8 1.5 0.9

1

3

7

8 10 Age (d)

14

21

28

1.548 1.546

63.20%

1.2

28

54.13%

0.550

0.6 4.05%

0.3 0 1

3

7

8 10 Age (d)

14

21

28

(b) Development of micro fracture toughness of hydrates phase in CNT samples Fig. 7.44 Development of micro fracture toughness of hydrates phase in OPC (3P) and a nanosilicastrengthened (NS) pastes, and b carbon nanotubes strengthen (CNT) pastes subject to unfrozen (control) and frozen conditions at the ages of 7 days, 3 days, and 1 day (Wei et al., 2021)

278

7 Testing and Analysis of Micro Fracture Properties

when frozen starts also has a minor effect on the fracture toughness of the hydrates phase. This suggests the superior performance of carbon nanotubes on improving the micro fracture toughness of the hydrates phase in the cementitious materials subject to the early-age frost attack.

7.6.4 Effect of Nanomaterials on Micro Fracture Properties of ITZ The development of fracture toughness of ITZ phase in the OPC, the NS, and the CNT sample under the conditions of unfrozen (control) and subject to frozen at the age of 1, 3, and 7 days is shown in Fig. 7.45. Similar to the development of hydrates fracture toughness, the pastes strengthened either by nanosilica or by carbon nanotubes also show improved fracture toughness of ITZ phases during the entire 28 days testing period, and more improvement is seen in CNT pastes than that in NS pastes. Under the frozen conditions and compared to the OPC paste, nanosilica is capable of improving the fracture toughness of ITZ at the age of 28 days by 45.8%, 68.3%, and 77.8% for the NS samples subject to frozen at the age of 7, 3, and 1 day, respectively. They are 94.7%, 141.2% and 187.3% for the CNT samples. The strengthen magnitude of the nanosilica particles on the ITZ is more than that on the hydrates (45.8%, 68.3% and 77.8% for ITZ phase versus 32.8%, 46.8% and 61.1% for hydrates phase). Whereas, the enhancement of carbon nanotubes on the fracture toughness of hydrates phase and ITZ phase is similar (94.7%, 141.2%, and 187.3% for ITZ phase versus 104.2%, 139.7%, and 181.1% for hydrates phase). The early-age frozen also hinders the development of the ITZ fracture toughness in the nanosilica-strengthened paste (NS) (Fig. 7.45a). The earlier the frozen, the more reduced fracture toughness at the age of 28 days. The reduction of ITZ fracture toughness at the age of 28 days is 6.86%, 14.43%, and 29.07% for the sample frozen at the age of 7, 3, and 1 day, respectively, when compared to the unfrozen NS samples. More frost damage occurs in ITZ phase compared to that in the hydrates phase (6.86%, 14.43%, and 29.07% for ITZ phase versus 5.30%, 11.54%, and 16.18% for hydrates phase), indicating that the effect of frost attack is more pronounced in vulnerable phase in nanosilica particles-strengthened paste. Similar to Fig. 7.44b, it is seen from Fig. 7.45b that the early-age freeze–thaw cycles and the age when the frozen starts also have a minor effect on the fracture toughness of the ITZ phase in the carbon nanotubes-strengthened paste (CNT). This suggests the superior performance of carbon nanotubes on improve the micro fracture toughness of both the ITZ phase and the hydrates phase in the cementitious materials. For the nanomaterials-strengthened paste, one interesting phenomenon is observed from Figs. 7.44 and 7.45 that higher increase of fracture toughness occurs in the NS samples than that of the CNT samples during the 7 days’ freeze–thaw cycling period. This characteristic holds true for both the hydrates phase (Fig. 7.44) and the ITZ phase (Fig. 7.45). This suggests the ability of strengthening by the nanosilica

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials 3P-7d-F NS-7d-F NS-Control

KC-ITZ (MPa•m½)

1.2 1

47.82%

0.8

0.831 0.570

35.67%

0.6 0.4

3P-3d-F

1.2

NS-3d-F NS-Control

1

0.8

0.2

0

0

1.4

7

8 10 Age (d)

14

21

3P-1d-F NS-1d-F NS-Control

1.2 1

0.776 0.461

65.29%

0.4

9.02%

3

0.888

101.26%

0.6

0.2 1

KC-ITZ (MPa•m½)

0.888

1.4

KC-ITZ (MPa•m½)

1.4

279

28

3.06%

1

3

7

8 10 Age (d)

14

21

28

0.888

0.8 422.39%

0.6 0.4

0.688 0.387

228.17%

0.2

-10.29%

0 1

3

7

8 10 Age (d)

14

21

28

(a) Development of micro fracture toughness of ITZ phase in NS samples 3P-7d-F CNT-7d-F CNT-Control 23.83%

KC-ITZ (MPa•m½)

1.2 1

1.110 20.90%

0.8

0.570

0.6

9.02%

1

1.112 1.120

46.38%

0.8

32.38%

0.6

0.461

3.06%

0.2

0.2

0

0 1

3

7

8 10 Age (d)

14

21

3P-1d-F CNT-1d-F CNT-Control

1.4 1.2

KC-ITZ (MPa•m½)

1.2

0.4

0.4

28

1

3

7

8 10 Age (d)

14

21

28

1.112

1 0.8

3P-3d-F CNT-3d-F CNT-Control

1.4 1.112

KC-ITZ (MPa•m½)

1.4

1.113 81.12%

0.6

61.92%

0.387

0.4 0.2

-10.29%

0 1

3

7

8 10 Age (d)

14

21

28

(b) Development of micro fracture toughness of ITZ paste in CNT samples Fig. 7.45 Development of micro fracture toughness of ITZ phase in OPC (3P) and a nanosilicastrengthened (NS) pastes, and b carbon nanotubes-strengthened (CNT) pastes subject to unfrozen (control) and frozen conditions at the ages of 7 days, 3 days, and 1 day (Wei et al., 2021)

280

7 Testing and Analysis of Micro Fracture Properties

particles even during the period of frost attack due to the extra contribution from the pozzolanic reaction of the nanosilica particles. The earlier the frozen age, the greater the magnitude of the fracture toughness increase during the 7 days’ freeze–thaw cycling period.

7.6.5 Effect of Nanomaterials on ITZ Thickness It was found that the thickness of ITZ affects the fracture properties of concrete, less ITZ thickness provides better fracture properties for concrete (Cwirzen & Penttala, 2005; Lizarazo-Marriaga et al., 2014). It is of significance to investigate the effect of the nanomaterials on the ITZ thickness to better understand their strengthening mechanism. Due to the fact that there is no significant difference between the fracture toughness of the ITZ and the hydrates phases at the age of one day, it is not possible to accurately distinguish the ITZ phase from the hydrates phase based on their fracture toughness. Therefore, this study discusses only the ITZ thickness in paste at the age of or greater than 3 days. The ITZ thickness quantification method has been proposed based on the fracture toughness distribution along the scratching path, which has been illustrated in Sect. 4.6.6. Under the unfrozen condition, the ITZ thickness development in the OPC, the NS, and the CNT paste is shown in Fig. 7.46. It is seen that the ITZ thickness in the NS paste is slightly less than that in the OPC paste, whereas the ITZ thickness in the CNT paste is slightly greater than that in the OPC paste. The addition of nanosilica can promote the pozzolanic reaction to form denser matrix (Gaitero et al., 2008; Jo et al., 2007), and also works as filler due to its much smaller size (three orders of magnitude less) compared to that of the clinker particles (Quercia & Brouwers, 2010). This leads to a thinner ITZ thickness in the NS sample. Previous studies show that the carbon nanotubes can improve the early-age hydration rate by acting as the nucleating sites to produce the denser hydrates (Marker et al., 2005). On the other hand, Marker and Chan (2009) and Wang et al. (2020) found that

3P-Control NS-Control Difference

4

1

5

0.8

4

3

0.6

2

0.4

1

0.2 0

0 3

7

8

10 14 Age (d)

21

28

(b)

3P-Control CNT-Control Difference

1 0.8

Difference (μm)

(a)

Difference (μm) Thickness of ITZ (μm)

Thickness of ITZ (μm)

5

3

0.6

2

0.4

1

0.2

0

0 3

7

8

10 14 Age (d)

21

28

Fig. 7.46 Development of ITZ thickness in a OPC versus NS paste, and b OPC versus CNT paste under the unfrozen condition (Wei et al., 2021)

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials

281

the carbon nanotubes are bundles distributed or adhered on the cement grains due to their high surface energy. Therefore, it can be deduced that the adhered carbon nanotubes might form a barrier between the unreacted cement particle and the products. This barrier might have some effects on the dense of ITZ, which leads to a greater ITZ thickness in the CNT paste. The decreasing difference of ITZ thickness over time is also found between the OPC and the CNT paste from Fig. 7.46b, and the difference diminishes at the age of 28 days. Figure 7.47 shows the development of ITZ thickness in the OPC, the NS, and the CNT paste subject to freeze–thaw cycles at the age of 7 and 3 days. A clear decreasing trend over time for ITZ thickness is observed for all samples, and the ITZ thickness is the greatest in CNT paste, followed by OPC paste, the smallest one is in NS paste. The period with missing curve is when the sample is subject to freeze–thaw cycles. The decreased ITZ thickness during the freeze–thaw period represents the ability of further hydration and densifying of the cementitious materials even under the frozen condition, which is also supported by the findings in Figs. 7.44 and 7.45 of the continuous increase of fracture toughness during the 7 days’ freeze–thaw cycling period. From Fig. 7.47, the largest ITZ thickness reduction of 0.48 µm and 0.67 µm is found in NS paste subject to the freeze–thaw cycles at the age of 7 days and 3 days, respectively. This verifies again that nanosilica particles promote the hydration and densifying of ITZ even during the freeze–thaw cycling period. The reduction of the ITZ thickness in the CNT paste during the freeze–thaw cycling period is less than that in the OPC paste, indicating that the addition of carbon nanotubes is not effective on forming denser structure in the ITZ during the frozen period. From the fact that the fracture toughness of ITZ in the CNT paste is much greater than that in the OPC paste (Fig. 7.45b), it is the carbon nanotubes themselves contribute to the fracture toughness increase of the ITZ phase. The ITZ thickness at the age of 28 days in the OPC, the NS, and the CNT pastes is compared for the different frozen conditions in Fig. 7.48. It is seen that the early-age frozen increases the ITZ thickness for all samples. The earlier the age at frozen, the greater the ITZ thickness. The ITZ thickness at the age of 28 days for samples subject to freeze–thaw cycles at the age of 1 day is twice of that of the unfrozen samples. (a)

4

3P-7d-F

5

NS-7d-F

4

CNT-7d-F

3

0 0

5

10

15 20 Age (d)

1

F-T cycles

0

25

30

CNT-3d-F

-0.67μm

2

-0.48μm F-T cycles

1

3P-3d-F NS-3d-F

-0.07μm -0.14μm

3

-0.19μm -0.22μm

2

(b)

Thickness of ITZ (μm)

Thickness of ITZ (μm)

5

0

5

10

15 20 Age (d)

25

30

Fig. 7.47 Development of ITZ thickness in OPC (3P), NS, and CNT samples subject to freeze–thaw cycles at the age of a 7 days, and b 3 days (Wei et al., 2021)

282

2.5 3P Thickness of ITZ ( μm)

Fig. 7.48 ITZ thickness at the age of 28 days for OPC, NS, and CNT pastes subject to unfrozen and frozen conditions at the age of 7 (7d-F), 3 (3d-F), and 1 (1d-F) day (Wei et al., 2021)

7 Testing and Analysis of Micro Fracture Properties

2 1.5

NS CNT

1 0.5 0 Control

7d-F

3d-F

1d-F

7.6.6 Strengthening Mechanism of Fracture Properties by Nanomaterials The increase of the micro fracture toughness of the ITZ and the hydrates phases in the nanomaterials-strengthened paste is quantified by comparing to that of the OPC paste. The results are plotted in Fig. 7.49. It is seen from Fig. 7.49 that the greater increase of the fracture toughness is found for the ITZ compared to that of the hydrates phase in the NS (nanosilica-strengthened) paste, whereas the increase of the fracture toughness of hydrates is greater than that of the ITZ phase in the carbon nanotubes-strengthened (CNT) paste. Meanwhile, the earlier the frost attack, the more significant the improving effect of the nanomaterials on the fracture toughness of the ITZ and the hydrates phases. The improvement of the nanosilica on the ITZ fracture toughness is larger than that on the hydrates fracture toughness under both the unfrozen and the frozen conditions. The reason might be that the nanosilica particles fill the more porous ITZ phase and at the same time react with the calcium hydroxide in ITZ to form CSH and further strengthen the ITZ. This is also seen from Fig. 7.49a that there is a peak at the age of 7 days, which is suspected to be strengthened by the early-age accelerated pozzolanic reaction of the nanosilica particles. This is consistent with the difference of the thickness of ITZ between the NS paste and the OPC (3P) paste (Fig. 7.46a), that the difference reaches maximum at the age of 7 days. The carbon nanotubes can improve both the fracture toughness of the hydrates and the ITZ phase, and are more effective in improving the fracture toughness of the hydrates phase, possibly due to their stronger bridging effect in hydrates (Fig. 7.50a). The bridging effect is significant during the early ages with the maximum fracture toughness improvement of 500% at the age of 1 day and drop to 185% at the age of 3 days. This is because of the dominant strengthening effect of the carbon nanotubes compared to the vulnerable matrix at early ages. It should be aware that the bridging effect depends very much on the bounding between the carbon nanotubes and the matrix. The bridging fails if carbon nanotubes are pull out from the matrix (Fig. 7.50b).

7.6 Strengthening Mechanism of Fracture Properties by Nanomaterials 800

800

NS-Control

KC increase (%)

(a)

KC increase (%)

600 400 200 0

800

5

10

15 20 Age (d)

25

200

0

5

10

15 20 Age (d)

30

ITZ Hydrates

KC increase (%)

KC increase (%)

25

CNT-7d-F

600

400

400 200

0

0 0

800

5

10

15 20 Age (d)

25

30

0

KC increase (%)

KC increase (%)

400 200 0 0 800

5

10

15 20 Age (d)

5

800

NS-3d-F

600

25

10

600

200 0

30

ITZ Hydrates

400 200

5

10

800

400

25

CNT-3d-F

0

NS-1d-F

600

15 20 Age (d)

0

30

KC increase (%)

KC increase (%)

ITZ Hydrates

400

800

NS-7d-F

200

(d)

600

30

600

(c)

CNT-Control

0 0

(b)

283

15 20 Age (d)

25

30

CNT-1d-F

600

ITZ Hydrates

400 200 0

0

5

10

15 20 Age (d)

25

30

0

5

10

15 20 Age (d)

25

30

Fig. 7.49 Increase of micro fracture toughness of hydrates phase and ITZ phase in NS and CNT samples relative to the OPC paste sample under the a unfrozen (control) and frozen conditions at the age of b 7 (7d-F), c 3 (3d-F), and d 1 (1d-F) day (Wei et al., 2021)

284

7 Testing and Analysis of Micro Fracture Properties

(b)

(a) Carbon nanotube

Carbon nanotube CSH

Failed carbon nanotube

CSH CSH

CSH

Fig. 7.50 SEM image of a carbon nanotubes bridging CSH, and b the failed carbon nanotubes in paste sample (Wei et al., 2021)

7.6.7 Overall Strengthening Effectiveness by Nanomaterials The overall effect of nanomaterials on the fracture properties in terms of the fracture toughness of the hydrates phase, the ITZ phase, and the thickness of ITZ is shown in Fig. 7.51 for pastes subjected to unfrozen and frozen at the age of 7, 3, and 1 day. The equilateral triangle in the middle represents the value of the above three properties of OPC paste, which are all assigned with value of 1. Any triangle with vertex within the middle equilateral triangle represents the lower property value than that of the OPC, and the vertex out of the middle equilateral triangle represents the higher property value than that of the OPC paste. It is seen that the carbon nanotubes have contributes significantly to improve the fracture toughness of both ITZ and hydrates phases, and the improvement is pronounced particularly for the paste subject to frozen at very early ages. The strengthening magnitude by the carbon nanotubes on fracture toughness of the ITZ and the hydrates is almost the same, indicating a superior performance of carbon nanotubes that they can improve both the individual and the overall fracture toughness of paste. Therefore, it is recommended to utilize carbon nanotubes to strengthen the cementitious materials which is suspect to be subject to the early-age frost attack. Compared to carbon nanotubes, the nanosilica is more effective in reducing the thickness of ITZ between the clinker and the surrounding hydrates. Moreover, the strengthening effect by nanosilica on fracture toughness of ITZ is better than that on hydrates.

7.7 Comparison of Fracture Properties Between Micro and Macro Scale It is well known that the cementitious materials possess multiscale and multiphase features. The properties of each phase at the microscale determine the macroscale

7.7 Comparison of Fracture Properties Between Micro and Macro Scale

(a) KC of ITZ

Thickness of ITZ

(c)

KC of ITZ

Thickness of ITZ

3P-Control NS-Control CNT-Control

KC of hydrates

3P-3d-F NS-3d-F CNT-3d-F

KC of hydrates

(b) KC of ITZ

Thickness of ITZ

(d) KC of ITZ

Thickness of ITZ

285

3P-7d-F NS-7d-F CNT-7d-F

KC of hydrates

3P-1d-F NS-1d-F CNT-1d-F

KC of hydrates

Fig. 7.51 Strengthening effect of nanosilica and carbon nanotubes at the age of 28 days on the thickness of ITZ, micro fracture toughness of ITZ and hydrates for a the unfrozen condition, b frozen at the age of 7 days, c frozen at the age of 3 days, and d frozen at the age of 1 days (Wei et al., 2021)

behavior (Bernard et al., 2003). From Figs. 7.31, 7.33 and 7.35, it is found that the development of the micro fracture properties of the hydrates and the ITZ phases and the development of the macro fracture properties of cement paste follow similar logarithmic trend with the curing age. To study the relationship between the macro and the micro fracture properties, the macro fracture properties of cement paste and the micro fracture properties of each phase for the unfrozen, 7d-FT, 3d-FT and 1dFT samples are analyzed. As shown in Fig. 7.52, the macro fracture properties of paste can be related to the micro fracture properties of the ITZ and the hydrates phases which were measured at the same age and the same curing conditions. The results show that the development of the macro fracture properties of the cement paste follows linearly with the development of the micro fracture properties of the ITZ and the hydrates phases. Moreover, the micro fracture toughness of the ITZ and

286

7 Testing and Analysis of Micro Fracture Properties y = 2x

(a)

50

y=x

0.8

40

y = 2.2345x 0.6 -0.1432

0.2

0.2

0.4 0.6 0.8 Macro KC (MPa•m½)

y = 0.4849x - 2.0326

10

Hydrates 0

y = 0.5x

20

ITZ

0

ITZ Hydrates

30

y = 1.7955x -0.1376

0.4

y=x

(b)

Micro ζC (J/m2)

Micro KC (MPa•m½)

1

y = 0.415x - 2.8772

0 1

0

10

20 30 40 Macro ζC (J/m2)

50

Fig. 7.52 Relationship between a macro and micro fracture toughness, and b macro and micro fracture energy under the same curing condition (Kong et al., 2021)

the hydrates phases is 1–2 times of the macro fracture toughness of the cement paste, whereas the micro fracture energy of these two phases is less than half of the macro fracture energy of the cement paste. Therefore, the macro fracture properties of the cement paste can be estimated by the micro fracture properties of each phase. The authors reviewed the fracture toughness of concrete, mortar, cement paste, as well as ITZ, hydrates, and clinker phase from the previous studies (Cao et al., 2019; Gautham & Sasmal, 2019; Hu et al., 2014; Nazerigivi et al., 2018; Qing et al., 2018; Xu et al., 2020). It is found that the fracture toughness of the cementitious materials decreases with the increasing scale. The fracture toughness of concrete, mortar, cement paste, ITZ, hydrates, and clinker are 0.22 MPa·m½ , 0.38 MPa·m½ , 0.52 MPa·m½ , 0.70 MPa·m½ , 0.79 MPa·m½ , and 1.94 MPa·m½ , respectively. The decreasing fracture toughness with the increasing scales is mostly due to the larger defects existing in the large-scale specimens. On the other hand, it is found that the macro fracture energy of the cement paste is larger than the micro fracture energy of each individual phase. This can be explained from the experimental point of view that the main crack may deviate from the initial direction of propagation in the beam samples during the three-point bending test due to the multiphase and microcracking nature of the cement paste. This will result in an actual large fracture area in the paste beam during the three-point bending test. However, the effect of the larger fracture area is not considered in Eq. (7.19) for calculating the fracture energy. Moreover, when the main crack propagates, the crack may develop into multiple branches (Karihaloo, 2003; Prasad et al., 2010). This phenomenon will also result in the increase of the fracture area, and the larger fracture area is still not considered in Eq. (7.19). In terms of frost damage, the relationship between the micro fracture properties loss and the macro fracture properties loss at the age of 28 days was analyzed. The macro fracture properties loss measured by the six three-point beam bending tests are related to the micro fracture properties loss measured by the six nanoscratching tests, as shown in Fig. 7.53. The results show that the macro fracture properties loss

7.8 Summary 60

287 50

y=x

y = 2x

(a)

y=x

(b)

50 Micro KC loss (%)

40

40

Micro ζC loss (%)

1d-FT 3d-FT

30

20 7d-FT

Hydrates

0 0

10

20 30 40 Macro KC loss (%)

50

1d-FT

20

3d-FT

10

ITZ

10

30

ITZ 7d-FT

0 60

0

10

Hydrates

20 30 40 Macro ζC loss(%)

50

Fig. 7.53 Relationship between a macro and micro fracture toughness loss, and b macro and micro fracture energy loss at the age of 28 days compared to the unfrozen condition (Kong et al., 2021)

is also linearly proportional to the micro fracture properties loss for the ITZ and the hydrates phases. The micro fracture toughness loss of the ITZ and the hydrates phases is about 2 times of the macro fracture toughness loss of the cement paste, and the micro fracture energy loss almost equals to the macro fracture energy loss.

7.8 Summary The existing methods for the fracture toughness measurement of the film materials, the rock materials, and the cementitious materials at the micro scales are reviewed in this chapter, including the LEM method, the energy method, and the nanoscratch method. The Lawn–Evans–Marshall (LEM) method by indentation is commonly used for measuring the fracture properties of the brittle materials, which applies to various types of crack propagations. LEM method requires accurate measurement of the crack length, which hinders its application for materials that might not show notable cracks upon failure, such as the quasi-brittle materials. The energy-based method solves the above problem, which calculates the fracture toughness based on the load–displacement curve obtained from the indentation test. The original energy-based method does not deal with the indentation curve with a holding stage. The improved energy-based method is proposed to cooperate with the standard NI test with a holding stage in the load–displacement curve, which is considered to be more desirable for quantifying the fracture toughness of materials tested by the NI test. As a different technique from the LEM method and the energy-based method, the nanoscratch method measures and calculates the fracture properties of individual phases based on the linear elastic fracture mechanics assumption, which is verified to perform well when characterizing the fracture properties of cementitious materials.

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The fracture toughness can be quantified for individual phases, including the ITZ phase between the unreacted grain and the surrounding hydrate. The carbon nanotubes can improve the fracture toughness of both hydrates and ITZ phase, particularly under the condition of subject to frozen at very early ages, and are more effective in improving the fracture toughness of the hydrates phase, possibly due to their stronger bridging effect in hydrates. The nanosilica is more effective in reducing the thickness of ITZ between the clinker and the surrounding hydrates. Moreover, the strengthening effect by the nanosilica on the fracture toughness of ITZ is better than that on the hydrates phase. The development of both the macro fracture properties of paste and the micro fracture properties of the hydrates and the ITZ phases follows logarithmic trend with the curing age. The fracture toughness of the cementitious materials increases with the decreasing size scale, and it is opposite for the fracture energy. A linear relationship exists between the macro fracture properties of the paste and the micro fracture properties of the ITZ and the hydrates phases. And the reduction of the fracture properties from the frost attack at the two scales is also linearly proportional.

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Chapter 8

Testing and Analysis of Micro Creep

Abstract Creep can be beneficial or detrimental for a concrete structure, the fundamental mechanism of which has not been fully understood due to the complex microstructures of cementitious materials. This chapter aims to summarize the current research efforts concerning the testing and analysis of the creep and the creep recovery of cementitious materials at both macro and micro scales. The instrumented indentation technique is used to evaluate the creep behavior of cementitious materials at both nano and micro scales. A strong analogy of the logarithmic creep behavior is found between the cementitious materials and the soils, and the creep modulus is an intrinsic parameter that reflects the long-term creep rate of cement paste. A dashpot with a variable viscosity that increases linearly with time can capture the logarithmic creep behavior. In addition, the ratio of the recoverable creep to the total creep can be obtained based on the measured results using two indentation schemes, i.e., the “loading-holding-unloading-reholding” scheme and the “loading-unloadingreholding” scheme, and the recoverable creep can account for up to 0.41 of the total creep. The content of this chapter is expected to provide an in-depth understanding of the testing and analysis of micro creep and creep recovery of cementitious materials. Keywords Characteristic time · Contact creep modulus · Creep recovery · Logarithmic creep behavior · Long-term creep rate · Loading strain rate

8.1 Introduction Creep refers to the time-dependent deformation development induced by a constant sustained stress, and creep recovery denotes the time-dependent deformation recovery when the external load is removed (Neville, 1971), which can be either beneficial or detrimental to a concrete structure. For example, creep can relax some of the restrained stress in concrete structures, and thus reduces the cracking risk. On the other hand, creep can cause a loss of prestress over time in structure, and thus reduces the serviceability of the structure. Since the first publication on the creep of concrete in 1907 (Hatt, 1907), the creep of concrete has been widely studied for more than 100 years, and many creep prediction models have been developed to calculate the time-dependent deformation of concrete structures, such as the ACI code, the fib © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_8

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model code, the GL2000 code, etc. However, the fundamental mechanism of creep has not been fully understood due to the complex microstructures of cementitious materials. It is widely known that cementitious materials are heterogeneous materials that exhibit creep properties at different scales ranging from concrete to cement paste, down to the individual hydration products, i.e., CSH gel. Recently, the NI technique has been proved a successful technique to evaluate the micro creep and creep recovery of cementitious materials at micro scale (Frech-Baronet et al., 2017, 2020; Gan et al., 2020; Vandamme & Ulm, 2009; Wei et al., 2017; Zhang et al., 2014). The creep property can be assessed based on the change of the indentation depth over the holding stage of the indentation test, while the creep recovery can be evaluated based on the recovery of the indentation depth over the reholding stage. By using a suitable indentation force, one can not only measure the creep property of the single phase (e.g., LD CSH, HD CSH, and clinker) at nanoscale, but also measure the homogeneous creep of cement paste at the micro scale, which can provide experimental supports to illustrate the creep mechanism of concrete structures. However, why the indentation technique enables the quantitative characterization of the long-term kinetics of the logarithmic creep of cement-based materials orders of magnitude faster than that by the macroscopic test is still unclear (Vandamme, 2018; Vandamme & Ulm, 2013), which would hinder the establishment of the multiscale model to predict the macro creep of cementitious materials. This chapter aims to summarize the current research efforts concerning the testing and analysis of the micro creep and the creep recovery of cementitious materials. First, the existing methods to measure the creep and creep recovery of cementitious materials at macro scale are reviewed. Then the method to quantify the micro creep and creep recovery of cementitious materials based on the indentation technique is introduced. Finally, the measured creep and creep recovery of cementitious materials by the indentation technique are recalled and discussed, which is expected to provide an in-depth understanding of testing and analysis of micro creep and creep recovery of cementitious materials.

8.2 Creep Test of Cementitious Materials at Macro Scale Up to date, many experimental methods have been developed at macro scale to investigate the creep and creep recovery of cementitious materials, which can be roughly divided into the uniaxial creep test (Atrushi, 2003; Chen et al., 2019; Ranaivomanana et al., 2013; Rossi et al., 2013a; Wei et al., 2018), the flexural creep test (GarcíaTaengua et al., 2014; Kassimi & Khayat, 2021; Liang & Wei, 2019a), the multiaxial creep tests (Charpin et al., 2018; Kim et al., 2005; Liu et al., 2002), and the restrained creep test (Gao et al., 2013; Liang & Wei, 2019b; Østergaard et al., 2001).

8.2 Creep Test of Cementitious Materials at Macro Scale

295

8.2.1 Uniaxial Compressive Creep Test The uniaxial compressive creep test is one of the most widely used methods to study the creep property of cementitious materials at macro scale (Atrushi, 2003; Cagnon et al., 2015; Chen et al., 2019; Rossi et al., 2013b). The common experimental setup for compressive creep measurement is shown in Fig. 8.1. In the uniaxial compressive creep test, at least two specimens should be prepared. The constant uniaxial compressive load is imposed on one of the specimens (the loaded one). Then the deformation of the loaded specimen is monitored and recorded by the deformationtype sensor (e.g., LVDT) during the measurement. Since the shrinkage deformation occurs simultaneously during the creep, another specimen (the unloaded one) is set free of loading to measure the shrinkage deformation. By assuming that the shrinkage deformation of the loaded specimen is the same as that of the unloaded specimen, one can obtain the stress-dependent deformation of the loaded specimen (i.e., the elastic plus the creep deformation) by subtracting the shrinkage deformation measured on the unloaded specimen from the measured total deformation on the loaded one. After the sustained load is removed from the loaded specimen, one can continue to monitor the deformation of this specimen, which can be used to evaluate the creep recovery. The typical force–time curve and the time-dependent deformation–time curve are displayed in Fig. 8.2. It should be noted that the creep property of cementitious materials depends on the applied stress level. Usually, the applied stress level should be less than 30–40% of the compressive strength, which ensures that the concrete exhibits a linear creep property, i.e., the creep deformation is proportional to the applied stress. When the applied stress increases during the creep test, nonlinear creep or damage will occur within the specimen, and the linear relationship between the creep deformation and the applied stress will no longer hold. Based on the measured compressive creep results, the creep property can be normally assessed by three terminologies, i.e., the creep compliance J(t, t 0 ), the specific creep C(t, t 0 ), and the creep coefficient ϕ(t, t 0 ). Creep compliance denotes Fig. 8.1 Experimental setup for compressive creep measurement (Cagnon et al., 2015)

8 Testing and Analysis of Micro Creep

Deformation

Stress

296

t0

t

creep creep recovery elastic time

time

Fig. 8.2 Illustration of a stress–time curve, and b time-dependent deformation–time curve during the creep test

the total strain εt (t) (including the elastic strain εe (t0 ) and creep strain εc (t)) at time t induced per unit stress σ0 (t0 ) imposed at time t 0 , which can be expressed as: J (t, t0 ) =

εe (t0 ) + εc (t) εt (t) = σ0 (t0 ) σ0 (t0 )

(8.1)

Specific creep denotes the creep strain εc (t) at time t induced per unit stress σ0 (t0 ) imposed at time t 0 , which can be expressed as: C(t, t0 ) =

εc (t) σ0 (t0 )

(8.2)

The creep coefficient denotes the ratio of the creep strain εc (t) at time t induced per unit stress σ0 (t0 ) imposed at time t 0 to the elastic strain εe (t0 ) caused by the imposed stress σ0 (t0 ), which can be expressed as: ϕ(t, t0 ) =

εc (t) εe (t0 )

(8.3)

Based on Eqs. (8.1) to (8.3), one can find the three terminologies satisfy the following equation: J (t, t0 ) =

1 1+ϕ(t, t0 ) +C(t, t0 )= E(t0 ) E(t0 )

(8.4)

8.2.2 Uniaxial Tensile Creep Test Uniaxial tensile creep test is another method to study the creep property of cementitious materials since characterizing the tensile creep behavior of concrete is of fundamental importance for tensile stress and cracking potential evaluations (Bissonnette et al., 2007; Garas et al., 2009; Rossi et al., 2013a; Yao & Wei, 2014). The experimental procedure of the uniaxial tensile creep test is similar to that of the uniaxial

8.2 Creep Test of Cementitious Materials at Macro Scale

Flail arm lever

297

Disks with each mass of 13 kg for loading

Sample Frame

Fig. 8.3 Experimental setup for tensile creep measurement (Rossi et al., 2013a)

compressive creep test. That is, at least two specimens should be prepared for the measurement to separate the shrinkage deformation from the creep deformation. One typical experimental setup for tensile creep measurement is shown in Fig. 8.3, which adopts a frame-type flail arm lever. The derivation of the tensile creep of cementitious materials based on the measured results can also be done according to Eqs. (8.1)–(8.4).

8.2.3 Flexural Creep Test Since many concrete structures or components (e.g., concrete pavement and concrete beams) are subject to flexural loading, it is necessary to investigate the creep property of cementitious materials under the flexural loading condition. Recently, several studies concerning the measurement of flexural creep of cementitious materials have been reported (García-Taengua et al., 2014; Ghezal & Assaf, 2016; Liang & Wei, 2018; Tailhan et al., 2013). In the flexural creep test, a concrete beam is subject to flexural load, then the deflection of the beam or the strains at the top and bottom surfaces of the beam is monitored and recorded, which is used to evaluate the flexural creep of cementitious materials. Figure 8.4 shows one typical experimental setup for flexural creep measurement. This test setup uses a four-point bending configuration. Concrete beams with the dimension of 50 mm in height, 50 mm in width, and 1220 mm in length, under the conditions of loaded and unloaded, were measured by their deflection using the LVDTs. The measuring locations were at the mid-span and the 350 mm from the mid-span. During the test, the loaded beam was subject to two loads of 10 kg symmetrically which were 200 mm away from the beam supports. Based on the measured deflection results, the creep coefficient of cementitious materials can be obtained according to the following equation:

298

8 Testing and Analysis of Micro Creep

Fig. 8.4 Experimental setup for flexural creep measurement (Liang & Wei, 2018)

ϕ(t, t0 ) =

f m−L (t, t0 ) − f ins−L (t0 ) f ins−L (t0 ) + f ins−U (t0 )

(8.5)

where f m−L (t, t0 ) is the measured deflection at time t of the loaded beamed with flexural load applied at the age of t 0 ; f ins−L (t0 ) is the instantaneous deflection caused by the external load P applied at the age of t 0 ; f ins−U (t0 ) is the instantaneous deflection caused by the self-weight G at loading age of t 0 . According to the classical beam theory (Bauchau & Craig, 2009), the instantaneous deflection caused by the external load and the self-weight can be calculated from Equations (8.6) and (8.7), respectively. ) ( Pa 3l 2 + 12b2 − 12lb (8.6) f ins−L (t0 ) = 24E c (t0 )I ( ) 5 4 G 3l 2 e2 4 3 2 2 3 2 l − b + 2lb − 6e b − l b + 6e lb − f ins−U (t0 ) = 24E c (t0 )I 16 2 (8.7) where, a is the distance between the load and the nearest beam support; b is the distance between the support and the nearest measuring location; e is the distance between the beam support and the nearest beam end; l is the distance between the two beam supports; I is the moment of inertia of concrete beam; E c (t 0 ) is the elastic modulus of concrete at time t 0 .

8.2.4 Restrained Creep Test Up to date, several researchers have employed the restrained test to investigate the creep property of cementitious materials (Gao et al., 2013; Kovler, 1994; Liang & Wei, 2019b; Østergaard et al., 2001). The restrained creep tests can be roughly classified into the uniaxially restrained creep test and the restrained ring creep test,

8.2 Creep Test of Cementitious Materials at Macro Scale

299

(a)

(b)

A

Portion of concrete ring

Steel ring Concrete ring Strain gauge Steel base

Compressive stress in radial direction

150 mm 304 mm 330 mm 406 mm

Tensile stress in circumferential direction

Section A-A

Fig. 8.5 Experimental setup for restrained creep measurement. a Uniaxially restrained creep test (Kovler, 1994), b restrained ring test for creep measurement (Liang & Wei, 2019b)

which are shown in Fig. 8.5. In the uniaxially restrained creep test, two specimens are mounted horizontally on the laboratory table, one for the restrained shrinkage test, and the other for the free shrinkage test. The stress generated in the restrained specimen is measured utilizing a load cell, which is used to assess the creep property of cementitious materials together with the measured elastic modulus according to Eqs. (8.8) to (8.10). ε(t) = εe (t) + εc (t) + εsh (t) = 0 σ (t) E(t)

(8.9)

σ (t) ϕ(t) E(t)

(8.10)

εe (t) = εc (t) =

(8.8)

where εe (t), εc (t), and εsh (t) are the elastic, the creep, and the shrinkage strains, respectively; εsh (t) is the measured free shrinkage strain; E(t) is the elastic modulus of concrete; ϕ(t) is the unknown creep coefficient. In the restrained ring test, as shown in Fig. 8.5b, the concrete ring is cast outside the steel ring. Due to the shrinkage of concrete, the outer surface of the steel ring is subject to uniform compressive stress and restrains the free shrinkage of the concrete ring. It is obvious that the radial stress is in compression and the circumferential stress is in tension in concrete ring. Therefore, concrete creep can change the stress

300

8 Testing and Analysis of Micro Creep

condition within the concrete ring and thus modifies the strain distribution within the steel ring. Then based on the measured circumferential strain of the steel ring by using the strain gauge, one can obtain the creep property of concrete by analytical method (Gao et al., 2013) or finite element method (Liang & Wei, 2019b). The advantage of the restrained creep test lies in that it can reflect the influence of the restrained condition normally occurring in the real concrete structures on the creep property. However, it should be noted that the creep property cannot be directly obtained from the restrained creep test as the uniaxial creep test does. This test is mainly used to study the early age creep of cementitious materials, it may not work well for the mature concrete since the shrinkage of the mature concrete is insignificant and is not able to induce a pronounced restrained condition for the inner steel ring and consequently, and less strain is measured for the creep property calculation.

8.2.5 Multiaxial Creep Test Although there exist a large number of studies on concrete creep under uniaxial loading, studies on concrete creep under the multiaxial loading are rather limited. The time-dependent stress and strain development in the multiaxial-loaded concrete structures are still unclear. Therefore, several studies have been conducted to investigate the creep property of cementitious materials under the multiaxial stress condition (Bernard et al., 2003; Charpin et al., 2018; Kim et al., 2005; Liu et al., 2002). Currently, the multiaxial creep tests can be classified into biaxial creep tests and triaxial creep tests according to the stress conditions applied during the creep test, which are shown in Fig. 8.6. In the biaxial creep test, the specimen is loaded in two orthogonal directions, and the time-dependent deformations in the corresponding directions are monitored and recorded by the LVDTs. In the triaxial creep test, the specimen is loaded in three directions, and similarly, the time-dependent deformations in these directions are recorded by LVDTs. Then, both the loading history and the measured deformation are used to evaluate the multiaxial creep property of cementitious materials. Unlike the uniaxial creep test, the interaction of creep properties in different directions should be considered carefully for the multiaxial creep test, which can be assessed by the creep Poisson’s ratio (Aili et al., 2016; Charpin et al., 2018). The creep Poisson’s ratio of concrete, usually defined as the ratio of the radial deformation to the axial deformation, can reflect the deformation interaction between the two orthogonal directions, which can also serve as an essential parameter to study the creep and the loss of prestress in multiaxially prestressed structures. Since the creep of cementitious materials would last for a very long time, to provide a reliable prediction of long-term creep, it is recommended that the creep test of cementitious materials should be carried out over hundreds of days (Le Roy, 1995, Jirásek & Dobrusk, 2012). With such a long duration of the creep measurement, the experimental conditions such as the temperature, the applied load, and the moisture content are difficult to control, which may increase the error of the measured results.

8.3 Quantifying Micro Creep and Creep Recovery of Cementitious …

301

Fig. 8.6 Experimental setup for multiaxial creep measurement. a Biaxial creep test (Charpin et al., 2018), b triaxial creep test (Bernard et al., 2003)

Another disadvantage of the macro creep tests lies in that they fail to give a direct assessment of the local creep property of the cementitious materials.

8.3 Quantifying Micro Creep and Creep Recovery of Cementitious Materials Based on Indentation Technique Measurement of creep and creep recovery of materials at micro scale by indentation technique is similar to that at macro scale by conventional compressive creep test. As shown in Fig. 8.7, the creep property of materials can be assessed based on the deformation development of specimen at macro scale or the change of indentation depth at micro scale under the constant force. The macro creep recovery of materials can be determined based on the deformation recovery of a specimen when the external force is totally removed, which is a little different from the measurement of micro creep recovery by indentation technique where a relatively small force should be kept and applied to the indenter during the creep recovery stage. This is because the relatively small indentation force can ensure the indenter is always in contact with the specimen during the creep recovery stage, and thus the recovery of the indentation depth can be monitored by the indentation equipment for evaluating the micro creep recovery of materials. One typical measured curve during the microindentation (MI) tests with a maximum indentation force of 4 N, the loading/unloading time of 10 s, the holding time of 300 s, and the reholding time of 625 s is shown in Fig. 8.8. It has been reported that thermal drift caused by the temperature variation may influence the measured results of indentation tests if it is not effectively suppressed during the test (Nohava et al., 2009). Therefore, one should be aware of the influence of the thermal drift on creep recovery of the tested materials. The thermal drift rate can be experimentally measured on the reference sample with little creep property (e.g., the fused silica) by using the loading–holding–unloading–reholding scheme.

micro scale

macro scale

302

8 Testing and Analysis of Micro Creep creep deformation

specimen

recovery of elastic deformation recovery of creep deformation

rigid indenter

increase of indentation depth caused by creep

recovery of indentation depth caused by elastic recovery

relatively small force to keep the indenter in contact with specimen recovery of indentation depth caused by creep recovery

Force (Deformation)

sample creep stage

creep recovery stage

creep recovery Force Deformation Time

Fig. 8.7 Illustration of measurements of creep and creep recovery at macroscopic and microscopic scales (Liang & Wei, 2020a)

Since the reference sample exhibits little creep, the change of the indentation depth occurring during the reholding stage is caused by the thermal drift, which can be used to determine the thermal drift rate of the indenter (i.e., the change of indentation depth per unit of unloading time). Liang and Wei (2020a) measured the thermal drift rate during the MI creep test by using the fused silica reference sample with the maximum indentation force of 300 mN, the results are shown in Fig. 8.9. It can be seen that the amplitude of all the measured thermal drift rates on the fused silica is smaller than 0.04 nm/s, it is therefore inferred that the depth change caused by the thermal drift during the holding stage (180 s) and the reholding stage (1000 s) of the MI tests conducted on cement pastes in this study is less than 7.2 nm and 40 nm, which is much smaller than the measured indentation depth on cement pastes [about 630– 5200 nm in Liang and Wei (2020a)]. The measured depth change over the reholding stage on cement pastes is mainly caused by the creep recovery of cement pastes. Therefore, one can conclude that the thermal drift rate has a negligible influence on the measured creep and creep recovery of cement pastes by MI tests. It should be noted that the average stress beneath the Berkovich indenter (i.e., the contact hardness) varies with the microstructures and the mechanical properties of cement pastes. Therefore, the micro creep and the creep recovery of cement pastes may not be well characterized by the change in indentation depth over the holding stage and the reholding stage, respectively. Similar to the contact creep function shown in Eq. (3.26), the recoverable contact creep can be expressed as (Liang & Wei, 2020a): Crecoverable (t) =

2ac ∆h recoverable creep (t) Pmax

(8.11)

8.3 Quantifying Micro Creep and Creep Recovery of Cementitious …

(a)

(b)

5000

5000

creep

creep recovery

4000

Indentation force Indentation depth

3000

20000 16000 12000

2000

8000

1000

4000

0 0

200

400

600

Indentation force (mN)

24000

Indentation depth (nm)

Indentation force (mN)

303

4000 3000

1000

unloading stage

loading stage

2000

0 1000

800

holding stage creep

Pmax=4000 mN

Precovery=12 mN reholding stage creep recovery

0 0

4000

8000

12000

16000

20000

Indentation depth (nm)

Time (s)

(d)

(c) 2000

2000

1600

1600

1200

1200

800

800

400

400

0

0 0

50

100

150

200

250

300

0

50

100

Time (s)

150

200

250

300

Time (s)

Fig. 8.8 Typical measured curves during the 4 N-10 s-300 s-10 s-625 s MI tests: a P–t curve and h-t curve, b P–h curve, c ∆hcreep (t)-t curve during the creep stage, and d ∆hrecovery (t)-t curve during the recovery stage (Liang & Wei, 2020b)

Thermal drift rate (nm/s)

0.06

upper bound=0.04 nm/s

0.04 average=0.0045 nm/s

0.02 0 1 -0.02

2

3

4

5 6 7 8 Measuring point

9

10

-0.04 -0.06

lower bound=-0.04 nm/s

Fig. 8.9 Measured thermal drift rate on the fused silica reference sample (Liang & Wei, 2020a)

304

8 Testing and Analysis of Micro Creep

where ∆hrecoverable creep (t) is the recovery of the indentation depth evolving over the reholding stage in the MI tests at time t.

8.4 Creep Property of Calcium Silicate Hydrate by NI It is generally accepted that the complex creep property of concrete originates from the viscous behavior of calcium silicate hydrates (CSH). To investigate the creep property of CSH, Vandamme and Ulm (2009) first conducted the statistical NI tests on cement pastes with w/c ratios of 0.15, 0.2, 0.3, and 0.4. The NI depth was selected to assess the phase properties of CSH clusters according to the scale separability conditions, i.e., the depth of indentation must be much larger than the elementary size of the CSH particle (≈5 nm). After obtaining the change in depth during the holding stage, Eq. (3.27) were used to fit the measured results to obtain the creep modulus, then the creep property of the individual phase can be obtained by conducting a deconvolution process using the maximum likelihood method via the Expectation– Maximization (EM) algorithm implemented in the EMMIX software. Based on the deconvolution results, the calcium silicate hydrates in the tested cement pastes can be classified into three phases, i.e., the LD CSH, HD CSH, and UHD CSH (ultrahighdensity calcium silicate hydrate). The typical indentation force–indentation depth curves as well as the change in depth–time curves for LD CSH, HD CSH, and UHD CSH are shown in Fig. 8.10. It can be seen that the change in depth during the holding stage for LD CSH, HD CSH, and UHD CSH decreases in turn, which suggests that the creep property of different types of CSH decreases in turn. Although different types of cement pastes were measured by indentation tests, the obtained mechanical properties (i.e., the indentation modulus and indentation hardness) and creep modulus of the three types of CSH seem independent of the mixture proportion of cement pastes, which are summarized in Table 8.1. However,

(a)

(b)

LD CSH HD CSH UHD CSH

LD CSH HD CSH UHD CSH Ln Fit

Fig. 8.10 Measured a indentation force–indentation depth curve, and b change of indentation depth–time curve during the holding stage for LD CSH, HD CSH, and UHD CSH (Vamdamme and Ulm 2009)

8.4 Creep Property of Calcium Silicate Hydrate by NI

305

Table 8.1 Deconvolution results of paste with w/c ratios of 0.15, 0.2, 0.3, and 0.4 (Vandamme & Ulm, 2013) Indentation modulus M (GPa)

Indentation hardness H (GPa)

Contact creep modulus C (GPa)

Packing density η (%)

LD CSH

22.39 ± 4.84

0.607 ± 0.155

112.2 ± 23.3

0.673 ± 0.046

HD CSH

34.82 ± 5.25

0.999 ± 0.214

182.5 ± 43.7

0.776 ± 0.037

UHD CSH

47.53 ± 6.63

1.625 ± 0.375

342.6 ± 85.0

0.863 ± 0.046

Fig. 8.11 Volume fractions of LD CSH, HD CSH, and UHD CSH, as determined from the deconvolution process for a selected number of materials that differ only in the w/c ratio (Vandamme & Ulm, 2009)

Volume fraction

Phase

w/c ratio

the volume fraction of the three types of CSH varies with the mixture proportion, as shown in Fig. 8.11. This indicates that the mixture proportion of cementitious materials doesn’t influence the intrinsic mechanical properties of CSH, and the different macroscopic properties of cementitious materials are related to the difference in the volume fraction of different types of CSH. Jone and Grasley (2011) also studied the creep property of hydration products at nanoscale by indentation. The w/c ratio of cement paste used in the NI test was 0.4. The maximum indentation force was set as 1.5 mN, and the holding time was set as 30 s. The probability density plots of the indentation depth change during the holding stage h(t) and the viscoelastic modulus E(t) were obtained and are plotted in Fig. 8.12. It can be seen that both h(t) and E(t) exhibit a trimodal response. This could be attributed to the varying quantities of different CSH, which is consistent with the findings by Vandamme and Ulm (2009). (b) E(t) (GPa)

(a)

h(t) (nm)

Fig. 8.12 Probability density plots of a h(t) and b E(t) showing the trimodal trend extracted from the NI tests with the darker areas indicating higher probability and the dashed lines indicating mean values (Jones & Grasley, 2011)

T (s)

T (s)

306

8 Testing and Analysis of Micro Creep

8.5 Influence of Experimental Protocol To evaluate the influence of different experimental protocols on the measured creep and creep recovery of cementitious materials, Liang and Wei (2020a, b) conducted various MI tests on cement paste samples. Two experimental schemes are designed (Fig. 8.13). The MI test with no holding stage at the maximum indentation force but a reholding stage under a relatively small force after the unloading stage is denoted as Scheme 1; while the MI test with a holding stage at the maximum indentation force and a reholding stage under a relatively small force after the unloading stage is denoted as Scheme 2. The detailed test parameters are summarized in Table 8.2.

8.5.1 Influence of Indentation Load To evaluate the influence of indentation load on the measured creep and creep recovery of cementitious materials, Liang and Wei (2020b) conducted MI tests on cement paste with w/c ratio of 0.3. The applied indentation forces were 2 N, 4 N, and 8 N, respectively.

assess the recoverable creep mainly over loading stage

Indentation force

Indentation force

Scheme 1

Indentation depth

Time

increase of indentation depth due to creep property

assess the creep property over holding stage assess the total recoverable creep over both loading and holding stages

Indentation force

Indentation force

Scheme 2

Time

recovery of indentation depth due to recoverable creep

recovery of indentation depth due to recoverable creep

Indentation depth

Fig. 8.13 Two schemes in the MI tests to assess the creep and creep recovery of cement paste at micro scale (Liang & Wei, 2020a)

8.5 Influence of Experimental Protocol

307

Table 8.2 Experimental schemes to measure the creep and creep recovery of cement pastes at micro scale by MI technique (Liang & Wei, 2020a) Scheme 1

Scheme 2

Value

Role

Value

Role

Loading rate (N/s)

0.8

0.8

Loading time (s)

10

Evaluating the recoverable creep over the loading stage

Holding load (N)

8

8

Evaluating the total recoverable creep over the loading and holding stages, and the irrecoverable creep over the holding stage

Hold time (s)

0

180

Unloading rate (N/s)

0.8

0.8

Unloading time (s)

10

10

Reholding load (mN)

40

40

Reholding time (s)

1000

1000

Experimental parameter Loading stage

Holding stage

Unloading stage

Reholding stage

10

The measured changes in indentation depth (∆hcreep (t)) during the holding stage from the MI tests with different maximum indentation forces are shown in Fig. 8.14a. As the indentation force increases, ∆hcreep (t) will increase. However, ∆hcreep (t) doesn’t increase linearly with the increasing indentation force. For example, when the maximum indentation force is increased from 2 to 4 N, ∆hcreep (t) will only increase by 37%. This could be explained by the fact that the relation between the indentation force and the indentation depth isn’t linear when the Berkovich indenter is used in the MI test (e.g., see Fig. 8.8b). Based on Eq. (3.26), the contact creep functions ([L(t)-L(0)]) can be easily obtained, which are shown in Fig. 8.14b. The indentation force seems to have little effect on the average [L(t)-L(0)] despite [L(t)-L(0)] obtained by a smaller indentation force exhibits a greater standard deviation. This indicates that the MI tests with the maximum indentation force greater than 2 N allow measuring the homogeneous creep property of cement paste. In addition, the nonlinear creep phenomenon can be negligible as the maximum indentation force increases, the reason lies in that the average stress beneath the indenter (i.e., the contact hardness) will not change significantly due to the self-similarity of the Berkovich tip (Vandamme & Ulm, 2013). The measured recovery of indentation depth (∆hrecovery (t)) during the reholding stage from the MI tests with different maximum indentation forces is shown in Fig. 8.14c. Similar to ∆hcreep (t), ∆hrecovery (t) will increase with the increasing indentation force. For instance, ∆hrecovery (625) is 422.3 nm, 645.5 nm, and 920.5 nm for the maximum indentation forces of 2 N, 4 N, and 8 N, respectively. Due to the fact that the measured thermal drift rate during the MI tests is not greater than 0.05 nm/s,

308

8 Testing and Analysis of Micro Creep

Fig. 8.14 a Measured change in indentation depth over the holding stage, b contact creep function, and c measured recovery of indentation depth in the MI tests with indentation forces of 2 N, 4 N, and 8 N (Liang & Wei, 2020b)

the change in indentation depth caused by the thermal drift after 625 s reholding is less than 0.05 × 625≈32 nm, which is quite smaller than ∆hrecovery (625). As a result, it can be concluded that ∆hrecovery (t) generated during the reholding stage is mainly caused by the creep recovery of cement paste. In addition, ∆hrecovery (t) doesn’t increase linearly with the maximum indentation force, which is similar to ∆hcreep (t). Gan et al. (2021) recently investigated the short-term creep recovery of microcantilever cement paste beams at micro scale by microindentation. The dimension of the microcantilever cement paste beam was about 300 μm × 300 μm × 1650 μm. Under the bending condition, the microcantilever beam will be ruptured when the indentation force reached 67–86 mN. Figure 8.15 shows the schematic diagram of the micro-bending test setup. The loading procedures are summarized in Table 8.3. The stress level is defined as the ratio of applied force to the fractured force, which is 30–92%. In the microcantilever beam indentation, the total displacement δ total (t) at the upper face of the fixed end of the beam can be expressed as:

8.5 Influence of Experimental Protocol

309

Fig. 8.15 Schematic diagram of microbending test setup (Gan et al., 2020)

Load Fixed nut Adhesive

Indenter Cantilever beam

Baseplate

Table 8.3 Summary of the loading procedures for microcantilever beam bending tests (Gan et al., 2021) Test series

w/c

Loading duration (s)

Holding load (mN)

Stress level (%)

Holding duration (s)

Second holding load (mN)

Second holding duration (s)

1

0.3

5

40

46

180

0.4

600

2

0.3

5

60

69

180

0.6

600

3

0.3

5

80

92

180

0.8

600

4

0.4

5

20

30

180

0.2

600

5

0.4

5

40

59

180

0.4

600

6

0.4

5

60

88

180

0.6

600

7

0.4

5

40

59

300

0.4

300

8

0.4

50

40

59

180

0.4

300

δtotal (t) = δe + δv + δ p + δd =

⏋ ┌ dp 3h σ (t) + ε (σ, t) + σ (t) + γdrift t creep 2L 2 E int σmax (8.12)

where δ e , δ v , δ p , and δ d are the elastic deformation(reversible), viscoelastic deformation (reversible), plastic deformation (irreversible), and thermal drift (irreversible), respectively; σ (t) is the stress applied to the sample up to time t, and σ (t) = Pdh/2I, P is the indentation force at time t, d is the measured distance between the load point and the fracture point, h is the side length of the square beam cross section, and I = h4 /12 is the moment of inertia; L is the measured distance between the load point and the fixed end of the microcantilever beam; E int is the intrinsic elastic modulus of cement paste; d p is the non-recoverable plastic displacement generated during the loading stage, σ max is the maximum stress, and σ max = Pmax dh/2I, γdrift is the drift rate (0.026 nm/s) determined from the tests of glass beams; εcreep (σ, t) is creep strain of cement paste, which can be expressed as: εcreep (σ, t) =

n ∑ ┌ ⏋ σ (ti ) − σ (ti−1 ) × C(t, t0 ) i=1

310

8 Testing and Analysis of Micro Creep

Fig. 8.16 Average creep compliances with scattering bars for a w/c ratio = 0.3 samples and b w/c ratio = 0.4 samples (Gan et al., 2021)

=

┌ ⏋ t − t0 β ) [σ (ti ) − σ (ti−1 )] × α( t1 i=1

n ∑

(8.13)

where σ (ti ) − σ (ti−1 ) is the stress increment at a specified time interval (t i –t i-1 ). The 0 β ) is time interval used in the study of Gan et al. (2021) is 0.1 s; C(t, t0 ) = α( t−t t1 the specific basic creep compliance, t 0 is the time when load is applied, t-t 0 is the holding time under constant load; t 1 is the reference time unit, i.e., 1 s. α and β are two fitting parameters. Figure 8.16 shows the measured creep compliances for cement pastes with w/c ratios of 0.3 and 0.4 tested by different indentation forces. It can be seen that the creep compliances of cement pastes with w/c ratios of 0.3 and 0.4 are independent of indentation force, which is consistent with the findings by Liang and Wei (2020b). Figure 8.17 displays the measured creep recovery evolution for cement pastes with w/c ratios of 0.3 and 0.4 tested by different indentation forces. Similarly, the recovery of indentation depth increases with increasing indentation force for both the two types of cement pastes, which reveals the same conclusion by Liang and Wei (2020b).

8.5.2 Influence of Loading Rate/Loading Time In the conventional creep test at the macro scale, the loading time should be as short as possible to avoid the significant development of creep during the loading stage. To assess the influence of the loading time on the measured micro creep and creep recovery of cementitious materials, Liang and Wei (2020b) conducted MI creep tests on cement pastes with a w/c ratio of 0.3. The loading times were set as 5 s, 10 s, 20 s, and 40 s, respectively. The other experimental parameters are shown in Table 8.2. Figure 8.18a shows the measured changes in indentation depth during the holding stage (∆hcreep (t)) in the MI tests with different loading times of 5 s, 10 s, 20 s, and

8.5 Influence of Experimental Protocol

311

Fig. 8.17 Measured creep recovery evolution curves at the second holding stage for a w/c ratio = 0.3 and b w/c ratio = 0.4 samples (Gan et al., 2021)

40 s. It is seen that ∆hcreep (t) decreases with increasing loading time. The possible reason is that more creep will develop during the loading stage with a greater loading time, and thus ∆hcreep (t) developed during the holding stage of the MI test is reduced.

Fig. 8.18 a Measured change in indentation depth over the holding stage, b contact creep function, and c measured recovery of indentation depth in the MI tests with loading times of 5 s, 10 s, 20 s, and 40 s (Liang & Wei, 2020b)

312

8 Testing and Analysis of Micro Creep

Figure 8.18b shows the contact creep functions ([L(t)-L(0)]) obtained from the MI tests with different loading times of 5 s, 10 s, 20 s, and 40 s. [L(t)-L(0)] decreases with increasing loading time, which is similar to ∆hcreep (t). This measured result reveals that the creep development of cement paste over the holding stage is influenced by the loading time. The measured recoveries of indentation depth (∆hrecovery (t)) over the reholding stage of 625 s from the MI tests with different loading times of 5 s, 10 s, 20 s, and 40 s are shown in Fig. 8.18c. Although ∆hrecovery (t) decreases slightly with the increasing loading time, the loading time has a minor influence on the creep recovery of cement paste. Gan et al. (2021) also studied the influence of loading time on the creep and creep recovery of cementitious materials by microcantilever beam indentation technique. The loading times were set as 5 s and 50 s, respectively. The other experimental parameters are shown in Table 8.3. By fitting Eq. (8.12) with the measured indentation force (P)-indentation displacement (δ) curve, one can determine the parameters in Eq. (8.13). Then the specific creep compliance can be obtained by dividing εcreep (σ, t) by the applied maximum stress σ max . Figure 8.19 shows the comparison of the specific creep compliance of the w/c ratio of 0.4 cement paste with loading time of 5 s with that with loading time of 50 s. It is seen that the specific basic creep compliance at the end of the loading stage increases with the increasing loading time. However, less creep is generated during the holding stage, which is similar to the findings in Liang and Wei (2020b). Figure 8.20 shows the measured creep recovery by Gan et al. (2021) at the second holding stage for different loading histories. It is seen from Fig. 8.20 that the creep recovery of cement pastes with w/c ratio of 0.4 increases slightly with the increasing loading time, which contradicts the measured results by Liang and Wei (2020b) where the creep recovery of cement paste with w/c ratio of 0.4 decreases slightly with the increasing loading time. The reason may lie in that the loading time has a Fig. 8.19 Comparison of creep compliance evolutions for two different loading durations (Gan et al., 2021)

8.5 Influence of Experimental Protocol

313

Fig. 8.20 Measured creep recovery at the second holding stage for different loading histories (Gan et al., 2021)

minor effect on the creep recovery of cement pastes, and the contradictive results may be caused by the different indentation methods and the different testing conditions.

8.5.3 Influence of Holding Time To assess the influence of the holding time on the measured creep and creep recovery of cementitious materials, Liang and Wei (2020b) performed MI creep tests on cement pastes with w/c ratio of 0.3. The holding times were set as 0 s, 180 s, and 300 s, respectively. The other experimental parameters are shown in Table 8.2. Figure 8.21a displays the measured changes in indentation depth during the holding stage with different holding times of 180 s and 300 s. It can be seen that ∆hcreep (t) measured from the MI test with holding time of 300 s is almost the same as that with 180 s holding time. Based on the measured changes in indentation depth during the holding stage, one can obtain the contact creep functions for the MI tests with holding times of 180 s and 300 s, which are shown in Fig. 8.21b. The holding time has little effect on the contact creep function of cement paste, which is similar to ∆hcreep (t). The measured recoveries of indentation depth (∆hrecovery (t)) over the reholding stage from the MI tests with different holding times of 0 s, 180 s, and 300 s are shown in Fig. 8.21c. It is seen that ∆hrecovery (t) with holding time of 180 s is almost the same as that from the MI test with holding time of 300 s. In addition, the MI test without a holding stage (0 s) also exhibits an obvious recovery of indentation depth, which is 63–66% of those with holding times of 180 s and 300 s. This indicates that a considerable portion of recoverable creep has developed during the loading stage, and the indentation depth caused by the creep deformation developing during the holding stage is partially recoverable.

314

8 Testing and Analysis of Micro Creep

Fig. 8.21 a Measured change in indentation depth, b contact creep function over the holding stage, and c measured recovery of indentation depth in the MI tests with holding times of 0 s, 180 s, and 300 s (Liang & Wei, 2020b)

8.5.4 Influence of the Experimental Protocols of MI Test on the Long-Term Creep Rate of Cement Paste This section will further discusses the influence of the MI experimental protocols on the long-term creep of cement paste. The existing findings have revealed that the creep modulus in Eq. (3.27) governs the long-term logarithmic creep rate of cementitious materials (Frech-Baronet et al., 2017; Liang et al., 2017; Nguyen et al., 2014; Vandamme & Ulm, 2009, 2013; Wei et al., 2017; Zhang et al., 2014). A lower creep modulus means that the cementitious materials creep faster. By fitting the measured contact creep function shown in Figs. 8.14b, 8.18b, and 8.21b with Eq. (3.27), the creep moduli of cement paste in the MI tests with different experimental protocols can be determined, which are shown in Fig. 8.22. It can be seen from Fig. 8.22 that the creep modulus of cement paste remains almost constant when the indentation force increases from 2 to 8 N. The reason lies in that the indentation force ranging from 2 to 8 N can measure the homogeneous property of cement paste. In addition, due to the self-similarity of the Berkovich indenter, the average stress beneath the indenter (i.e., contact hardness) is nearly the same between different maximum indentation forces, which will therefore result in no nonlinear creep or damage when indentation force of the MI tests is increased.

8.5 Influence of Experimental Protocol

315

Fig. 8.22 Creep moduli of cement paste with w/c ratio of 0.3 obtained from the MI tests with different experimental protocols (Liang & Wei, 2020b)

As shown in Fig. 8.18b, although a lower contact creep function is measured from the MI test with a longer loading time, the loading time has little effect on the creep modulus, i.e., the loading time will not affect the long-term creep rate. As a result, it can be inferred that the loading time only affects the characteristic time of cement paste, i.e., the time when the short-term creep switches to the longterm creep. The characteristic times of cement paste obtained from the MI tests with loading times of 5 s, 10 s, 20 s, and 40 s are shown in Fig. 8.23. It can be seen that the fitted characteristic times from the MI test with maximum indentation force of 2 N are almost the same as that from the MI test with maximum indentation force of 4 N under the same loading time, because the homogeneous creep property of cement paste is measured under the indentation forces of both 2 N and 4 N. On the other hand, the characteristic time (τ ) increases with increasing loading time, which indicates that the loading time affects the time when the short-term creep switches to the long-term creep. In addition, it can be seen from Fig. 8.22 that the holding time appears to have no effect on the creep modulus of cement paste when the holding time ranges from 40 to 300 s, which indicates that the creep rate of the tested materials under the indentation force is almost the same despite the holding time. This is because a 2.0 Characteristic time (s)

Fig. 8.23 Comparison of the characteristic times of cement paste from the MI tests with different loading times under indentation forces of 2 N and 4 N (Liang & Wei, 2020b)

1.6

2N 4N

1.2 0.8 0.4 0.0 5s

10s 20s Loading time

40s

316

8 Testing and Analysis of Micro Creep

maximum indentation of 2–4 N is sufficent to measure the homogenous property of cement paste and the stress conditions of cement paste at the time when the holding stage starts are almost the same for different holding times. In conclusion, the creep modulus of cement paste obtained from all the MI tests seems not to be influenced by the loading time, the holding time, and the maximum indentation force. The average creep modulus of all the MI tests shown in Fig. 8.22 is 209.1 GPa, which indicates that creep modulus is an intrinsic parameter characterizing the long-term creep rate of cement paste. The slight difference in the creep modulus between different MI tests is probably caused by the microstructural fluctuation of cementitious materials.

8.5.5 Influence of Loading Strain Rate Since the creep property of cementitious materials is time-dependent, the loading strain rate would influence the creep development of cementitious materials. In the , h is the indentation indentation test, based on the concept of true strain (ε = dh h depth), the loading strain rate is defined as: ε˙ =

dh 1 dε = dt h dt

(8.14)

Liang et al. (2017) conducted continuous stiffness measurement (CSM) on cement pastes with w/c ratios of 0.3, 0.4, and 0.5 to investigate the influence of loading strain rate on the viscoplastic depth measured in the CSM test. The curing ages of the cement paste sample with w/c ratio of 0.3, 0.4, and 0.5 were about 1 year, 4 years, and 4 years, respectively. The experimental strain rates were 0.01 s−1 , 0.05 s−1 , 0.1 s−1 , and 0.5 s−1 , respectively. The maximum indentation depth was 30 μm. The holding time during the maximum indentation force was 10 s. The excitation frequency of the added harmonic motion was 45 Hz, and the amplitude of the sinusoidal depth was 2 nm by default. Six indents were made on each paste sample under each strain rate level. The P–h curve (shown in Fig. 8.24a) during the loading stage will be used to examine the effect of the loading strain rate. Due to the fact that the indentation depth obtained from the CSM test is composed of the elastic depth (he ) and the viscoplastic depth (hvp ), he which is independent of the loading strain rate should be excluded from the recorded indentation depth during the CSM test. The process is illustrated in Fig. 8.24b. When the tested materials are linearly elastic, homogeneous, and isotropic, the elastic indentation depth obtained from the MI test using a conical indenter can be expressed as Eq. (8.15), which is a function of the applied force (P), the indentation modulus (M), and the half-cone angle (θ ) (Su et al., 2013). / he =

πP 2M tan θ

where θ = 70.23o for a Berkovich indenter.

(8.15)

8.5 Influence of Experimental Protocol

317

Fig. 8.24 a Typical measured P–h curve and b determination of hvp from the total and elastic deformation during loading stage from (a) (Liang et al., 2017)

Then the viscoplastic depth (hvp ) can be calculated as the difference between the indentation depth (h) and the elastic depth (he ) during the loading stage: h vp = h − h e

(8.16)

The hvp -t curves under the strain rates of 0.01 s−1 , 0.05 s−1 , 0.1 s−1 , and 0.5 s−1 during the loading stage for the cement paste with w/c ratio of 0.3 (curing age = 1 year) are shown in Fig. 8.25. As the loading strain rate increases, hvp will decrease. This is because the tested sample under a greater strain rate will take less time to reach the maximum indentation depth, thus the hvp caused by the creep property of cementititous materials develops less. Both the strain rate and the w/c ratio of cementitious materials affect the hvp P curves. The hvp -P curves during the loading stage under the four strain rates (0.01 s−1 −1 , 0.05 s−1 , 0.1 s−1 , and 0.5 s) for cement paste with w/c ratios of 0.3, 0.4, and 0.5 are shown in Fig. 8.26. Each curve is the average of the 6 measured results on the tested cement paste. Table 8.4 summarizes the typical average hvp and the corresponding standard deviations of the 6 curves at the indentation forces of 2 N, 5 N, and 7 N for each strain rate. It is the same as expected that hvp increases when the indentation force increases, but decreases with increasing strain rate under 20 strain rate 16 hvp (μm)

Fig. 8.25 hvp -t curves under different strain rates of 0.01 s−1 , 0.05 s−1 , 0.1 s−1 , and 0.5 s−1 during loading stage for cement paste with w/c ratio = 0.3 (Liang et al., 2017).

12

0.01 0.1

0.05 0.5

end of the harmonic motion

8 w/c ratio =0.3

4 0 0

200

400

600 800 1000 1200 1400 Time (s)

318

8 Testing and Analysis of Micro Creep

any given load. Similar results were reported for soils based on the constant rate of strain consolidation test and the long-term consolidation test at macro scale where the viscoplastic strain decreases with increasing strain rate under any given stress (Leroueil et al., 1985; Watabe et al., 2008). The reason lies in that the viscoplastic depth would evolve more at a lower strain rate. The strain rate has a more significant effect on the cement paste with a lower w/c ratio. It can be seen from Fig. 8.26 that a much stronger dependence of the hvp -P curves on the strain rate is observed in the cement paste with w/c ratios of 0.3. While the hvp -P curves for the cement pastes with w/c ratios of 0.4 and 0.5 seems to be affected less by the strain rate. It is obvious that the cement paste with w/c ratio of 0.3 contains larger amount of unhydrated phase as compared to that of the cement pastes with w/c ratios of 0.4 and 0.5. In addition, the curing age of cement paste with w/c ratio = 0.3 (1 year) is relatively shorter than that of the two cement pastes with w/c ratio of 0.4 and 0.5 (4 years), which will result in a lower degree of hydration with more unhydrated phase left. This may suggest that the unhydrated phase play a role in the strain rate effect on the hvp -P curves. The reason may be that the stress redistribution between the creep phase (CSH) and the non-creep phase (unhydrated clinker) is more significant in cement paste containing more unhydrated clinker, which will enhance the strain rate effect.

0 0 2 4 6 8 10 12 14 16 18

2

4

P (N) 6

8

10

12 0.01 0.05 0.1 0.5

strain rate

w/c ratio = 0.3

hvp (μm)

(c)

0 0 2 4 6 8 10 12 14 16 18

2

4

(b)

hvp (μm)

hvp (μm)

(a)

0

2

0 2 4 6 8 10 12 14 16 18

P (N) 6

4

P (N) 6

strain rate

8

10

12 0.01 0.05 0.1 0.5

w/c ratio = 0.4

8

strain rate

10

12 0.01 0.05 0.1 0.5

w/c ratio = 0.5

Fig. 8.26 Influence of strain rate on hvp -P curves of cement paste with different w/c ratios: a w/c ratio = 0.3 (curing age = 1 year), b w/c ratio = 0.4 (curing age = 4 year), and c w/c ratio = 0.5 (curing age = 4 year) (Liang et al., 2017)

8.6 Creep Recovery

319

Table 8.4 Average hvp and the corresponding standard deviations of the 6 curves at the indentation forces of 2 N, 5 N, and 7 N under each strain rate for cement pastes with w/c ratios of 0.3, 0.4, and 0.5 (Liang et al., 2017) w/c ratio Viscoplastic depth hvp (μm) 0.3

0.4

0.5

Strain rate (s−1 )

0.01

0.05

0.1

0.5

P=2N

5.95 ± 0.67

4.95 ± 0.58

4.49 ± 0.61

4.15 ± 0.89

P=5N

11.58 ± 0.58 8.83 ± 0.34

7.98 ± 0.48

6.91 ± 0.46

P=7N

14.91 ± 0.32 10.29 ± 0.31 10.05 ± 0.48 8.43 ± 0.49

P=2N

6.09 ± 0.67

5.69 ± 1.23

5.58 ± 0.69

4.66 ± 1.56

P=5N

9.44 ± 0.79

8.88 ± 0.67

8.84 ± 0.49

7.94 ± 0.65

P=7N

11.52 ± 0.94 10.85 ± 0.80 10.51 ± 0.51 9.68 ± 0.61

P=2N

7.44 ± 1.82

P=5N

11.61 ± 1.38 10.79 ± 1.11 10.41 ± 0.89 9.63 ± 0.55

7.04 ± 1.74

6.95 ± 0.82

6.28 ± 0.85

P=7N

13.74 ± 0.83 12.62 ± 0.52 12.15 ± 0.71 11.30 ± 0.50

8.6 Creep Recovery 8.6.1 Ratio of Recoverable Creep to the Total Creep of Cement Paste at Micro Scale and Its Comparison to the Macroscopic One As shown in the previous sections, a considerable portion of the recoverable creep has developed during the loading stage, and the creep deformation occurring over the holding stage is also partially recoverable. The ratio of the recoverable creep to the total creep of cement paste at micro scale will be calculated based on the measured changes in indentation depth over the loading and holding stages along with the recovery of indentation depth over the reholding stage, which are listed in Table 8.5. It can be seen that a longer holding duration will result in an increase in the depth change evolving over the holding stage (∆h holding-stage creep ) under the same indentation force. However, the holding duration seems to have little effect on the total recoverable indentation depth (∆h recoverable creep ), which suggests that the recoverable creep occurring during the holding stage may be insignificant after 40-s holding. In order to calculate the ratio of the recoverable creep to the total creep of cement paste at micro scale, the total depth change (∆h total creep ) is assumed to be caused by the creep property of cement paste, which consists of two components, i.e., one developing during the loading stage (∆h loading-stage creep ) and the other developing during the holding stage (∆h holding-stage creep ). In this book, it is assumed that ∆h loading-stage creep is mainly induced by the recoverable creep that is equal to the recovery of indentation depth measured from the MI test without a holding stage due to the fact that the irrecoverable creep that develops during the relatively short loading stage (i.e.,

300

180

300

180

10

10

10

10

7

9

14

16

180

10

2

40

10

6

8

4

4

2

2

2

Test Loading Holding Indentation No. time (s) time (s) force (N)

10

10

10

10

10

10

Unloading time (s)

625

625

625

625

625

625

Reholding time (s)

633.3

410.4

410.4

287.8

287.8

287.8

∆hloading-stage creep (nm)

1851.4

1442.0

1396.8

1132.7

1023.6

746.3

∆hholding-stage creep (nm)

920.5

624.6

645.5

442.8

422.3

425.6

0.37

0.34

0.36

0.31

0.32

0.41

∆hrecoverable creep (nm) Rrecoverable

Table 8.5 Summary of the ratio of the recoverable creep to the total creep of cement paste with w/c ratio of 0.3 at micro scale assessed by MI test (Liang & Wei, 2020b)

320 8 Testing and Analysis of Micro Creep

8.6 Creep Recovery

321

only 10 s) is minor. ∆h holding-stage creep is the change in indentation depth occurring during the holding stage, which is composed of both recoverable and irrecoverable indentation depths. The total recoverable indentation depth (∆h recoverable creep ) can be obtained from the reholding stage in the MI test with a holding stage. Therefore, the ratio of the recoverable creep to the total creep of cement paste at micro scale can be expressed as: Rrecoverable =

∆h recoverable creep ∆h recoverable creep = ∆h total creep ∆h loading-stage creep + ∆h holding-stage creep

(8.17)

As shown in Table 8.5, the creep recovery/total creep ratio of cement paste assessed from the MI test at micro scale ranges between 0.31 and 0.41. The average creep recovery/ total creep ratio is 0.35, which indicates that the recoverable creep is an important micro creep component of cementitious materials. In addition, the creep recovery/total creep ratio tends to decrease as the holding time increases. Next, the creep recovery/total creep ratio of cement paste at micro scale will be compared with that obtained from the macroscopic creep and creep recovery of concrete. The ratio of the creep recovery to the total creep of concrete obtained from the macroscopic uniaxial and multiaxial compressive creep tests by Neville et al (1983) are shown in Table 8.6. Similar to the measured result at micro scale, the creep recovery/total creep ratio of concrete at macro scale decreases with increasing loading duration. The comparison of the ratios of the recoverable creep to the total creep of cementitious materials at micro scale and those at macro scale obtained from Reference (Neville et al., 1983) are shown in Fig. 8.27. As shown in Fig. 8.27, the creep recovery/ total creep ratio is higher under the multiaxial stress condition. For example, the average creep recovery/ creep ratio obtained from the uniaxial compressive creep tests is 0.23, while it is 0.30, 0.31, and 0.25 in stress directions 1, 2, and 3 of the multiaxial compressive creep tests, respectively. Moreover, the average creep recovery/creep ratio obtained from the MI tests is 0.35, which is comparable with that obtained by the macroscopic multiaxial compressive creep tests. This is because the speciemens used in the MI test and the multiaxial creep test are under multiaxial stress conditions. The above results suggest that the MI test can evaluate both the recoverable and the irrecoverable creep of cementitious materials at micro scale. However, it should be noted that the average creep recovery/creep ratio obtained from the MI tests is slightly greater than that obtained from the macroscopic multiaxial compressive creep tests, which may be rleated to the difference in stress level in the MI test and the macroscopic multiaxial compressive creep tests. Gan et al. (2021) also investigated the recoverable creep/total creep ratios of cementitious materials at micro scale by microcantilever beam indentation technique. It should be noted that although the creep recovery ratio in the work by Gan et al. (2021) was defined as the ratio between the total creep recovery displacement and the total creep displacement, the displacements at the loading and the unloading stages were calculated based on Eq. (8.12), which is different from that by Liang and Wei (2020b).

12.5

98

320

282

312

13.1

6.2

42

165

10.1↑

35

94

257

262

11.7

12.2

28

28

235

10.0

28

170

160

5.0

7.1

28

28

97

4.1

28

130

122

5.4

5.6

22

Total creep (× 10–6 )

Applied stress (MPa)

0.21

0.03

0.20

0.23

0.26

0.25

0.26

0.26

0.25

0.33

0.22

0.27

Ratio of creep recovery to total creep

Uniaxial compressive creep test

28

Duration of load (days)

5.2

13.4

12.8

8.4

13.2

12.6

12.5

11.2

10.0

4.1

4.0

6.7

3.6

13.9

13.6

5.6

9.5↑

7.2

11.4

9.7

7.7

3.4

3.8

6.5

0

9.1↑

13.5↑

10.1↑

1.8

0

2.4

2.1

3.1

4.3

3.9

6.3

112

275

260

157

270

250

240

235

200

105

70

102

287

267

82

72

120

210

190

140

80

67

92

70

Direction 2

Direction 1

σ3

σ1

σ2

Total creep (×10–6 )

Applied stress (MPa)

Multiaxial compressive creep test

0.33

10

167

0.27

0.24

0.19

0.27

−35 92

0.31 0.33

−30 −65

0.33

0.31

0.29

0.39

25

20

113

65

0.29

−30 47

Direction 1

Direction 3

0.25

0.22

0.33

0.31

0.27

0.30

0.31

0.36

0.25

0.40

0.24

0.50

Direction 2

1.50

0.24

0.27

0.14

0.35

0.16

0.20

0.12

0.31

0.35

0.31

0.25

Direction 3

Ratio of creep recovery to total creep

Table 8.6 Summary of the ratio of the creep recovery to the total creep of concrete in macroscopic uniaxial and multiaxial compressive creep tests taken from the reference (Neville et al., 1983)

322 8 Testing and Analysis of Micro Creep

8.6 Creep Recovery 0.8 Recoverable creep/ total creep

Fig. 8.27 Recoverable creep/total creep ratios of cementitious materials at micro scale (this book) to those at macro scale obtained from (Neville et al., 1983)

323

The solid lines denote the average recoverable creep/ total creep ratio

0.6

0.35 0.4

0.23

0.30

0.31

0.25

0.2

0

micro scale

macro scale

MI test

Uniaxial Multiaxial Multiaxial Multiaxial test test 1 test 2 test 3

Figure 8.28 shows the average creep recovery ratios obtained from different testing series by Gan et al. (2021). It should be noted that the second holding duration is 300 s and 50 s for test series 7 and 8, respectively, while it is 600 s for other test series. As shown in Fig. 8.28, most of the recovery ratios range from 70 to 90%, which is much higher than the average recovery ratios (0.23–0.35, shown in Fig. 8.27) reported by Liang and Wei (2020b). The reason may lie in that the indentation method employed by Liang and Wei (2020b) is different from that by Gan et al. (2021); furthermore, the derivation methods of the recoverable and irrecoverable displacements of the test materials are different, which would lead to the different results.

Fig. 8.28 Calculated recovery ratios for different testing series (Gan et al., 2021)

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8 Testing and Analysis of Micro Creep

Fig. 8.29 Rheological model composed of four Kelvin–Voigt elements to simulate the recovery of indentation depth of cement paste over the reholding stage (Liang & Wei, 2020b)

8.6.2 Modeling of Recovery of Indentation Depth of Cement Paste at Micro Scale As for the measured macro creep of concrete, the macroscopic recoverable creep of concrete can be well modelled by a Kelvin–Voigt element consisting of a spring and a dashpot in parallel (Bažant et al., 2012; Giorla & Dunant, 2018; Hilaire et al., 2014; Mallick et al., 2017). Whether the Kelvin–Voigt element can be used to model the recovery of indentation depth will be assessed in this section. Similar to the macroscopic recoverable creep, the recovery of indentation depth over the reholding phase is assumed to be fitted with the following equation: ∆h recoverable creep (t)=

N ∑ ( ) ai 1 − e−bi t

(8.18)

i=1

where ai and bi are the amplitude of the depth recovery and the retardation time of the ith Kelvin–Voigt element, respectively, which can be determined by fitting the measured results with Eq. (8.18); N is the number of the Kelvin–Voigt elements. It is shown that the measured depth recovery over the reholding stage can be well modeled by four Kelvin–Voigt elements (shown in Fig. 8.29) after several fitting trials. The comparison of the measured and the fitted depth recovery over the reholding stage of MI tests with different indentation forces and holding durations is shown in Fig. 8.30, which shows that the measured depth recovery is in good agreement with the fitted one. This indicates that the Kelvin–Voigt element shown in Fig. 8.29 can capture well the recovery of the indentation depth induced by the micro creep recovery of cemenitious materials.

8.6.3 Creep Recovery Rate of Cement Paste at Micro Scale Glucklich (1959) and Illston (1965) have shown that the creep recovery of concrete measured at macro scale exhibits a rapid devolvement followed by a slow one. The micro creep recovery rate of cement paste measured by MI tests will be examined in this section. By derivating Eq. (8.18) with respect to time, the creep recovery rate of cement paste over the reholding stage (∆h˙ recoverable (t)) can be easily expressed as:

8.6 Creep Recovery

325

Fig. 8.30 Measured and fitted depth recovery of cement paste over the reholding stage for the MI tests with indentation forces of a 2 N, b 4 N, and c 8 N (Liang & Wei, 2020b)

∆h˙ recoverable creep (t)=

N ∑

ai bi e−bi t

(8.19)

i=1

The depth recovery rates of cement paste over the reholding stage for the MI tests with no holding stage and a holding stage of 180 s under the maximum indentation forces are shown in Fig. 8.31. It can be seen that the depth recovery rate of the MI creep test with a holding stage of 180 s is greater than that with no holding stage. This is reasonable because some recoverable creep will also develop during the holding stage, which will also recover during the reholding stage and thus increases the depth recovery rate. It can be seen from Fig. 8.31 that the depth recovery rate for all the MI tests decreases bilinearly with increasing reholding time, i.e., the micro depth recovery rate of cement paste exhibits an initial rapid linear decrease followed by a slow linear decrease, which is similar to the experimental creep recovery results of concrete at macro scale (Glucklich, 1959; Illston, 1965). This indicates that the MI tests used in this book can measure the micro creep recovery of cementitious materials. It is noted that the transition time from the initial rapid period to the slow one is 40–80 s, which confirms the finding in Sect. 8.6.1 that the recoverable creep may be insignificant after a holding time of 40 s during the holding stage. The above transition time by the MI test at micro scale is quite smaller than that at macro scale (usually ranging from days to years). The very high stress level beneath the indenter

326

8 Testing and Analysis of Micro Creep

in the MI tests may explain such a difference between the macro and microtransition times. Based on the measured creep recovery results by the microcantilever beam indentation technique, Gan et al. (2021) also observed similar results concerning the recovery rate of indentation depth of cement paste, which is shown in Fig. 8.32. As can be seen in Fig. 8.32, the creep recovery exhibits a rapid devolvement followed by a slow one. Generally, the rate of creep recovery gradually decreases as the recovery time increases. Similar to the results shown in Fig. 8.31, there are clearly two different attenuation speeds observed in the samples with different w/c ratios of 0.3 and 0.4 measured by the microcantilever beam indentation technique and the corresponding transition point is 15–20 s. 10

2N-10s-180s-10s

1 0.1 0.01 rapid linear decrease stage

0.001 0

100

200

slow linear decrease stage

300 400 Time (s)

Depth recovery rate (nm/s)

(c)

500

10

(b)

2N-10s-0s-10s

Depth recovery rate (nm/s)

Depth recovery rate (nm/s)

(a)

600

4N-10s-0s-10s

0.1 0.01 rapid linear decrease stage

0.001 0

700

10

4N-10s-180s-10s

1

slow linear decrease stage

100 200 300 400 500 600 700 Time (s)

8N-10s-0s-10s 8N-10s-180s-10s

1 0.1 0.01 rapid linear decrease stage

0.001 0

100

200

300 400 Time (s)

slow linear decrease stage 500

600

700

Fig. 8.31 Recovery rate of indentation depth of cement paste over the reholding stage for the MI tests with indentation forces of a 2 N, b 4 N, and c 8 N (Liang & Wei, 2020b)

w/c=0.3, 40mN w/c=0.3, 60mN w/c=0.3, 80mN

Stage 1

Stage 2

327 Recovery rate per unit stress (nm/s/MPa)

Recovery rate per unit stress (nm/s/MPa)

8.7 Influence of Mixture and Environment

w/c=0.4, 20mN w/c=0.4, 40mN w/c=0.4, 60mN

Stage 1

Second hold time (s)

Stage 2

Second hold time (s)

Fig. 8.32 Measured evolutions of recovery rate per unit stress for cement pastes with different w/c ratios by microcantilever beam indentation (Gan et al., 2021)

8.7 Influence of Mixture and Environment 8.7.1 Influence of Mixture Proportion of Cementitious Materials The existing findings reveal that the mixture proportions have a great influence on the creep and creep recovery of cementitious materials (Bažant & Jirásek, 2018; Neville et al., 1983). To evaluate the influence of the water-to-cement ratio of cement paste on the measured creep and creep recovery by indentation test, Liang and Wei (2020a) conducted MI tests on cement pastes with w/c ratios of 0.3, 0.4, and 0.5. The curing age was 7 days. Two MI schemes were designed, which are the same as those shown in Fig. 8.13. The detailed test parameters are summarized in Table 8.2. Figure 8.33 shows the contact creep function of cement pastes obtained from the MI results based on Eq. (3.26). Under the same curing age, the contact creep function of the cement pastes increases with the increasing w/c ratio, indicating that cement pastes with a higher w/c ratio will creep more at micro scale. This is in agreement with the macro creep results of cement paste with different w/c ratios (Liang & Wei, 2019a, 2019b; Neville et al., 1983). Figure 8.34 shows the recovery of indentation depth of cement pastes with different w/c ratios during the reholding stage. Both the MI tests with no holding stage and those with a holding stage of 180 s show an obvious recovery of indentation depth, which remain nearly constant after 1000 s recovery. In addition, the indentation depth recovers more rapidly in the MI tests with no holding stage than that with a holding stage. The measured depth recovery by the MI tests with no holding stage is 62.8–64.9% of that measured with a holding stage of 180 s, indicating that 62.8–64.9% of recoverable creep evolves over the loading stage. And the creep indentation depth developed during the holding stage is partially recoverable. Besides, as shown in Fig. 8.34, the magnitude of depth recovery of cement pastes increases with

328

8 Testing and Analysis of Micro Creep

Fig. 8.33 Measured contact creep function of cement pastes with different w/c ratios at the age of 7 days (Liang & Wei, 2020a)

∆ hrecoverable creep (t) (nm)

the increasing w/c ratio, suggesting that a higher w/c ratio will contribute to more recoverable creep of cement pastes. Vandamme and Ulm (2013), and Zhang et al.(2014) have proved that the contact creep function obtained from the indentation tests can be used to evaluate the longterm creep of cementitious materials, and the long-term creep rate mainly depends on the creep modulus. A greater creep modulus will result in a lower long-term creep rate. Based on the measured contact creep function and Eq. (3.27), the contact creep moduli of cement pastes with different w/c ratios can be determined, which are shown in Fig. 8.35. It can be seen that a high w/c ratio of cement paste will result in a low creep modulus under the same curing age, which suggesting that a higher w/c ratio contributes to increasing the long-term creep rate of cement pastes at micro scale. It has been shown in Fig. 8.34 that a greater recovery of indentation depth over the reholding stage will be observed for the cement paste with a higher w/c ratio. The magnitude of recoverable contact creep of cement pastes with different w/c ratios

Fig. 8.34 Measured recovery of indentation depth of cement pastes with different w/c ratios at the age of 7 days (Liang & Wei, 2020a)

8.7 Influence of Mixture and Environment 150 Creep modulus (GPa)

Fig. 8.35 Fitted creep modulus of cement pastes with different w/c ratios at the age of 7 days from the contact creep function (Liang & Wei, 2020a)

329

Curing age =7 days 100

50

0 0.3

0.4 w/c ratio

0.5

is summarized in Fig. 8.36. It can be seen from Fig. 8.36 that the magnitude of the recoverable contact creep of cement paste increases with increasing w/c ratio. For instant, the magnitude of recoverable contact creep of cement paste with 7–day curing increase by 83.4% when the w/c ratio increases from 0.3 to 0.5. This indicates that the w/c ratio is an important factor influencing the recoverable creep of cement pastes. Next, the ratio of the recoverable creep to the total creep, which is expressed as the ratio of recovery of indentation depth caused by the recoverable creep (∆hrecoverable creep ) to the total change in indentation depth caused by both recoverable creep and irrecoverable creep (∆htotal creep ), is used to further investigate the micro recoverable creep of cement pastes. Based on the measured indentation depth of cement pastes in Table 8.7, the ∆hrecoverable creep /∆htotal creep ratios of cement pastes with different w/c ratios can be obatined, which are shown in Fig. 8.37. As shown in Fig. 8.37, the ∆hrecoverable creep /∆htotal creep ratio is 19.3–25.8%, which will decrease with increasing w/c ratio. This experimental result reveals that the proportion of recoverable creep in the total creep is larger in the cement pastes with a lower w/c ratio at micro scale, which is in agreement with the findings at macro scale (Meyers, 1967; Neville, 1960; Staley & Peabody, 1946). In fact, the existing findings show that concrete with curing ages of several years may have a recoverable creep/ total creep ratio as high as 0.8 at macro scale (Neville et al., 1983). 60 Curing age =7 days [L(180)-L(0)]recoverable (×10-6/MPa)

Fig. 8.36 Magnitude of recoverable contact creep of cement pastes with different w/c ratios at the age of 7 days in the MI tests with a holding time of 180 s (Liang & Wei, 2020a)

50 40 30 20 10 0 0.3

0.4 w/c ratio

0.5

4997.8 ± 610.9

664.1 ± 39.7

715.2 ± 38.6

w/c = – 0.3 (3 years)

–

–

–

w/c = 0.3 (7 days)

w/c = 0.4 (7 days)

w/c = 0.5 (7 days)

760.3 ± 37.5

3368.1 ± 208.2

628.6 ± 23.4

–

w/c = 0.3 (1 day)

5214.4 ± 422.5

1851.4 ± 58.3

4152.0 ± 250.2

790.3 ± 87.5

Scheme 1

MI tests

1216.5 ± 81.5

1101.8 ± 91.3

1057.1 ± 46.9

924.0 ± 21.7

1249.7 ± 90.4

5974.7

5713.0

4032.2

2480.0

4942.3

0.204

0.192

0.262

0.373

0.253

Scheme 2 ∆htotal = Rrecoverable = ∆h1holding stage creep (0) (nm) ∆h1reholding stage creep (1000) ∆h2holding stage creep (180) ∆h2reholding stage creep (1000) ∆h1reholding stage creep (1000) + ∆h2reholding creep stage (1000)/∆htotal creep ∆h2holding stage creep (180) (nm) (nm) (nm) (nm)

Table 8.7 Summary of the change in indentation depth over the holding stage and the recovery of indentation depth over the reholding stage of cement pastes in the MI tests (Liang & Wei, 2020a)

330 8 Testing and Analysis of Micro Creep

Fig. 8.37 Ratio of the indentation depth caused by recoverable creep to that caused by total creep of cement pastes with different w/c ratios at the age of 7 days (Liang & Wei, 2020a)

331

∆hrecoverable creep /∆htotal creep

8.7 Influence of Mixture and Environment

0.5 Curing age =7 days 0.4 0.3 0.2 0.1 0.0 0.3

0.4 w/c ratio

0.5

Zhang et al. (2014) studied the influence of the mixture proportion of cementitious materials on the micro creep property by MI. The maximum indentation force was 20 N, and the holding stage was 300 s. The measured results are shown in Fig. 8.38. It can be seen that a greater the w/c ratio will result in a greater magnitude of the creep strain, and the addition of silica fume contributes to reducing the creep strain. Specimens with different clinkers (cement clinker from Saint Vigor with a specific surface of 0.35m2 /g and specific gravity of 3.18 g/cm3 , denoted as SV clinker; cement clinker from Saint-Pierre-la-Cour with a specific surface of 0.45m2 /g and specific gravity of 3.11 g/cm3 , denoted as LC clinker) but with the same mix proportions (i.e., the same w/c ratio of 0.38) exhibited almost the same creep property. And the trends observed on the creep properties of cement pastes from MI test are qualitatively comaprable to those observed by the macroscopic uniaxial compression test (Zhang et al., 2014). For both the MI creep test and the macroscopic uniaxial compression creep test, the creep function is well characterized by a logarithmic function of time after a transient period. The amplitude of the rate of this logarithmic kinetics of creep is controlled by the creep modulus (i.e., the uniaxial creep modulus in the macro uniaxial creep test and the contact creep modulus in the MI creep test). By performing homogenization from the cement paste scale using the estimated contact creep modulus, the predicted uniaxial creep moduli of concrete are in excellent agreement with the measured ones as shown in Fig. 8.39. This indicates that the minutes-long MI can be used to study the long-term creep of cementitious materials. Gan et al. (2020) also investigated the influence of mixture proportion on the short-term creep of cementitious materials by microbending tests on the miniaturized cantilever beams (as shown in Fig. 8.15). The determination of specific basic creep compliance based on the measured results of microbending tests can refer to Eqs. (8.12) and (8.13). Based on the measured results, one can obtain the specific basic creep compliance of cement pastes with w/c ratios of 0.3, 0.4, and 0.5 with respect to the logarithmic time, which is shown in Fig. 8.40. It can be seen that the specific basic creep compliance of cement paste increases with the increasing w/c ratio, which is consistent with the findings obtained by Liang and Wei (2020a), as shown in Fig. 8.33.

332

8 Testing and Analysis of Micro Creep

Fig. 8.39 Uniaxial creep modulus measured by uniaxial compression creep experiments on concrete versus predicted uniaxial creep modulus of concrete based on MI creep experiments performed on cement paste (Zhang et al., 2014)

Predicted contact creep modulus Ci,con (GPa)

Fig. 8.38 Contact creep functions of different cement paste samples obtained by MI (Zhang et al., 2014)

Measured uniaxial creep modulus Cu,con (GPa)

Fig. 8.40 Specific basic creep compliance with respect to the logarithmic time (Gan et al., 2020)

8.7 Influence of Mixture and Environment

(b)

L(t)-1/M0 (×10-6/MPa)

L(t)-1/M0 (×10-6/MPa)

(a)

333

time (s)

time (s)

Fig. 8.41 Measured contact creep compliance function for FA0, FA20, and FA40 specimens (w/b = 0.40; Pmax = 3 N, holding time = 180 s) tested at the age of a 3 days and b 90 days (Mallick et al., 2019). Note FA0, FA20, and FA40 denote the cement paste specimens containing 0, 20, and 40% fly ash by weight of the binder, respectively; SC and DC denote the sealed condition and dried condition, respectively

Mallick et al. (2019) investigated the influence of fly ash content on the creep development of cement pastes by MI. Cement paste specimens containing 0, 20, and 40% fly ash (by weight of binder) were tested at 1, 3, 7, 28, and 90 days. It is found that the influences of the fly ash content on creep are related to the loading age, which is shown in Fig. 8.41. The creep deformations of cementitious materials loaded at early ages increase with the increasing content of fly ash. However, the creep difference diminished when the loading age is beyond 28 days, which can be accounted for by the effect of the delayed pozzolanic reactions.

8.7.2 Influence of Microstructures of Cementitious Materials Microstructures of cementitious materials are one of the most important factors that affect creep development. Several researchers have conducted indentation tests to evaluate the influence of microstructures of cementitious materials on the creep properties, which are shown as follows. Since the molecular structure and the chemical composition of CSH are not constant, this variability is expected to affect the viscoelasticity of cementitious materials. Hunnicutt et al. (2016) conducted stress relaxation NI tests on the synthesized CSH and CASH. The displacement–time curve during the stress relaxation test is shown in Fig. 8.42a. An indentation depth of 175 nm was made in the specimen within 1 s. The stress relaxation was monitored under this indentation depth for 60 s, and finally, the specimen was unloaded within 1 s. Figure 8.42b shows the normalized stress relaxation results. It is seen that the normalized stress relaxation function of CASH is greater than that of CSH, which indicates that the substitution of silicon with aluminum in CASH produces less viscous property. The authors suggested that

334

8 Testing and Analysis of Micro Creep (b)

(a)

1.0

CSH CASH

175

Normalized force

Displacement (nm)

0.95 0.9 0.85 0.8 0.75 0.7 0.65

1

Time (s)

61 62

0

10

20

30

40

50

60

Time (s)

Fig. 8.42 a Stress relaxation indentation test, b normalized stress relaxation of CSH and CASH showing 99.8% confidence intervals (Hunnicutt et al., 2016)

this phenomenon is related to the addition of aluminum increasing the aluminosilicate mean chain length and the crosslinking between layers (Hunnicutt et al., 2016). The viscoelastic response of CASH appears to result from layers or particles continuously moving relative to each other under external stress. When CASH has a longer mean chain length and exhibits crosslinking, it has fewer particles. The reduction in particle number decreases the relative particle movement and thus reduces the viscous property of cementitious materials. Wei et al. (2017) studied the influence of the microstructures of cement pastes on the creep development by MI test. The maximum indentation force was set as 0.5 N, and the holding time was 180 s. Four typical microstructures were tested for indentation creep measurement. The BSE images of the four typical microstructures and the corresponding mechanical and creep properties are summarized in Table 8.8. The indented area covering porous large pore is defined as Type I. While the indented area covering porous smaller pore than that in Type I is defined as Type II. Type III refers to the area that covers a large number of hydration products and a small amount of unreacted clinker grains and pores, and Type IV denotes the indented area that is mainly composed of unreacted clinker grain. The influence of microstructure on the micro creep of cementitious materials is analyzed by the depth increment ∆h(t) during the holding period and the contact creep function [L(t)-L(0)]. Figure 8.43 shows the measured ∆h(t) and [L(t)-L(0)] of the four typical microstructures. It is noted that the development of ∆h(t) does not correspond well to the microstructure characteristic. The greatest ∆h(t) is found for Type II, the ∆h(t) of Type III is close to that of Type IV. However, Type I with more and larger pores reveals the least ∆h(t), which is contradictory to the expectation that porosity causes a greater ∆h(t) value. The reason lies in that the average stress beneath the indenter (i.e., the contact hardness) varies with the microstructures (see Table 8.8, H = 0.25, 0.64, 0.70, 0.73 GPa for microstructures I, II, III, and IV, respectively), low stress will induce less deformation, and the creep property of the four microstructures cannot be compared directly based on ∆h(t). As shown in Fig. 8.43b, the contact creep function calculated according to Eq. (3.26) corresponds well to the microstructure. A larger porosity leads to more creep. Type I exhibits the greatest [L(t)-L(0)], followed by Types II, III, and IV. The

8.7 Influence of Mixture and Environment

335

Table 8.8 Characteristics of indents made on typical microstructures under indentation load of 0.5 N (Wei et al., 2017) Type Microstructure image

Characteristics

Mechanical creep properties

I

The indented zone is composed of large-size pores; the phase fractions of the void, hydration product, and clinker are 21.1%, 65.8%, and 13.1%, respectively; the characteristic length of the indent is about 66.4 μm

E = 20.8 GPa H = 0.25 GPa C = 135.0 GPa τ = 0.19 s

II

Most of the hydrated phase are covered by the indented zone; the phase fractions of the void, hydration product, and clinker are 2.3%, 95.1%, and 2.6%, respectively; the characteristic length of the indent is about 47.1 μm

E = 26.2 GPa H = 0.64 GPa C = 181.1 GPa τ = 0.49 s

III

A certain amount of clinkers are covered by the indented zone; the phase fractions of the void, hydration product, and clinker are 3.5%, 83.7%, and 12.8%, respectively; the characteristic length of the indent is about 45.0 μm

E = 28.3 GPa H = 0.70 GPa C = 184.3 GPa τ = 0.44 s

IV

The indented zone is mainly composed of the unhydrated clinker; the phase fractions of the void, hydration product, and clinker are 2.0%, 52.8%, and 45.2%, respectively; the characteristic length of the indent is about 40.7 μm

E = 33.9 GPa H = 0.73 GPa C = 192.4 GPa τ = 0.34 s

average contact creep function of all the indents obtained in the MI test, [L(t)-L(0)]ave , is almost equal to that of Type III, which is the most common microstructure found in hardened paste under the maximum indentation force of 0.5 N. Figure 8.43b further validates that the contact creep function [L(t)-L(0)] can quantify the micro creep of hardened paste. L(t) bears as much information as the uniaxial creep compliance J(t) for the macroscopic creep test, which is a material property that depends neither on the indenter geometry nor on the indentation force.

336

8 Testing and Analysis of Micro Creep

Fig. 8.43 Influence of microstructure (Types I, II, III, IV as shown in Table 8.8) on a the increment of indentation depth during holding and on b the contact creep function for w/c ratio = 0.3 paste at the age of 5 months under the indentation load of 0.5 N (Wei et al., 2017)

Hu et al. (2020) conducted MI tests on the CSH compacts (w/c ratio of 0.45) with different Ca/Si ratios (0.8–2.0) to investigate the influence of mixture proportion on the creep property. The comparisons of MI results with different magnitudes of loads applied are summarized in Table 8.9. It is seen form Table 8.9 that the CSH compacts with higher Ca/Si ratio reveal lower creep. A higher porosity will result in a lower contact creep modulus and a lower characteristic time, which corresponds to larger creep. This finding is in agreement with the previous studies by MI test (Nguyen et al., 2012, 2014; Pourbeik et al., 2013), as shown in Fig. 8.44. The contact creep modulus of CSH with similar porosity as in a hardened cement paste (close to 30%) is about 180 GPa. Niewiadomski and Stefaniuk (2020) assessed the creep development of the cement matrix of the self-compacting concrete modified with the addition of nanoparticles (i.e., SiO2 , TiO2 , and Al2 O3 nanoparticles) using the indentation method. The creep property of cementitious materials was then evaluated by using the creep coefficient C IT , which is expressed as: CI T =

h2 − h1 × 100% h1

(8.20)

where h1 and h2 are the indentation depth at the end of the loading stage and at the end of the holding stage, which are illustrated in Fig. 8.45. The measured creep coefficients of different cement pastes are summarized in Table 8.10. The type of used nanoparticles seems to affect the creep coefficient C IT in comparison to the reference series (S0). It is found that the addition of nanoparticles does not significantly influence the creep of the cement matrix.

Ca/Si-1.5

0.46

0.34

0.45

403.76 ± 13.80 395.15 ± 16.09 372.68 ± 11.59 226.52 ± 27.76 209.57 ± 29.07 200.59 ± 30.45 393.76 ± 38.80 386.63 ± 18.78 372.63 ± 13.13 317.75 ± 9.45 307.48 ± 24.25 295.85 ± 6.04

16.23 ± 0.37

15.05 ± 0.45

14.61 ± 0.41

7.82 ± 0.64

6.91 ± 0.76

6.87 ± 0.89

14.60 ± 0.95

14.09 ± 0.38

13.71 ± 0.25

12.84 ± 0.23

11.97 ± 0.60

11.66 ± 0.16

500

1000

2000

500

1000

2000

500

1000

2000

500

1000

2000

0.31

Ca/Si-0.8

Indentation hardness (GPa)

Young’s modulus (GPa)

Loads (mN)

Porosity

Specimen

Table 8.9 Comparison of MI results between different mixture proportions (Hu et al., 2020)

168.51 ± 2.96

166.83 ± 8.42

174.46 ± 4.60

204.68 ± 5.46

207.00 ± 6.78

221.36 ± 14.84

83.74 ± 11.26

94.70 ± 11.83

86.14 ± 23.11

139.37 ± 5.55

139.83 ± 6.78

132.66 ± 17.19

Contact creep modulus (GPa)

0.17 ± 0.01

0.19 ± 0.01

0.18 ± 0.01

0.18 ± 0.01

0.19 ± 0.01

0.18 ± 0.02

0.28 ± 0.03

0.20 ± 0.03

0.91 ± 0.28

0.31 ± 0.02

0.22 ± 0.02

0.31 ± 0.02

Characteristic time (s)

8.7 Influence of Mixture and Environment 337

338

8 Testing and Analysis of Micro Creep

Fig. 8.45 Illustration of the indentation depth–time curve (Niewiadomski & Stefaniuk, 2020)

Indentation depth (nm)

Fig. 8.44 Creep modulus versus porosity for CSH (C/S = 0.6, 0.8, 1.0, 1.2, and 1.5), cement paste, and hydrated C3 S by MI (Nguyen et al., 2014)

h2 h1

Creep Unloading Loading t2 t1 Indentation time (s)

Table 8.10 Mean values and standard deviations of the creep coefficient (Niewiadomski & Stefaniuk, 2020) Concrete series

S0

S1

S2

S3

μCIT [%]

8.49

8.96

8.32

9.06

σ CIT [%]

1.46

1.36

1.26

1.03

σ CIT /μCIT

0.172

0.152

0.151

0.114

Note S 0 , S 1 , S 2 , and S 3 denote pastes with no nanoparticles, 4.0% SiO2 , 4.0% TiO2 , and 4.0% Al2 O3

8.7.3 Influence of Curing Age of Cementitious Materials Curing age is also one of the most important factors that affect the creep and creep recovery of cementitious materials. To evaluate the influence of curing age, Liang and Wei (2020a) performed MI tests on cement paste with a w/c ratio of 0.3. The curing ages of cement pastes were 1 day, 7 days, and 3 years, respectively. The

8.7 Influence of Mixture and Environment

339

Fig. 8.46 Measured contact creep function of cement pastes with w/c ratio of 0.3 at different curing ages (Liang & Wei, 2020a)

contact creep function measured at different curing ages is shown in Fig. 8.46. It is seen from Fig. 8.46 that the contact creep function of cement paste increases as the curing age decreases. For instance, the average contact creep function of cement paste with curing age of 1 day is 107.7 × 10–6 /MPa, which is 3.8 times of that of cement paste with curing age of 3 years (28.1 × 10–6 /MPa). A similar result has been reported for concrete at macro scale (Hanson, 1953; Neville et al., 1983), Hanson (1953) has also found that the average creep function of the concrete with curing age of 2 days can be 3.73 times of that of concrete with curing age of 1 year. The recovery of indentation depth of cement pastes with w/c ratio of 0.3 at the age of 1 day, 7 days, and 3 years during the reholding stage is displayed in Fig. 8.47. The magnitude of the depth recovery of cement pastes decreases with increasing curing age under the same indentation force, indicating that the amount of recoverable creep of cement pastes will decrease when the curing age is increased. Figure 8.48 shows the creep moduli of cement pastes with different curing ages. It is seen from Fig. 8.48 that the creep modulus of cement pastes increases when the curing age of the cement paste is increased, indicating a shorter curing age will result in the increase in the long-term creep rate of cement pastes at micro scale. The possible reason lies in that the aging of CSH, i.e., the polymerization process of the silicates in CSH (Jennings, 2004) becomes more significant in the cement paste with a longer curing age, which makes the cement pastes become denser and stiffer; therefore, a lower long-term creep rate is obtained. The magnitude of the recoverable contact creep of cement pastes with different curing ages is shown in Fig. 8.49. As the curing age of cement paste increases, the magnitude of the recoverable contact creep decreases. For instance, the magnitude of the recoverable contact creep of cement paste with w/c ratio of 0.3 decreases by 60.9% when the curing age is increased from 1 day to 3 years, suggesting that curing age is also an important factor affecting the recoverable creep of cement pastes. Based on the measured change of indentation depth during the holding stage and the reholding stage shown in Table 8.7, the ∆hrecoverable creep /∆htotal creep ratios of cement pastes with different curing ages can be easily determined, which are shown

8 Testing and Analysis of Micro Creep

∆h recoverable creep (t) (nm)

340

Fig. 8.47 Measured recovery of indentation depth of cement pastes with w/c ratio of 0.3 at different curing ages (Liang & Wei, 2020a) 250 Creep modulus (GPa)

Fig. 8.48 Fitted creep modulus of cement pastes with w/c ratio of 0.3 at different curing ages from the contact creep function (Liang & Wei, 2020a)

w/c ratio=0.3

200 150 100 50 0 1 day

60 [L(180)-L(0)]recoverable (×10-6/MPa)

Fig. 8.49 Magnitude of recoverable contact creep of cement pastes with w/c ratio of 0.3 at different curing ages in the MI tests with a holding time of 180 s (Liang & Wei, 2020a)

7 days Curing age

3 years

w/c ratio=0.3

50 40 30 20 10 0 1 day

7 days Curing age

3 years

Fig. 8.50 Ratio of the indentation depth caused by recoverable creep to that caused by total creep of cement pastes with w/c ratio of 0.3 at different curing ages (Liang & Wei, 2020a)

341

∆hrecoverable creep /∆htotal creep

8.7 Influence of Mixture and Environment

0.5

w/c ratio=0.3

0.4 0.3 0.2 0.1 0.0 1 day

7 days Curing age

3 years

in Fig. 8.50. As shown in Fig. 8.50, the ∆hrecoverable creep /∆htotal creep ratio of the tested cement paste is 25.3–37.2%, which will increase with increasing curing age. The result displayed in Fig. 8.50 indicates again that the proportion of the recoverable creep in the total creep is larger in stiffer cement pastes with a lower w/c ratio at micro scale.

8.7.4 Influence of Temperature The macroscopic experimental results show that temperature is an important factor that affects the creep development of cementitious materials (Alogla & Kodur, 2020; Nasser & Neville, 1965; Vidal et al., 2015). However, the effect of thermal activation on the creep mechanisms of cementitious materials is not fully understood. FrechBaronet et al. (2020) investigated the thermal effect on the creep rate of cement pastes with different water-to-binder ratios of 0.20, 0.42, and 0.55 by MI technique. MI creep tests were performed at a maximum indentation force of 8 N. Both the loading time and the unloading time were set as 5 s. And the holding time was set as 300 s. The MI tests were performed with the MCT3 microindenter with a Berkovich tip at 10 °C, 23 °C, 40 °C, and 60 °C. The test temperature was controlled by a closed-loop Peltier module, which was increased at a rate of 0.2 °C/minutes after the specimen was put into the MCT3 microindenter. After the temperature reached the target value, it will be maintained for 10 h before testing, which was expected to be enough to assure a stable and uniform temperature distribution within the specimen. During the test, the typical temperature distribution on a sample surface monitored by the employed high-precision infrared camera (Fluke TiX500) is shown in Fig. 8.51, which shows that the actual temperature is almost the same as the set one. Figure 8.52 shows the mean creep compliance function for the five mix designs, i.e., cement paste with w/b ratio of 0.2 replacing cement with 10% of silica fume (0.2-SF), pure cement paste with w/b ratio of 0.2 (0.2), pure cement paste with w/b ratio of 0.42 (0.42), pure cement paste with w/b ratio of 0.55 (0.55), and pure cement paste with w/b ratio of 0.55 that was kept in a small water container for

342

8 Testing and Analysis of Micro Creep

Fig. 8.51 Thermal images of cement paste with w/b = 0.42 and heated at a 40 °C and b 60 °C (Frech-Baronet et al., 2020)

7 days (0.55-W), respectively. In general, temperature has little effect on the MI creep curves. Furthermore, as shown in Fig. 8.53, the creep modulus doesn’t show an obvious variation with the test temperature. The above findings are different from those at macro scale (Alogla & Kodur, 2020; Nasser & Neville, 1965; Vidal et al., 2015). Based on the measured creep results at both micro and macro scales, Frech-Baronet et al. (2020) suggested that the influence of the thermal activation is on the characteristic times rather than the long-term logarithmic creep rate (i.e., creep modulus). Therefore, the temperature won’t affect the asymptotic logarithmic rate of basic creep. However, further work should be conducted to investigate the thermo-activation effect on the short-term creep mechanisms.

8.7.5 Influence of Relative Humidity It is well known that relative humidity (RH) is another important factor that affects the creep development of cementitious materials at macro scale (Bazant & Chern, 1985; Nastic et al., 2019; Vandewalle, 2000; Wei et al., 2020). However, the influence of relative humidity on the micro creep of cementitious materials is still unclear. To investigate the influence of relative humidity, Frech-Baronet et al. (2017) conducted MI creep and relaxation tests on cement paste with w/c ratio of 0.6 at different relative humidities. The illustration of the MI creep and the relaxation tests is shown in Fig. 8.54. The microindenter was enclosed in a hermetic chamber to control the relative humidity. The hermetic enclosure was connected in series to an Erlenmeyer flask containing a saturated salts solution and a pump in a closed-loop system. Four different stable relative humidities were achieved in the chamber (i.e., 18%, 33%, 55%, 75%, respectively) using saturated solutions of different salts (i.e., lithium chloride at 84.25 g/100 mL; magnesium chloride at 54.3 g/100 mL, magnesium nitrate at 125.0 g/100 mL, potassium chloride at 34.4 g/100 mL). Both the creep test and the relaxation test were conducted in the study by Frech-Baronet et al. (2017). Based on the measured results, the contact creep modulus (C) and the contact relaxation modulus (R) can be expressed as:

8.7 Influence of Mixture and Environment

343

Fig. 8.52 Mean creep compliance function for a 0.20-SF, b 0.20, c 0.42, d 0.55, and e 0.55-W cement pastes at different temperature by MI (Frech-Baronet et al., 2020) Fig. 8.53 Contact creep modulus C i for different temperatures and water-to-binder ratios (Frech-Baronet et al., 2020)

344

(a)

8 Testing and Analysis of Micro Creep creep stage

(b)

relaxation stage

Fig. 8.54 Typical indentation force–indentation depth curve for a creep test and b relaxation test (Frech-Baronet et al., 2017)

C = lim

1 ˙ L(t)t

(8.21)

R = lim

1 ˙ M(t)t

(8.22)

t→∞

t→∞

where, M(t) is the contact relaxation modulus of the indented material. Figure 8.55 shows the relationship between the creep modulus and the relaxation modulus with relative humidity. It can be seen that the higher the relative humidity is, the higher creep or relaxation rate of the cement paste is. The results agree with previous works on macroscopic samples (Bazant & Chern, 1985; Nastic et al., 2019; Vandewalle, 2000; Wei et al., 2020). The mechanism of the relative humidity effect on the creep of cementitious materials can be illustrated in Fig. 8.56. The capillary suction in the capillary pores will increase as the RH within the specimen drops, which will result in an instantaneous increase in the disjoining pressure on the contact sites of the CSH gels (Bažant et al., 1997). In addition, the increase in the transversal compressive disjoining pressure may increase internal friction of CSH sheets, which will reduce their sliding capacity and thus a low long-term creep rate is observed. Besides, the microcracking drying effects may be another explanation to the measured results. Recently, Suwanmaneechot et al. (2020) also investigated the creep behavior of CSH under different drying relative humidities by MI tests. The cement paste samples are cured in lime-saturated water for 1 year and then placed into chambers with controlled humidity for three years. Before MI measurement, samples are dried and then stored in the original chamber for at least two weeks to control the target relative humidity. The measured contact creep functions of cement pastes under different relative humidities are shown in Fig. 8.57. It can be seen from Fig. 8.57 that the contact creep function of cement pastes L55 and H55 (made from low heat Portland cement and high early strength Portland cement, respectively, with a w/c ratio of 0.55) increases with increasing relative humidity. By fitting the contact creep function with Eq. (3.27), the creep modulus and the characteristic time of the cement pastes under different relative humidities can be obtained, which are shown in Fig. 8.58. As shown in Fig. 8.58, the creep modulus decreases with increasing

8.8 Implications of Logarithmic-Type Creep Development of Cement …

345

Fig. 8.55 a Creep contact modulus (C i ) as function of RH; b relaxation contact modulus (Ri ) as function of RH (Frech-Baronet et al., 2017). Note All samples are cured at 100% RH for 1 month and at 50% RH for 2 months. F–C 1 Days denotes the force-controlled test subject to 1 day of RH recurring with holding time of 300 s; F–C 7 Days denotes the force-controlled test subject to 7 days of RH recurring with holding time of 300 s; F–C 1000 s denotes the force-controlled test subject to 1 day of RH recurring with holding time of 1000 s; DC 1 Days denotes the displacement-controlled test subject to 1 day of RH recurring with holding time of 300 s

S S Pc

CSH gel

τ Pc

Gel contact points

CSH sheet

Fig. 8.56 Schematic representation of the effect of capillary pressure on the contact forces of the cement paste gel and, in turn, on the friction of the sliding CSH (Frech-Baronet et al., 2017)

relative humidity, while the characteristic time exhibits an opposite trend with the relative humidity. This indicates that the creep rate of cementitious materials will increase when the relative humidity increases, and the characteristic time at which creep starts exhibiting long-term logarithmic kinetics will be delayed as the relative humidity increases.

8.8 Implications of Logarithmic-Type Creep Development of Cement Pastes at Micro Scale As shown in Fig. 8.59, the micro long-term creep development of cementitious materials has been reported to possess a logarithmic-type feature for both NI and

100%RH

95%RH 90%RH 80%RH 70%RH 60%RH ~11%RH

Time (s)

Contact creep function (×10-6/MPa)

(a)

8 Testing and Analysis of Micro Creep

Contact creep function (×10-6/MPa)

346

(b) 100%RH

95%RH 85%RH 79%RH 58%RH 40%RH 33%RH 11%RH

Time (s)

Fig. 8.57 Measured and fitted contact creep functions of the hardened cement paste samples dried at different relative humidities, a H55, b L55 (Suwanmaneechot et al., 2020)

Fig. 8.58 Creep properties obtained from the analysis of the hardened cement paste dried at different relative humidities: a contact creep modulus, b characteristic time (Suwanmaneechot et al., 2020)

MI tests. The logarithmic-type feature can be well captured by Eq. (3.27) where the logarithmic-type creep kinetics is governed by both the creep modulus and the characteristic time. Based on the findings by Vandamme and Ulm (2009), the longterm creep rate of cementitious materials can be characterized by the creep modulus, while the time at which creep starts exhibiting logarithmic kinetics is controlled by the characteristic time. This section will further discuss the logarithmic-type creep development of cement pastes at the micro scale. Liang and Wei (2020a) employed a dashpot element with a variable viscosity to model the micro creep of cement pastes, which is shown in Fig. 8.59. Obviously, the relation between the stress and strain of the dashpot element shown in Fig. 8.59 can be written as:

8.8 Implications of Logarithmic-Type Creep Development of Cement …

347

Fig. 8.59 Rheological model to characterize the logarithmic-type creep of cementitious materials during the holding stage of MI test (Liang & Wei, 2020a)

σ0 = ηir (t)

dε dε = (η0 + kt) dt dt

(8.23)

where σ0 and ε are the constant stress and the viscous strain of the dashpot element, respectively; ηir (t)=η0 +kt denotes the viscosity of the dashpot element at time t, η0 is the initial viscosity at the time when the external force is applied, and k is the growth rate of viscosity with time due to the microstructure change of the cementitious materials. By transforming Eq. (8.23), the following equation can be derived: dε dt = η0 + kt σ0

(8.24)

When Eq. (8.24) is integrated on the both sides, the following equation can be obtained: ⎧t 0

dt = η0 + kt

⎧ε(t) 0

dε σ0

(8.25)

Based on Eq. (8.25), one can obtain the specific creep (C(t)) of cement paste at micro scale that is modeled by the dashpot element shown in Fig. 8.59, which is written as: ( ) t 1 ε(t) 1 η0 +kt (8.26) = ln = ln 1 + C(t) = σ0 k η0 k η0 /k According to Eq. (3.27) and Eq. (8.26), one can easily express the creep modulus and the characteristic time as followings:

348

8 Testing and Analysis of Micro Creep

⎧

C =k

/ τ = η0 k

(8.27)

Clearly, the growth rate of viscosity with time is equal to the creep modulus according to Eq. (8.27). Moreover, the initial viscosity of the dashpot is related to the creep modulus and the characteristic time, which is expressed as: η0 = C × τ

(8.28)

As shown in Fig. 8.59, the ability of cementitious materials to creep at micro scale is controlled by the initial viscosity of the dashpot element. The greater initial viscosity means that cementitious materials creep less at micro scale. It can be inferred that the magnitude of the initial viscosity depends greatly on the microstructures of the cementitious materials, which will be influenced by the mixture proportions and the curing age of cementitious materials. Based on the measured contact creep functions of cement pastes obtained from the MI creep tests by Liang and Wei (2020a), the initial viscosity of cement pastes under different w/c ratios and curing ages can be obtained according to Eq. (8.28), which is shown in Fig. 8.60. As shown in Fig. 8.60, the initial viscosity decreases with increasing w/c ratio of the cement paste, but increases with the curing age of the cement paste. This suggests that cement paste with a lower w/c ratio or a longer curing age will exhibit less creep. Moreover, a linear relationship between the initial viscosity and w/c ratio of cement pastes can be observed in the log–log scale coordinates system. A linear relation in the log–log scale coordinates system is also seen between the initial viscosity and the curing age of cement pastes. In order to validate the above relation between the initial viscosity and the curing age of cement pastes, the MI creep results measured on cement pastes by Wei et al. (2017) were used to determine the initial viscosity according to Eq. (8.28). The w/c ratio of the cement pastes was 0.3, which were under the sealed curing condition for 5 months and 1 year. It can be seen from Fig. 8.60b that the initial viscosity of the cement pastes used by Wei et al. (2017) can be well captured by the fitted one. This proves that a linear relation of initial viscosity and the curing age in the log–log scale coordinates system can be used to determine the initial viscosity of cement pastes at any other curing ages. Next, the relationship between the initial viscosity and the microstructures change of cement pastes will be discussed. It has been proposed that the creep of cementitious materials depends on the aging of CSH, i.e., the polymerization process of the silicates in CSH, which is affected by many factors including curing age, temperature and humidity, etc. (Jennings, 2004). To determine the degree of polymerization of silicates in cement paste, the trimethylsilylation method or 29 Si NMR can be employed (Hirljac et al., 1983; Jennings, 2004). Hirljac et al. (1983) used the trimethylsilylation method to determine the degree of polymerization of silicates in cement paste, and the weight percent of the silicate polymer with different hydration time at different temperatures is shown in Fig. 8.61. As shown in Fig. 8.61, the polymerization of

8.9 Summary

349

Fig. 8.60 Relation between the initial viscosity of cement pastes and a w/c ratio and b curing age (Liang & Wei, 2020a)

Fig. 8.61 Relation between the weight percent of silicate polymer and the hydration time under different temperatures of 5, 25, and 65 °C (Hirljac et al., 1983)

the silicates increases with increasing hydration time, and a high temperature will contribute to a faster polymerization process. Furthermore, the relation between the weight percent of silicate polymer and the hydration age is linear in a semi-log coordinate system, which is similar to the results shown in Fig. 8.60. Therefore, it is inferred that the increase in initial viscosity due to the increase of curing age is attributed to the progressive polymerization of the silicates in CSH. As the degree of polymerization increases, the CSH globules will become more tightly packed, which will slow down the creep development of the cementitious materials.

8.9 Summary The existing research findings were reviewed in this chapter concerning the testing and analysis of creep and creep recovery of cementitious materials. First, the methods to measure the creep and creep recovery of cementitious materials at macro scale were introduced, which include the uniaxial compressive creep measurement, the

350

8 Testing and Analysis of Micro Creep

uniaxial tensile creep measurement, the flexural creep test, the restrained test, and the multiaxial creep measurement. These methods usually require a relatively longer testing time from several days to dozens of years and cannot give a local estimation of creep and creep recovery of cementitious materials when compared to the micro scale testing methods. Recently, NI has become a popular technique to evaluate the local creep and creep recovery of cementitious materials at micro scale due to its attractive advantage of fast measurement. By using suitable indentation force, one can not only measure the creep property of single phase in cementitious materials (e.g., LD CSH and HD CSH) at nanoscale, but also measure the homogeneous creep property of cement paste at micro scale based on the loading–holding–unloading–reholding experimental scheme, which can provide experimental supports to illustrate the creep mechanism of concrete structures. The loading strain rate influences the development of the viscoplastic depth developing during the loading stage of the indentation test. The viscoplastic depth decreases with increasing strain rate under any given load. Similar results were reported for soils based on the constant rate of strain consolidation test and the longterm consolidation test at macro scale where the viscoplastic strain decreases with increasing strain rate under any given stress. This indicates that a strong analogy exists between this logarithmic creep behavior of CSH and that of soils. The ratio of the recoverable creep to the total creep can be obtained based on the measured results using the two indentation schemes, i.e., the “loading–holding– unloading–reholding” scheme and the “loading–unloading-reholding” scheme. The ratio of the recoverable creep to the total creep can be up to 0.41, which suggests that the recoverable creep is an important creep component of cement paste at micro scale. The loading duration has a minor effect on the creep recovery of cement paste. The recovery of the indentation depth can be well captured by the Kelvin–Voigt element. The creep modulus fitted from the contact creep function is an intrinsic parameter that reflects the long-term creep rate of cement paste, which is not affected by the MI experimental protocols (such as indentation load, loading time, and holding time). A dashpot with a variable viscosity that increases linearly with time can capture the long-term logarithmic creep behavior of cementitious materials at micro scale. The initial viscosity of the dashpot element characterizes the ability of cement pastes to creep at micro scale. The greater the initial viscosity is, the less creep potential cement pastes have.

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Meyers, B. L. (1967). Time-dependent strains and microcracking of plain concrete. Cornell University. Nasser, K. W., & Neville, A. M. (1965). Creep of concrete at elevated temperatures. Journal Proceedings, 62(12), 1567–1580. Nastic, M., Bentz, E. C., Kwon, O. S., Papanikolaou, V., & Tcherner, J. (2019). Shrinkage and creep strains of concrete exposed to low relative humidity and high temperature environments. Nuclear Engineering and Design, 352, 110154. Neville, A. M. (1960). Recovery of creep and observations on the mechanism of creep of concrete. Applied Scientific Research, 9(1), 71. Neville, A. M., Dilger, W. H., & Brooks, J. J. (1983). Creep of plain and structural concrete. Construction Pr. Neville, A. M. (1971). Creep of concrete: plain, reinforced, and prestressed. Nguyen, D. T., Alizadeh, R., Beaudoin, J. J., & Raki, L. (2012). Microindentation creep of secondary hydrated cement phases and C-S–H. Materials and Structures, 46(9), 1519–1525. Nguyen, D. T., Alizadeh, R., Beaudoin, J. J., Pourbeik, P., & Raki, L. (2014). Microindentation creep of monophasic calcium–silicate–hydrates. Cement and Concrete Composites, 48, 118–126. Niewiadomski, P., & Stefaniuk, D. (2020). Creep assessment of the cement matrix of selfcompacting concrete modified with the addition of nanoparticles using the indentation method. Applied Sciences, 10(7), 2442. Nohava, J., Randall, N. X., & Conté, N. (2009). Novel ultra nanoindentation method with extremely low thermal drift: Principle and experimental results. Journal of Materials Research, 24(3), 873–882. Østergaard, L., Lange, D. A., Altoubat, S. A., & Stang, H. (2001). Tensile basic creep of early-age concrete under constant load. Cement and Concrete Research, 31(12), 1895–1899. Pourbeik, P., Alizadeh, R., Beaudoin, J. J., Nguyen, D. T., & Raki, L. (2013). Microindentation creep of 45 year old hydrated Portland cement paste. Advances in Cement Research, 25(5), 301–306. Ranaivomanana, N., Multon, S., & Turatsinze, A. (2013). Tensile, compressive and flexural basic creep of concrete at different stress levels. Cement and Concrete Research, 52, 1–10. Rossi, P., Tailhan, J. L., & Le Maou, F. (2013a). Comparison of concrete creep in tension and in compression: Influence of concrete age at loading and drying conditions. Cement and Concrete Research, 51, 78–84. Rossi, P., Tailhan, J. L., & Le Maou, F. (2013b). Creep strain versus residual strain of a concrete loaded under various levels of compressive stress. Cement and Concrete Research, 51, 32–37. Staley, H. R., & Peabody, D. (1946). Shrinkage and plastic flow of pre-stressed concrete. Journal Proceedings, 42(1), 229–244. Su, C., Herbert, E. G., Sohn, S., LaManna, J. A., Oliver, W. C., & Pharr, G. M. (2013). Measurement of power-law creep parameters by instrumented indentation methods. Journal of the Mechanics and Physics of Solids, 61(2), 517–536. Suwanmaneechot, P., Aili, A., & Maruyama, I. (2020). Creep behavior of CSH under different drying relative humidities: Interpretation of microindentation tests and sorption measurements by multi-scale analysis. Cement and Concrete Research, 132, 106036. Tailhan, J. L., Boulay, C., Rossi, P., Le Maou, F., & Martin, E. (2013). Compressive, tensile and bending basic creep behaviours related to the same concrete. Structural Concrete, 14(2), 124–130. Vandamme, M. (2018). Two models based on local microscopic relaxations to explain long-term basic creep of concrete. Proceedings of the Royal Society A, 474(2220), 20180477. Vandamme, M., & Ulm, F. J. (2009). Nanogranular origin of concrete creep. Proceedings of the National Academy of Sciences, 106(26), 10552–10557. Vandamme, M., & Ulm, F. J. (2013). Nanoindentation investigation of creep properties of calcium silicate hydrates. Cement and Concrete Research, 52, 38–52. Vandewalle, L. (2000). Concrete creep and shrinkage at cyclic ambient conditions. Cement and Concrete Composites, 22(3), 201–208.

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Vidal, T., Sellier, A., Ladaoui, W., & Bourbon, X. (2015). Effect of temperature on the basic creep of high-performance concretes heated between 20 and 80 °C. Journal of Materials in Civil Engineering, 27(7), B4014002. Watabe, Y., Udaka, K., & Morikawa, Y. (2008). Strain rate effect on long-term consolidation of Osaka bay clay. Soils and Foundations, 48(4), 495–509. Wei, Y., Liang, S., & Gao, X. (2017). Indentation creep of cementitious materials: Experimental investigation from nano to micro length scales. Construction and Building Materials, 143, 222– 233. Wei, Y., Wu, Z., Huang, J., & Liang, S. (2018). Comparison of compressive, tensile, and flexural creep of early-age concretes under sealed and drying conditions. Journal of Materials in Civil Engineering, 30(11), 04018289. Wei, Y., Huang, J., & Liang, S. (2020). Measurement and modeling concrete creep considering relative humidity effect. Mechanics of Time-Dependent Materials, 24(2), 161–177. Yao, X., & Wei, Y. (2014). Design and verification of a testing system for strength, modulus, and creep of concrete subject to tension under controlled temperature and humidity conditions. Construction and Building Materials, 53, 448–454. Zhang, Q., Le Roy, R., Vandamme, M., & Zuber, B. (2014). Long-term creep properties of cementitious materials: Comparing microindentation testing with macroscopic uniaxial compressive testing. Cement and Concrete Research, 58, 89–98.

Chapter 9

Multiscale Prediction of Elastic Modulus of Cementitious Materials

Abstract This chapter discusses the existing multiscale models to predict the elastic modulus of cementitious materials across scales by the micromechanics-based homogenization method, including the Mori–Tanaka method, the self-consistent method, the generalized self-consistent method, and the differential method. To predict the elastic properties, the cementitious materials are usually divided into 3 to 5 scales, and the volume fraction of different phases at different scales can be determined from both the experimental and the theoretical methods. Since the interactions between different phases are not the same among the different homogenization methods, the predicted results may vary accordingly. The imperfect interface is an important factor that affects the accuracy of the multiscale prediction of the elastic property of cementitious materials. However, limited research has been conducted to investigate the actual mechanical properties and thickness of the imperfect interface between different phases in cementitious materials, which deserve systematic research in the future. Ketwords Imperfect interface · Micromechanics-based homogenization · Multiscale model · Scale division of cementitious materials · Volume fraction

9.1 Introduction Currently, there exist three methods to determine the elastic modulus of cementitious materials, i.e., the experimental method, the analytical method, and the numerical method. The experimental test is thought to be the most reliable method to evaluate the elastic modulus of cementitious materials, however, it is time-consuming and requires intensive labor compared with the other two methods. The analytical method usually obtains the elastic modulus of cementitious materials by some types of expression. Micromechanics-based homogenization is now the most widely used and the most popular analytical method to assess the elastic modulus of cementitious materials. Other analytical methods include the empirical equation fitted by experimental results, e.g., the relation between the elastic modulus and the compressive strength (CSA Standard A23.3–04, 2004). However, it should be kept in mind that analytical methods may not give an exact evaluation of the elastic modulus of all © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Wei et al., Mechanical Properties of Cementitious Materials at Microscale, https://doi.org/10.1007/978-981-19-6883-9_9

355

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9 Multiscale Prediction of Elastic Modulus of Cementitious …

types of cementitious materials (e.g., the empirical equation may not work well for the cementitious materials that are quite different from those used to obtain the empirical equation), which sometimes may lead to a non-negligible error. The numerical method to obtain the elastic modulus of cementitious materials is usually based on the finite element method (FEM) (Bary et al., 2009; Huang et al., 2013; Stefan et al., 2010; Zhang et al., 2014). By properly modeling the microstructure of cementitious materials, one can easily obtain the elastic modulus based on FEM results. The FEM can allow a precise evaluation of the influence of microstructures on the elastic modulus, however, accurate modeling of the complex microstructures of cementitious materials is a tough task with high computational cost. Though the prediction of the mechanical properties of cementitious materials is challenging by the analytical and the numerical methods due to the unrevealed microstructure of such materials, the related attempts to investigate the mechanical properties of cementitious materials have never ceased (Bernard et al., 2003; Nadeau, 2003; Sanahuja et al., 2007; Stefan et al., 2010; Huang et al., 2013; Hu & Li, 2014). This chapter is dedicated to summarizing the current research efforts concerning the multiscale prediction of elastic modulus of cementitious materials by micromechanics-based homogenization. First, the existing multiscale models to divide cementitious materials into several scales are reviewed. Then, the methods to calculate the volume fraction of individual phase at different scales are introduced, followed by the summary of the commonly-used homogenization methods based on the micromechanics. Finally, the predicted elastic properties of cementitious materials by micromechanics-based homogenization methods are discussed. This chapter is expected to provide an in-depth understanding of multiscale prediction of elastic modulus of cementitious materials by micromechanics-based homogenization.

9.2 Multiscale Model of Cementitious Materials To upscale the mechanical properties of cementitious materials based on micromechanics, it is necessary to define the multiscale model of cementitious materials, which should account for several key features including the material organogram, and the number and type of material constituents (Königsberger et al., 2021). Recently, many multiscale models have been proposed to upscale the elastic modulus (Bernard et al., 2003; Honorio et al., 2016; Sanahuja et al., 2007), the creep (Pichler et al., 2008), and the compressive strength (Königsberger et al., 2018) of cementitious materials. However, the multiscale models proposed by different researchers vary with the upscaled properties. To upscale the elastic modulus of cementitious materials from CSH scale to concrete scale, Bernard et al. (2003) divided the cementitious materials into four scales, i.e., CSH matrix, cement paste, mortar, and concrete, which is shown in Fig. 9.1. CSH matrix is composed of two types of CSH, i.e., LD CSH and HD CSH. Cement paste consists of the homogenous CSH, large CH crystals, aluminates, clinker, and capillary pore filled with water. Cement paste and sand as well as

9.2 Multiscale Model of Cementitious Materials

357

Level Ⅳ 10-2~10-1m

Concrete as a composite material: aggregates with ITZ embedded in homogeneous mortar matrix.

Level Ⅲ -3 -2 10 ~10 m

Mortar: sand particles embedded in a homogeneous cement paste matrix. ITZ must be considered as a separate phase.

Level Ⅱ -6 -4 10 ~10 m

Cement paste: homogeneous CSH with large CH crystals, aluminates, cement clinker, water. Some capillary porosity might be present depending on w/c ratio, and percolation threshold is defined at this scale.

Level Ⅰ -8 -6 10 ~10 m

Two types of CSH with different elastic properties. Exact morphology and volume fractions may vary depending on w/c ratio.

Fig. 9.1 Multiscale model of cementitious materials to upscale elastic modulus, taken from Bernard et al. (2003)

the ITZ between these two phases form the mortar. At the highest scale, concrete is composed of mortar, coarse aggregate, and the ITZ around it. In the multiscale model shown in Fig. 9.1, all the inclusion phases are assumed to be spherical. It is noted that although the ITZ is characterized in the multiscale model shown in Fig. 9.1, it was not considered during the multiscale prediction due to a critical lack of data concerning the development of mechanical properties of ITZ. To reflect the microstructures of cementitious materials more precisely, Sanahuja et al. (2007) divided cementitious materials into 5 scales, i.e., HD/LD CSH gel scale, inner/outer product scale, cement paste scale, mortar scale, and concrete scale, which are shown in Fig. 9.2. At the lowest scale, both HD CSH gel and LD CSH gel consist of a polycrystalline arrangement of the spherical pores and the oblate solid CSH bricks. It should be noted that the aspect ratio of the solid CSH bricks in LD CSH gel is 10–4 while it is 0.12 for HD CSH gel (resulting from fitting the elastic modulus of cementitious materials). The microstructures of the inner/outer products consist of a matrix made of the HD CSH/LD CSH gel, the oblate portlandite inclusions with an aspect ratio amounting to 0.1, and the spherical inclusions representing other hydrates including AFm, AFt, calcium aluminate, hydrogarnet, etc. At the cement paste scale, the outer products serve as the matrix material, while the unhydrated clinkers serve as inclusions with a spherical shape, and the unhydrated clinker will be coated by the inner products. At the mortar scale, the cement paste serves as the matrix material and the sand serves as the inclusion phase with a spherical shape. At the highest scale, the concrete is composed of the mortar (matrix) and the spherical coarse aggregate (inclusion).

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9 Multiscale Prediction of Elastic Modulus of Cementitious …

Fig. 9.2 Multiscale model of cementitious materials to upscale elastic modulus (Sanahuja et al., 2007), taken from Königsberger et al. (2021)

Recently, Honorio et al. (2016) divided cementitious materials into four scales to upscale the elastic modulus, i.e., the hydrate foam scale, the cement paste scale, the mortar scale, and the concrete scale, which are shown in Fig. 9.3. The hydrate foam consists of a hydrate matrix and the spherical inclusions representing capillary pores. The cement paste is made of a hydrate-foam matrix and the spherical inclusions representing clinker grains. The mortar is composed of the cement paste (matrix phase) and the sand (spherical inclusions). Then the mortar (matrix phase) and the coarse aggregate (spherical inclusion) form the concrete. This model, where all the inclusions are assumed to be spherical has also been employed to upscale the thermal (Honorio et al., 2018) and the electromagnetic (Honorio et al., 2020) properties of cementitious materials, which can give a satisfactory prediction of the thermal and the electromagnetic properties. To upscale the elastic modulus of cementitious materials by considering the imperfect interface between the inclusion and the matrix, for simplicity, the authors of this

Fig. 9.3 Multiscale model of cementitious materials to upscale elastic modulus (Honorio et al., 2016), taken from Königsberger et al. (2021)

9.2 Multiscale Model of Cementitious Materials

359

book divided cementitious materials into four scales (Liang et al., 2017), which are similar to the multiscale model developed by Bernard et al. (2003). As shown in Fig. 9.4, cementitious materials can be divided into CSH matrix scale, cement paste scale, mortar scale, and concrete scale. Since each scale is separated from the next one by at least one order of length magnitude, it is possible to predict the effective elastic properties of cementitious materials (Zaoui, 2002). All inclusion phases are assumed to be spherical. At the lowest scale, the CSH matrix comprises LD CSH (matrix scale) and HD CSH (inclusion phase). The homogenized CSH, together with the portlandite crystals (CH), the unhydrated clinker, and the capillary pore forms the cement paste. At this scale, the homogenized CSH serves as the matrix material, while the other phases serve as inclusions. The mortar at Level III is a three-phase composite material composed of the cement paste, the fine aggregate, and the interfacial transition zone (ITZ) surrounding the fine aggregate. The cement paste is considered as the matrix material and the sand serves as the inclusion. At the highest scale, the concrete is also a three-phase composite material and is composed of mortar, coarse aggregate, and the ITZ surrounding the coarse aggregate. The mortar is considered as the matrix material and the coarse aggregate serves as the inclusion. To upscale the creep property, Pichler et al. (2008) divided the cementitious materials into four scales, i.e., the CSH/anhydrous-cement scale (Level I), the cement paste scale (Level II), the mortar scale (Level III), and the macroscale (Level VI), which is shown in Fig. 9.5. There are two material constituents at CSH/anhydrous-cement scale, i.e., the anhydrous cement and the porous CSH. The anhydrous cement is made of four main clinkers, including C2 S, C3 S, C3 A, and C4 AF. The porous CSH consists of pore, water, and solid CSH that is composed of LD CSH and HD CSH. At Level II of this multiscale model, cement paste consists of porous CSH serving Level II: cement paste scale (10-6~10-4m) capillary pore CH CSH matrix

Level IV: concrete scale (10-2~10-1m) ITZ coarse aggregate mortar

clinker ITZ

HD CSH

LD CSH Level I: CSH matrix scale (10-8~10-6m)

cement paste sand

Level III: mortar scale (10-3~10-2m)

Macroscale (continuum)

Fig. 9.4 Multiscale model of cementitious materials to upscale elastic modulus, taken from Liang et al. (2017)

360

9 Multiscale Prediction of Elastic Modulus of Cementitious … CSH Ⅰb-1.

Ⅰ. CSH/anhydrous-cement scale 10-8 ~ 10-6m

LD CSH

HD CSH

Ⅰa. anhydrous cement

Ⅰb-2. C3S

C4AF

C2S C3A

air

porous CSH

water

Ⅱ. cement paste scale -6 -4 10 ~ 10 m gypsum CSH2 portlandite CH monosulfate C4ASH12 ettringite C6AS3H32 C3(A,F)H6

Ⅲ. mortar scale -2 10 m aggregate

Ⅳ. macroscale -1 10 m Fig. 9.5 Multiscale model of cementitious materials to upscale creep, taken from Pichler et al. (2008)

as the matrix phase and the inclusion phases such as spherical gypsum, monosulfate, portlandite, and entringite. Then the homogeneous cement paste (matrix) and the spherical aggregate (inclusion) form the mortar. At the macroscale, cementitious materials will be assumed to be homogeneous. Königsberger et al. (2018) divided cementitious materials into three scales to upscale the compressive strength and the elastic modulus, i.e., hydrate foam scale, cement paste scale, and mortar/concrete scale, which are shown in Fig. 9.6. This model has also been used to upscaled the creep property of cementitious materials (Königsberger et al., 2021). At the lowest scale, the microstructures of hydrate foam consist of a polycrystalline arrangement of spherical capillary porosity and hydrate-gel needles, which are isotropically oriented in space. The aspect ratio of the hydrate-gel needles is set equal to infinity(∞). The cement paste consists of hydrate foam (matrix) and spherical unhydrated clinker (inclusion). At the highest scale, the concrete is composed of cement paste (matrix) and spherical sand/coarse aggregate (inclusion). In summary, due to the complexity of the microstructure of cementitious materials, there is no consensus on the scale division of cementitious materials. It can be seen from Figs. 9.1, 9.2, 9.3, 9.4, 9.5 and 9.6 that the number of the divided scales and the composition phases in each scale depend on the upscaled property of cementitious materials. Usually, cementitious materials can be divided into three to five scales. Although the types and the geometry of the constituents at each scale may vary for

9.3 Determination of Volume Fraction of Individual Phase …

361

Fig. 9.6 Multiscale model of cementitious materials to upscale compressive strength and elastic modulus, taken from Königsberger et al. (2021)

different upscaling problems, it has been reported that the above multiscale models can achieve a good match between the predicted results and the measured results (Bernard et al., 2003; Honorio et al., 2016; Königsberger et al., 2018; Liang et al., 2017; Pichler et al., 2008; Sanahuja et al., 2007).

9.3 Determination of Volume Fraction of Individual Phase at Different Levels The volume fractions of different phases in cementitious materials are very important input parameters for the multiscale modeling. The hydration of cement is a very complex chemical process that highly depends on the reaction condition (e.g., temperature, relative humidity, age, and raw materials). To date, varieties of experimental methods have been adopted to quantify the microstructures and the volume fractions of different phases in cementitious materials (Kringos et al., 2013). For example, the volume fraction of pore may be evaluated by SEM, N2 -BET, MIP, NMR, X-ray CT, etc. The applications of different methods to quantify the porosity from nanometer scale to millimeter scale are summarized in Fig. 9.7. However, due to the multiple phases at each scale, the determination of the volume fraction of each phase by the experimental methods would be labor-consuming and inefficient. In addition, none of them alone can determine the volume fractions of phases at all scales due to the very complex microstructures of cementitious materials. X-ray CT has been reported to be very useful to determine the microstructures of cementitious materials. As shown in Fig. 9.8, x-ray CT is classified into the macro-CT, the micro-CT, and the nano-CT, which can be used to quantify not only the volume fraction of phases (e.g., capillary pore, air void, unhydrated clinker, sand, and coarse aggregate, as shown in Fig. 9.9) but also the spatial distribution of the corresponding phases at different scales (Chung et al., 2019). For example, to evaluate the effective modulus of cementitious materials by homogenization method, Huang et al. (2013) employed the micro-CT to determine the volume fraction of the unhydrated clinker in cement paste; Park et al. (2020) employed the micro-CT to evaluate the geometry and the orientation distribution function of the pores in air-entrained cement pastes.

362

9 Multiscale Prediction of Elastic Modulus of Cementitious …

Method

1 nm

10 nm

Size range 10 μm 100 nm 1 μm

100 μm

1 mm

Mercury intrusion porosimetry Gas adsorption Water absorption Helium pycnometry Thermoporometry Nuclear magnetic resonance Small-angle scattering Optical microscopy Electron microscopy

Fig. 9.7 Summary of different methods to quantify the pore phase in cementitious materials

Concrete

Resolution Scale

Foamed concrete

Cement paste

Cement paste

Plessis et al. (2016) Chung et al. (2018) Kim et al. (2019) Bossa et al. (2014) 0.65 μm 63.5 nm 0.1 mm 16 μm Macro-scale CT

Micro-scale CT

Nano-scale CT

Fig. 9.8 Measuring scales of different X-ray CT scanners (Chung et al., 2019)

Numerical model is another popular method to determine the volume fraction of individual phase in cementitious materials. Many models such as the HYMOSTRUC model (Van Breugel, 1991), the CEMHYD3D model (Bentz, 1997), the μic model (Navi & Pignat, 1996), and the DUCOM model (Maekawa et al., 1999) as shown in Fig. 9.10, are available to simulate the hydration process and the microstructures of cementitious materials. The HYMOSTRUC model can simulate the development of properties of Portland cement, e.g. the development of hydration and microstructures, the volume changes of the hardening cement pastes, and the effect of geometrical changes of the microstructure on the creep of the hardening concrete. The CEMHYD3D model that uses the discretization approach is used to simulate the development of properties of cement-based materials, e.g., the microstructure development of hardening cement paste, the adiabatic temperature rise of the Portland cement and the silica fume blended concrete, and the diffusivity of chloride ions in the silica fume blended cement paste. The μic model can provide a simulation platform on which different theories concerning cement hydration could be explicitly

9.3 Determination of Volume Fraction of Individual Phase … Pore-related Different pore size

without AE

363 Multi-phase microstructure (paste scale)

with AE

Pores Hydration products Unhydrated cement grains (Zhang and Jivkov 2014)

Pores

(Kim et al. 2012) (Chung et al. 2016a)

Pores in insulating cement

Pores in light-weight aggregate (Huang et al. 2013)

(Chung et al. 2016b)

(Lu et al. 2017)

Hydration products

Unhydrated cement grains

Pores

Multi-phase microstructure (concrete scale) void coarse aggregate mortar crack

concrete specimen before fatigue test

concrete specimen after fatigue test

Fig. 9.9 Reconstructed 3D structures of cement-based materials (Chung et al., 2019; Skarzynski et al., 2019)

modeled and studied. Besides, the μic model can provide an effective means to reconstruct numerical microstructures resulting from the complex processes that occur at the level of individual particles. Therefore, the microstructures can be analyzed for the calculation of different properties, such as the mechanical and the transport properties of cement. The DUCOM model can evaluate both the early-age properties of hardening concrete (such as the cement hydration heat and thermal conduction, pore structure formation, and moisture equilibrium) and the durability of concrete (e.g., chloride ion transport, carbonation, and corrosion of steel re-bar and calcium ion leaching). In the latest updated DUCOM model, the effect of the limestone powder and the reaction of silica fume are incorporated. Although numerical models could also obtain the volume fractions of different phases at all scales, they might deviate from the actual ones due to the complicated hydration reaction of cementitious materials. Currently, there exists no universal method to determine the volume fraction of individual phase at different levels of cementitious materials. A combination of experimental, numerical, and theoretical methods may be expected to give an accurate assessment of the volume fraction of individual phase at different scales.

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9 Multiscale Prediction of Elastic Modulus of Cementitious …

(a) HYMOSTRUC model (by TuDelft)

(b) CEMHYD3D (by NIST)

(c) μic model (by EPFL)

(d) DUCOM model (by U-Tokyo)

Fig. 9.10 Hydration models to simulate the hydration process and microstructure of cementitious materials

9.3.1 Volume Fraction at CSH Scale At the CSH matrix scale, it is of importance to determine the volume fractions of the two types of CSH, i.e., the LD CSH and the HD CSH. Currently, both the theoretical method and the experimental method can be used to assess the volume fractions of LD CSH and HD CSH. Bernard et al. (2003) employed the hydration kinetics model to achieve this goal. It is assumed that the two types of CSH are the reaction products of the hydration of C3 S and C2 S. LD CSH and HD CSH correspond to the outer products and the inner products, respectively. Thus, LD CSH is formed during the nucleation and the growth process, and HD CSH is formed during the diffusion-controlled hydration reaction. The volume fraction of the LD CSH (V LD CSH (t)) and the HD CSH (V HD CSH (t)) at time t can be expressed as Eq. (9.1) and Eq. (9.2), respectively. ( < > ) C3 S ξC∗3 S − ξC∗3 S − ξ C3 S (t) + VLD CSH (t) = VCSH ( < > ) C2 S + VCSH ξC∗2 S − ξC∗2 S − ξ C2 S (t) +

(9.1)

> > C3 S < C2 S < ξ C3 S (t) − ξC∗3 S + + VCSH ξ C2 S (t) − ξC∗2 S + VHD CSH (t) = VCSH

(9.2)

where, ξC∗3 S and ξC∗2 S are the critical hydration degree, which corresponds to a critical thickness of the hydration products formed around the C3 S and C2 S clinker grains, and the value of ξC∗3 S and ξC∗2 S can be set as 0.6 (Springenschidt, 1994); ξ C3 S (t) and

9.3 Determination of Volume Fraction of Individual Phase …

365

C3 S ξ C2 S (t) are the hydration degree of C3 S and C2 S clinker grains at time t; VCSH and C2 S VCSH are the asymptotic values of the volume occupied by the reaction products / C3 S = Vc0 of C3 S and C2 S in the CSH (V CSH ), respectively, VCSH ρ m

/

M

ρc m C3 S MC3 S , ρCSH / MCSH

C2 S VCSH =

C2 S C2 S Vc0 cρCSH , Vc0 is the initial cement volume, ρc and ρCSH are the mass densities / MCSH of the cement and the CSH, respectively; m C3 S and m C2 S are the mass fractions of C3 S and C2 S in cement, respectively; MC3 S , MC2 S , and MCSH are the molar mass of C3 S, C2 S, and CSH, respectively; MC3 S = 228.32 g/mol, MC2 S = 172.24 g/mol, and MCSH = 227.2 g/mol, respectively (Bernard et al., 2003). Then the relative volume fractions of LD CSH (f LD CSH (t)) and HD CSH (f HD CSH (t)) can be calculated according to Eqs. (9.3) and (9.4).

f LD CSH (t) =

VLD CSH (t) VLD CSH (t) + VHD SH (t)

(9.3)

f HD CSH (t) =

VHD CSH (t) VLD CSH (t) + VHD CSH (t)

(9.4)

Another theoretical model to quantitatively calculate the volume fractions of LD CSH and HD CSH is the Jennings-Tennis model (Tennis & Jennings, 2000), which has also been employed in the multiscale model by Sanahuja et al. (2007), Huang et al. (2013), and Liang et al. (2017). According to the Jennings-Tennis model, the ratio of the mass of LD CSH to the total mass of the two types of CSH is given by: m LD CSH (t) = 3.017α H (t)(w/c) − 1.347α H (t) + 0.538

(9.5)

where, mLD CSH is the mass fraction of LD CSH in the total CSH; α H (t) is the degree of hydration; w/c is the water to cement ratio of the cement paste. By using the mass densities of LD CSH and HD CSH, the volume fractions of LD CSH and HD CSH can be calculated as: . CSH (t)ρHD CSH f LD CSH (t) = ρLD CSH +mmLDLDCSH (t)(ρHD CSH −ρLD CSH ) (9.6) f HD CSH (t) = 1 − f LD CSH (t) where, ρLD CSH and ρHD CSH are the mass densities of LD CSH and HD CSH, respectively. Recently, NI has also been a feasible method to quantify the volume fractions of LD CSH and HD CSH experimentally. By deconvolution of the measured mechanical properties (e.g., elastic modulus and contact hardness) of a huge number of indents made on cementitious materials, one can obtain not only the mechanical property of each phase, but also the area fraction of each phase in cementitious materials, which can be considered the same as the volume fraction of each phase. Figure 9.11 shows the elastic modulus frequency histogram for CSH based on the NI measurement conducted by Constantinides and Ulm (2004). It can be seen that the volume fractions

366

9 Multiscale Prediction of Elastic Modulus of Cementitious …

Fig. 9.11 Elastic modulus frequency histogram for CSH by NI (Constantinides & Ulm, 2004)

of LD CSH and HD CSH are about 70% and 30%, respectively, which can match well with the calculated results by the hydration kinetics model (Bernard et al., 2003).

9.3.2 Volume Fraction at Cement Paste Scale In neat Portland cement pastes, the stoichiometric reactions for the four main clinkers of ordinary Portland cement are normally used to determine the volume fractions of different phases at cement paste scale (Tennis & Jennings, 2000): ⎧ C3 S + 5.3H → 0.5C3.4 S2 H8 + 1.3C H ⎪ ⎪ ⎪ ⎪ ⎪ C2 S + 4.3H → 0.5C3.4 S2 H8 + 0.3C H ⎪ ⎪ ⎪ ⎪ ⎨ C4 AF + 2C H + 10H → 2C3 ( A, F)H6 ⎪ C3 A + 3C S H2 + 26H → C6 AS 3 H32 ⎪ ⎪ ⎪ ⎪ ⎪ C3 A + 0.5C6 AS 3 H32 + 2H → 1.5C4 AS H12 ⎪ ⎪ ⎪ ⎩ C3 A + C H + 12H → C4 AH13

(9.7)

The average degree of hydration (ξ ) of cement paste can be determined based on the degree of hydration of the four clinkers, which can be expressed as: Σ

m x ξx ξ= Σ mx x

(9.8)

x

where, x ∈ {C 3 S, C 2 S, C 3 A, C 4 AF}; mx is the mass fraction of the x phase. Based on Eq. (9.7), one can obtain the volume fraction of each phase at cement paste scale. For example, the volume fraction of CSH matrix (i.e., C 3.4 S 2 H 8 ) can be determined by the hydration of C 2 S and C 3 S: ] [ m C S 0.5MC3.4 S2 H8 m C S 0.5MC3.4 S2 H8 + ξC2 S (t) 2 f C3.4 S2 H8 (t) = ξC3 S (t) 3 ρ (9.9) MC3 S ρC3.4 S2 H8 MC2 S ρC3.4 S2 H8

9.3 Determination of Volume Fraction of Individual Phase …

367

where, MC3 S , MC2 S , and MC3.4 S2 H8 are the molar masses of C 3 S, C 2 S and C 3.4 S 2 H 8 , respectively; ρC3.4 S2 H8 is the density of C 3.4 S 2 H 8 ; ρ is the average density of cement paste, which can be expressed as: ρ = Σ mx x

ρx

1 +

w/ c ρH

Σ

mx

(9.10)

x

where, ρx and ρ H are the density of the x phase and water, respectively; w/c is the water-to-cement ratio. Recently, many researchers have employed the above stoichiometric reactions for cement clinkers to determine the volume fraction information at cement paste scale during the property upscaling (Bernard et al., 2003; Honorio et al., 2020; Pichler et al., 2008). Figure 9.12a and b display the evolution of the volume fraction of each phase at cement paste scale determined according to the stoichiometric reactions for cement clinkers. Based on Fig. 9.12, one can obtain the volume fraction of each phase at cement paste scale at a specific degree of hydration, i.e., a specific curing age. Another popular theoretical model to obtain the volume fraction information at cement paste scale is the classical Powers hydration model (Powers, 1962). Due to its reliability and its ease of implementation, the Powers hydration model has also been widely used to assess the volume fraction information (Honorio et al., 2020; Huang et al., 2013; Igarashi et al., 2004). According to the Powers model, the volume fractions of the capillary pore, the hydration products, and the unhydrated clinker satisfy the following equation: fp =

w/c − 0.36α H w/c + 0.32

Fig. 9.12 Variation of volume fractions at cement paste scale as a function of the overall degree of hydration. a cement paste with w/c ratio of 0.5 (Bernard et al., 2003). Note CSHa and CSHb denote the LD CSH and HD CSH, respectively); b cement paste with w/c ratio of 0.48 (Pichler et al., 2008)

368

9 Multiscale Prediction of Elastic Modulus of Cementitious …

w/c=0.3

w/c=0.4

w/c=0.5

Fig. 9.13 Volume fraction of different phases obtained from the Powers models as a function of the degree of hydration for cement pastes with various w/c ratios of 0.3, 0.4, and 0.5. (Honorio et al., 2020)

0.32(1 − α H ) w/c + 0.32 f p + fu + fh = 1

fu =

(9.11)

where, f p , f h , and f u are the volume fraction of the capillary pore, the hydration products, and the unhydrated clinker, respectively; α H is the degree of hydration. Figure 9.13 shows the volume fractions of different phases obtained with the Powers models as a function of the degree of hydration for cement pastes with various w/c ratios. As can be seen in Fig. 9.13, the volume fractions of both pores (chemical shrinkage + water) and the unhydrated clinker decrease with increasing degree of hydration, and the volume fraction of the hydration products increases with increasing degree of hydration. However, it is noted that the Powers hydration model cannot further divide the hydration products into CSH matrix, CH, and calcium aluminate hydrates, etc. Therefore, to reflect the microstructures at cement paste scale more accurately, one should combine the Powers hydration model and other methods. For example, Huang et al. (2013) assumed that the hydration products are composed of CSH matrix and CH and employed a simplified theoretical model to determine the volume fractions of CSH and CH. By assuming that cement clinkers only consist of C 3 S and C 2 S and these two clinkers share the same degree of hydration (α H ), the volumes of CH and the CSH matrix can be calculated as: ⎧ NCH MCH ⎪ ⎪ ⎨ VCH = ρCH (9.12) M N ⎪ ⎪ ⎩ VCSH = CSH CSH ρCSH where, ρCH and ρCSH are the mass densities of CH and the CSH matrix, respectively, and ρCSH = f LD ρLD + f HD ρHD ; MCH and MCSH are the molar masses of CH and the CSH matrix, respectively; NCH and NCSH are the numbers of moles of CH and the CSH matrix from one unit volume of cement, respectively, which can be expressed

9.3 Determination of Volume Fraction of Individual Phase …

369

as: .

NCH = α H (1.3NC3 S + 0.3NC2 S ) NCSH = α H (0.5NC3 S + 0.5NC2 S )

(9.13)

where, NC3 S and NC2 S are the numbers of moles of C 3 S and C 2 S per unit volume of cement, respectively, which can be expressed as: .

NC 3 S = NC 2 S =

f C3 S ρC3 S MC 3 S f C2 S ρC2 S MC 2 S

(9.14)

where, ρC3 S and ρC2 S are the mass densities of C 3 S and C 2 S, respectively; MC3 S and MC2 S are the molar masses of C 3 S and C 2 S, respectively; f C3 S and f C2 S are the volume fractions of C 3 S and C 2 S, respectively. Then the volume fractions of CH and the CSH matrix ( f CH and f CSH ) can be calculated as: . ( ) f CH = VCSHVCH 1 − fu − f p +VCH ( ) (9.15) VCSH f CSH = VCSH 1 − fu − f p +VCH The experimental method can also contribute to determining the volume fractions of CH from that of hydration products. For example, Liang and Wei (2020) used Powers hydration model to obtain the volume fractions of the capillary pore, the unhydrated clinker, and the hydration products. Then the volume fraction of CH in the hydration products was determined by thermogravimetric analysis.

9.3.3 Volume Fraction at Mortar Scale At mortar scale, cementitious materials usually can be divided into cement paste, sand, and the interfacial transition zone (ITZ) around the sand. Commonly, the volume fraction of individual phase at mortar scale can be calculated based on the mix-design parameters. The volume fraction of sand in mortar ( f s ) can be calculated based on the mix-design parameters, which is expressed as (Bernard et al., 2003): fs =

m s /ρs Vs = Vw0 + Vc0 + Vs m w /ρw + m c /ρc + m s /ρs

(9.16)

where, V s is the volume of sand in mortar; V w0 and V c0 are the initial volume of water and cement in mortar, respectively; mw , mc , and ms are the initial mass contents of water, cement, and sand in the mortar, respectively; ρ w , ρ c , and ρ s are the mass densities of water, cement, and sand, respectively.

370

9 Multiscale Prediction of Elastic Modulus of Cementitious …

ITZ is an important factor that affects the mechanical properties of cementitious materials. The previous studies reveal that the ITZ properties are influenced by various factors which include w/c ratio, curing age and condition, particle size of clinker, supplementary cementitious materials, and addition of nanomaterials (Elsharief et al., 2003; Gao et al., 2014; Leemann et al., 2006; Rangaraju et al., 2010; Wu et al., 2015; Xu et al., 2017). SEM/BSE is a feasible method to quantify the thickness of ITZ in cementitious materials. Scrivener and Laugesen (2004) concluded that the thickness of ITZ between the fine aggregate and the cement paste lies between 10 and 20 μm based on the SEM/BSE results. However, values larger than 20 μm were also reported by researchers (Gao et al., 2013a, 2013b). Once the ITZ thickness (t ITZ-s ) is obtained, the volume fraction of ITZ around the sand with the radius of r s (f ITZ-s ) can be approximately calculated by: f ITZ−s = f s

(rs + tITZ−s )3 − rs3 rs3

(9.17)

Then the volume fraction of cement paste (f paste ) in mortar can be expressed by: f paste = 1 − f ITZ−s − f s

(9.18)

9.3.4 Volume Fraction at Concrete Scale Analogously, the volume fraction of coarse aggregate in concrete ( f ca ) can be calculated based on the mix-design parameters: f ca =

m ca /ρca Vca = Vw0 + Vc0 + Vs + Vca m w /ρw + m c /ρc + m s /ρs + m ca /ρca

(9.19)

where, V ca , mca , and ρ ca are the volume fraction, the mass content, and the mass density of coarse aggregate in concrete, respectively. The calculation of the volume fraction of ITZ around the coarse aggregate (f ITZ-ca ) and the mortar (f mortar ) is similar to Eqs. (9.17) and (9.18), which can be expressed as: f ITZ−ca = f ca

3 (rca + tITZ−ca )3 − rca 3 rca

(9.20)

where, r ca is the radius of coarse aggregate; t ITZ-ca is the thickness of ITZ around the coarse aggregate with radius of r ca . f mortar = 1 − f ITZ - ca − f ca

(9.21)

9.4 Perfect Versus Imperfect Interface Conditions

371

9.4 Perfect Versus Imperfect Interface Conditions The properties of the interface are usually weaker than those of the two adjacent phases, which have a great impact on the overall mechanical properties of composite materials (Achenbach & Zhu, 1989; Gao, 1995; Yanase & Ju, 2012). Currently, many theoretical models have been established to simulate the interface properties, which can be classified into the interphase model and the spring-layer interface model, as shown in Fig. 9.14. In the interphase model, the interface is modeled by a thin layer of a third material with specified thickness embedded between the matrix phase and the inclusion. While in the spring-layer interface model, the thickness of the interface is not considered, the mechanical properties of the interface are simulated by layers of normal and tangential spring elements. In general, the interphase model can reflect the real situation more accurately, but it may bring a huge computational cost when combined with the finite element model. The spring-layer interface model can ease the calculation, which is easy to be implemented in the analytical and numerical simulation. However, it is quite difficult to determine the properties of spring elements directly. Recently, some attempts have been performed to consider the imperfect interface between the aggregate and the paste in both numerical models and analytical models. For example, Zhang et al. (2014) have employed the finite element method to consider the interface between aggregate and cement paste and the interface between fiber and cement paste. As shown in Fig. 9.15, the interfaces are modeled by shell elements. However, it should be noted that the interface layer is usually much thinner than the other phases, the meshing of the interface will result in large computing costs. Grondin and Matallah (2014) have proposed a methodology to consider the effect of the ITZs on the mechanical properties of concrete at the mesoscopic scale. An effective mixed interphase (EMI) around each aggregate was defined to reduce the computational cost (Fig. 9.16). As can be seen in Fig. 9.16, the EMI with a thickness of δ int is formed with ITZ and a part of the bulk cement paste, which together with the aggregate (diameter = ∅agg ) forms the inclusion phase (diameter = ∅agg + 2δ int )

(a) Interphase model

(b) Spring layer interface model

Fig. 9.14 Models describing the interface between different phases in cementitious materials (Zhu et al., , 2018)

372

9 Multiscale Prediction of Elastic Modulus of Cementitious …

in concrete. The rest of the bulk cement paste will serve as the matrix phase of concrete. The properties of EMI are homogenized by both the ITZs and the cement paste properties according to the volume fractions of ITZs and the cement paste in EMI. Since the thickness of the EMI is increased, the minimum element size of the FE model is increased, which can reduce the total number of elements and the computing cost.

(a) Aggregate

Aggregate element

Cement

ITZ element

(b) Fiber

Fiber element

Concrete ITZ element Fig. 9.15 Finite element discretization of composite ellipsoids surrounded by ITZ shells at a microscale and b mesoscale (Zhang et al., 2014)

Φagg

Range of EMI δint Paste in range of EMI

Φagg

δint

Homogenization of EMI

Aggregate ITZ Model of concrete at mesoscale

Bulk paste

Model of selected ITZ and paste in the range of EMI

EMI

Bulk paste Homogenization for forming the EMI

Fig. 9.16 Finite element model of the representative volume element (RVE) of concrete considering interphases between aggregates and matrix (Grondin & Matallah, 2014)

9.4 Perfect Versus Imperfect Interface Conditions

g d ef

d g f

373

Elastic properties

e r a

a

c

c b

b Fig. 9.17 Depiction of generalized self-consistent model for interface (Nadeau, 2003)

The analytical method is another popular method to consider the influence of the interface property on cementitious materials. For example, Nadeau (2003) developed an analytical model (self-consistent model) to predict the elastic modulus of concrete by considering the interfacial transition zone between cement paste and aggregate, which is shown in Fig. 9.17. In this model, the local elastic property of ITZ varies along the radial direction. a denotes the radius of the inclusion particle, b is the radius of the concentric sphere, which is determined to maintain the relative proportions of all constituents, and c is the thickness of the region within the cement paste with gradient material properties. d is the elastic properties of the inclusion, e is the elastic properties through the ITZ, f is the effective elastic properties from the previous (smaller) scale, and g is the effective material properties at the scale under consideration. At a given scale, quantities of a − f are the inputs to the model, and g is the quantity to be computed. Zheng et al. (2012) also employed a similar model, i.e., a n-layered spherical inclusion model, to consider the inhomogeneous property of ITZ, and the validity of the model was verified with three independent sets of experimental data. To ease the computation of the multiscale model considering the interface property between different phases, Qu (1993) developed a spring-layers analytical model with vanishing thickness, which has been widely used to simulate the interface property in fiber-reinforced materials (Qu, 1993), asphalt concrete (Gao et al., 2015), and cement concrete (Liang et al., 2017). In this model, it is assumed that the interfacial traction is still continuous, but displacement discontinuity may occur at the interface. The interface conditions are given by: .

[ ( ) ( )] .σi j n j [≡ (σi j )S + −( σi j)]S − n j = 0 .u i ≡ u i S + − u i S − = ηi j σ jk n k

(9.22)

( ) ( ) where, n j is the unit outward normal vector of the interface S, σi j S + and σi j S − are the values of σi j (x) as x approaches from outside and inside of the ( ) the interface ( ) inclusion, respectively, so are u i S + and u i S − ; x is the position vector; .σi j and .u i are the difference of the stresses and displacements as x approaches the interface

374

9 Multiscale Prediction of Elastic Modulus of Cementitious …

n

+

+

σij(S ) or ui(S ) σij(S-) or ui(S-)

matrix normal spring kN=1/ω

perfect interface kT → ∞ kN → ∞

d

solid inclusion (Level I: HD-CSH; Level II: CH, clinker; Level III: sand; Level IV: coarse aggregate)

tangential spring kT=1/π

imperfect interface kT = finite value kN = finite value

Fig. 9.18 Schematic of the tangential and normal interface conditions between the matrix and the inclusion (Liang et al., 2017)

from outside to inside of the inclusion, respectively. ηi j is a second-rank tensor representing the compliance of the interface, it is clear that ηi j = 0 corresponds to the perfect interface, while ηi j → ∞ represents the completely-debonded condition. Figure 9.18 shows the schematic of the tangential and normal interface conditions between the matrix and the inclusion. The interface is characterized by springlayers of vanishing thickness in the tangential and normal directions. The property of interface characterized by a second-order tensor ηi j can be expressed as (Qu, 1993): ηi j = π δi j + (ω − π )n i n j

(9.23)

where, δi j is the Kronecker delta; π and ω represent the compliance in the tangential and normal directions of the interface, respectively. In the previous studies (Gao et al., 2015; Liang et al., 2017; Qu, 1993), to meet the compatibility requirements, the interface condition with ω = 0 and π /= 0 is assumed. Clearly, this constitutive characterization of the interface allows relative tangential sliding between the two surfaces but no separation or interpenetration in the normal direction. Furthermore, π = 0 stands for the perfectly bonded interface and π → ∞ represents the completely debonded interface in the tangential direction. An interface with π ∈ [0, ∞] denotes an imperfect interface.

9.5 Micromechanics-Based Homogenization Method

375

9.5 Micromechanics-Based Homogenization Method In this section, a brief review of the existing homogenization methods including the Mori-Tanaka method (Mori & Tanaka, 1973), the self-consistent method, the generalized self-consistent method, and the differential method is provided. Within the framework of continuum micromechanics, the effective elastic properties of composite materials can be determined by assuming the existence of a representative volume element (RVE), which is shown in Fig. 9.19. To obtain the homogeneous properties of the composites structures based on the chosen RVE, it is required that the characteristic length of RVE (L) should be much less than that of the composite structures (H), i.e., L