NANOINDENTATION RELAXATION STUDY ANDMICROMECHANICS OF CEMENT-BASED MATERIALS

Mémoire

Nicolas Venkovic

Maîtrise en génie civilMaître ès sciences (M.Sc.)

Québec, Canada

© Nicolas Venkovic, 2016

Résumé

Ce travail évalue le comportement mécanique des matériaux cimentaires à différentes échellesde distance. Premièrement, les propriétés mécaniques du béton produit avec un bioplastifi-ant à base de microorganismes efficaces (EM) sont etudiées par nanoindentation statistique,et comparées aux propriétés mécaniques du béton produit avec un superplastifiant ordinaire(SP). Il est trouvé que l’ajout de bioplastifiant à base de produit EM améliore la résistance desC–S–H en augmentant la cohésion et la friction des nanograins solides. L’analyse statistiquedes résultats d’indentation suggère que le bioplastifiant à base de produit EM inhibe la précip-itation des C–S–H avec une plus grande fraction volumique solide. Deuxièmement, un modèlemulti-échelles à base micromécanique est dérivé pour le comportement poroélastique de la pâtede ciment au jeune age. L’approche proposée permet d’obtenir les propriétés poroélastiquesrequises pour la modélisation du comportoment mécanique partiellement saturé des pâtes deciment viellissantes. Il est montré que ce modèle prédit le seuil de percolation et le module deYoung non drainé de façon conforme aux données expérimentales. Un metamodèle stochas-tique est construit sur la base du chaos polynomial pour propager l’incertitude des paramètresdu modèle à travers plusieurs échelles de distance. Une analyse de sensibilité est conduite parpost-traitement du metamodèle pour des pâtes de ciment avec ratios d’eau sur ciment entre0.35 et 0.70. Il est trouvé que l’incertitude sous-jacente des propriétés poroélastiques équiva-lentes est principalement due à l’énergie d’activation des aluminates de calcium au jeune ageet, plus tard, au module élastique des silicates de calcium hydratés de basse densité.

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Abstract

This work assesses the mechanical behavior of cement-based materials through different lengthscales. First, the mechanical properties of concrete produced with effective microorganisms(EM)-based bioplasticizer are investigated by means of statistical nanoindentation, and com-pared to the nanomechanical properties of concrete produced with ordinary superplasticizer(SP). It is found that the addition of EM-based bioplasticizer improves the strength of C–S–Hby enhancing the cohesion and friction of solid nanograins. The statistical analysis of indenta-tion results also suggests that EM-based bioplasticizer inhibits the precipitation of C–S–H ofhigher density. Second, a multiscale micromechanics-based model is derived for the poroelasticbehavior of cement paste at early age. The proposed approach provides poroelastic propertiesrequired to model the behavior of partially saturated aging cement pastes. It is shown that themodel predicts the percolation threshold and undrained elastic modulus in good agreementwith experimental data. A stochastic metamodel is constructed using polynomial chaos ex-pansions to propagate the uncertainty of the model parameters through different length scales.A sensitivity analysis is conducted by post-treatment of the meta-model for water-to-cementratios between 0.35 and 0.70. It is found that the underlying uncertainty of the effectiveporoelastic proporties is mostly due to the apparent activation energy of calcium aluminateat early age and, later on, to the elastic modulus of low density calcium-silicate-hydrate.

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Contents

Résumé iii

Abstract v

Contents vii

List of Tables ix

List of Figures xi

Abstract xiii

1 Introduction 11.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I Nanoindentation study of calcium silicate hydrates in concrete pro-duced with effective microorganisms-based bioplasticizer 5

2 Partial introduction 9

3 Materials 113.1 Bulk preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Surface preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Methods 134.1 Nanoindentation relaxation analysis . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Packing density distribution and strength properties . . . . . . . . . . . . . . . 234.3 Energy activated relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Cluster analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Results and discussion 355.1 Nanoindentation relaxation tests . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Cluster analysis based on indentation modulus and hardness . . . . . . . . . . . 355.3 Assessment of packing distributions, strength and relaxation properties of C–S–H 42

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5.4 Cluster analysis of C–S–H phases based on indentation modulus, hardness andactivation volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Comparison with the macroscopic behavior . . . . . . . . . . . . . . . . . . . . 56

6 Partial conclusion 59

II Uncertainty propagation of a multiscale poromechanics-hydrationmodel for poroelastic properties of cement paste at early-age 61

7 Partial introduction 65

8 Materials 67

9 From the general inclusion problem of Eshelby to microporomechanics 699.1 Generalized inclusion problem of Eshelby . . . . . . . . . . . . . . . . . . . . . 699.2 Homogenization scheme of Mori and Tanaka . . . . . . . . . . . . . . . . . . . . 749.3 Self-consistent homogenization scheme . . . . . . . . . . . . . . . . . . . . . . . 809.4 Applications to microporomechanics . . . . . . . . . . . . . . . . . . . . . . . . 81

10 A multiscale poromechanics-hydration model 8710.1 Hydration model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 Multiscale poromechanics model . . . . . . . . . . . . . . . . . . . . . . . . . . 90

11 Polynomial chaos expansion and post-processing 9911.1 Polynomial chaos representation . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.2 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

12 Model input parameters 10512.1 Phase composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.2 Kinetic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.3 Elastic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.4 Microstructure parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

13 Results and discussion 10913.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.2 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11113.3 Global sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11413.4 Probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

14 Partial conclusion 121

Conclusion 123

A Matrix representation 125

Notation, conventions and identities 125A.1 Matrix representation of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Matrix representation of tensorial operations . . . . . . . . . . . . . . . . . . . 126A.3 Transformation of matrix representations . . . . . . . . . . . . . . . . . . . . . 126

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Bibliography 129

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List of Tables

3.1 Mix designs of self-compacting concrete produced either with EM-based bioplasti-cizer or superplasticizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 Parameters of the shape functions for the Berkovich and spherical indenters. . . . . 16

5.1 Summary statistics of the cluster analysis based on the indentation modulus andhardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Converged strength properties assessed by inverse analysis of the packing densitydistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Summary statistics of the cluster analysis of C–S–H phases for the sample withMapefluid N-200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4 Summary statistics of the cluster analysis of C–S–H phases for the sample withIH Plus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.5 Macroscopic compressive strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.1 Major oxides composition of cement PCCB9402 (Boumiz et al., 1996) . . . . . . . 67

12.1 Quantitative phase composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.2 Kinetic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.3 Apparent activation energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.4 Elastic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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List of Figures

1.1 Collapse of Koror-Babeldaob Bridge due to excessive delayed deformation, Republicof Palau (see Burgoyne and Scantlebury Burgoyne and Scantlebury (2006)). . . . . 2

4.1 Geometry of indentation tests with conical and spherical probes, see Vandamme(2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Penetration depth history of the indenter. . . . . . . . . . . . . . . . . . . . . . . . 184.3 Effect of viscosity on the measurement of contact stiffness. . . . . . . . . . . . . . . 214.4 Schematic representation of an indentation (adapted from Oliver and Pharr (1992)). 234.5 Energy activated process of a flow unit. . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Vacancy migration of flow units as described by the Eyring model applied to viscous

flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.7 Potential trough of a flow unit with and without shearing applied force. . . . . . . 284.8 Zhurkov dashpot and simple energy activated bodies. . . . . . . . . . . . . . . . . . 31

5.1 Non-dimensional penetration and load relaxation curves measured on the sampleproduced with Mapefluid N-200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Non-dimensional penetration and load relaxation curves measured on the sampleproduced with IH Plus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Summary of the cluster analysis based on indentation modulus and hardness. (a)Scatter-plot of indentation modulus and hardness with classification of indents and90% confidence ellipsoids. (b) BIC values for different sizes of mixture. . . . . . . . 38

5.4 Scatter-plots and classifications of (a) indentation modulus and (b) indentationhardness with non-dimensional relaxation at 600 s. . . . . . . . . . . . . . . . . . . 40

5.5 Histogram of classification uncertainty for the indents attributed to cluster 4 or 5. 415.6 Scatter-plots and scaling of packing density with the indentation modulus. . . . . . 435.7 Scatter-plots and scaling of packing density with the indentation hardness. . . . . . 435.8 Scatter-plots and scaling of packing density with the activation volume. . . . . . . 445.9 Scatter-plots and scaling of packing density with the characteristic time of relaxation. 445.10 Convergence of the cohesion of the solid phase of C–S–H obtained by inverse anal-

ysis with an increasing number of indents. . . . . . . . . . . . . . . . . . . . . . . . 465.11 Convergence of the friction coefficient of the solid phase of C–S–H obtained by

inverse analysis with an increasing number of indents. . . . . . . . . . . . . . . . . 475.12 Scatter-plots and scaling of the indentation modulus with the indentation hardness. 485.13 Scatter-plots and scaling of the indentation modulus with the activation volume. . 485.14 Scatter-plot and scaling of the indentation hardness with the activation volume. . . 495.15 Partial summary results of the cluster analysis based on indentation modulus,

hardness and activation volume. Model distributions of indentation hardness. . . . 50

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5.16 Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. Model distributions of indentation modulus. . . . 51

5.17 Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. Model distributions of activation volume. . . . . . 52

5.18 Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. BIC values for different sizes of mixtures. . . . . . 52

5.19 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationmodulus with the packing density of the sample produced with Maplefluid N-200. . 53

5.20 Scatter-plots, classifications, confidence ellipsoids and scaling of the activation vol-ume with the packing density of the sample produced with Maplefluid N-200. . . . 54

5.21 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationmodulus with the packing density of the sample produced with IH Plus. . . . . . . 54

5.22 Scatter-plots, classifications, confidence ellipsoids and scaling of the activation vol-ume with the packing density of the sample produced with IH Plus. . . . . . . . . 55

5.23 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationhardness with packing density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9.1 2D heterogeneous representation of the Eshelby equivalent eigenstrain problemwith inelastic deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9.2 2D homogeneous representation of the Eshelby equivalent eigenstrain problem withinelastic deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

10.1 Multiscale representation of the microstructure of cement paste, adapted from Con-stantinides (2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

13.1 Model predictions of the volume fractions for w/c=0.50. . . . . . . . . . . . . . . . 11013.2 Model predictions of the undrained elastic modulus and experimental data from

Boumiz et al. (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11113.3 Model predictions of the undrained Poisson’s ratio and experimental data for

w/c=0.40 from Boumiz et al. Boumiz et al. (2000). . . . . . . . . . . . . . . . . . . 11213.4 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11313.5 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11413.6 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11513.7 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11613.8 Pairwise correlations of the poroelastic properties with the percolation threshold. . 11713.9 Sensitivity analysis of the percolation threshold - First Sobol’ indices. . . . . . . . 11713.10Sensitivity analysis of the drained elastic moduls - First order Sobol’ indices. . . . 11813.11Sensitivity analysis of the Biot-Willis parameter - First order Sobol’ indices. . . . . 11913.12Smoothed probability density function of the drained elastic modulus as a function

of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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Foreword

Two articles were inserted in this document. We give here the dates at which the articleswere submitted, accepted and published. We also explain how the content of the originallypublished versions was adapted here. The role and contribution of the student is also describedfor each publication and some information is provided about the co-authors.

The first part of the document is a reproduction of an article that was submitted to Cement& Concrete Composites (Elsevier) the October 12th, 2012. The article was accepted afterrevision on December 6th, 2013 and eventually published as follows:

Venkovic N., Sorelli L. and Martirena F. (May 2014). Nanoindentation study of calcium silicatehydrates in concrete produced with effective microorganisms-based bioplasticizer. Cement &Concrete Composites, Volume 49: Pages 127–139.

The co-authors of the article are presented as follows. Luca Sorelli is associate professor atLaval Universty. Fernando-Martirena is director of the Center for Research and Developmentof Structures and Materials (CIDEM) at the University Central Marta Abreu of las Villas,Cuba. The student, first author of the article, proceeded to the surface preparation of thesamples as well as the nanoindentation experiments on each sample. The student also analyzedthe results and wrote the article. The samples were provided by Professor Martirena toProfessor Sorelli. The methodology section of the article, here in Chapter 4, was extendedto include additional information about the indentation analysis and the relaxation modelcalibrated experimentally. Also, several minor editions were made to the original text.

The second part of the document is a reproduction of an article that was submitted to Prob-abilistic Engineering Mechanics (Elsevier) the January 3rd, 2012. The article was acceptedafter revision on December 18th, 2012 and eventually published as follows:

Venkovic N., Sorelli L., Sudret B., Yalamas T. and Gagné R. (April 2013). Uncertainty prop-agation of a multiscale poromechanics-hydration model for poroelastic properties of cementpaste at early-age. Probabilistic Engineering Mechanics, Volume 32: Pages 5–20.

The other co-authors of the article are presented as follows. Bruno Sudret is professor at theChair of Risk, Safety & Uncertainty Quantification of the Swiss Federal Institute of Tech-

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nology in Zurich (ETHZ), Switzerland. Thierry Yalamas is general manager at PHIMECAEngineering, France. Richard Gagné is professor at the University of Sherbrooke and adjunctdirector of the Research Center on Concrete Infrastructures (CRIB). The student, first authorof the article, developed and implemented the poromechanical-hydration model. The studentalso performed the uncertainty and sensitivity analysis and wrote the article. The method-ology section of the article was extended through Chapter 9 where some elementary resultsof the problem of Eshelby are presented and applied to the derivation of common equationsof microporomechanics. Also, some equations were corrected and several minor editions weremade to the text.

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Chapter 1

Introduction

1.1 Problem Statement

Cement-based materials manufactured in deformable structures evolve over time andundergo delayed deformations. Even free of applied mechanical loads, concrete structuresare subjected to thermal dilations and time-dependent volumetric deformations (shrinkage orswelling) due to internal chemical processes and changes in the water content of the material’sporous network. Beyond the extent of these processes, when a load is applied on a concretestructure, the underlying material deforms instantly and continues to deform over time (orcreep) as the load is sustained. Conversely, sustained kinematic constraints lead up to stressreleases (relaxation) in the material eventually triggering stress redistributions in a structure.We refer to these phenomena as time effects.

Time effects can affect the deflection of beams over long periods of time and increase thesettlement of columns in tall buildings. When not properly taken into account, these phe-nomena can lead to structural disorders such as cracking, excessive deflection and differentialsettlement. Similarly, pre-stressed concrete structures such as beams, slabs, box girders andbridges sequentially built in stages can be widely affected by time-dependent deformations.Beyond their instantaneous mechanical response, these structures can undergo stress relax-ations inducing losses of pre-stress leading to the development of complex stress redistributionsnot considered in the original structural design. Pressure vessels and undersea shells are alsohighly sensitive to creep and shrinkage as they can trigger significant geometrical changesin the structure leading up to buckling and other instabilities. The Koror-Babeldaob (KB)Bridge used to connect two Pacific islands of the Republic of Paula is an infamous exampleof concrete structure that faced major disorders due to design mis-predictions of time effects.This bridge, built in 1977, was sequentially assembled by prestressing a box girder span of241 m (Yee, 1979). Even from the begining of the construction, the structure was subjectedto shrinkage, creep and loss of prestress leading to an increase of the midspan deflection overtime. By 1990, the deflection reached 1.2 m and the serviceability of the bridge was compro-

1

(a) Excessive deflection of KB Bridge (b) Collapse of KB Bridge

Figure 1.1: Collapse of Koror-Babeldaob Bridge due to excessive delayed deformation, Repub-lic of Palau (see Burgoyne and Scantlebury Burgoyne and Scantlebury (2006)).

mised (see Fig. 1.1a). In 1995, the deflection had increased so much that it was decided toperform remediation works in order to correct some of the sag and prevent further deflection(McDonald et al., 2004). In 1996, the bridge suddendly collapsed (see Fig. 1.1b) under neg-ligible traffic load and with no apparent external trigger (Pilz, 1997; McDonald et al., 2004;Burgoyne and Scantlebury, 2006). Even though the failure has still not been satisfactorilyexplained, it is clear that unexpected delayed deformations are at the origin of the structuraldisorders undergone by the bridge (Burgoyne and Scantlebury, 2006). The discrepancy be-tween the measurements and the delayed deflection computed for the design was pointed outby Bažant et al. (2008). Although these predictions were based on a model approved by theEuropean Concrete Committee (European Concrete Committee (Comité européen du béton,CEB), 1972), the model of creep and shrinkage approved by the ACI Committee 209 (ACICommittee 209, 1972, 2008) would have predicted similar deformations (Bažant et al., 2008).Bažant et al. (2010) concluded that none of the current models for creep and shrinkage aresatisfactorily predictive. The authors also highlighted the real necessity to improve creep (orrelaxation) and shrinkage predictions from concrete composition.

1.2 Research Motivation

Predictive models for the behavior of concrete structures ideally rely on the numericalsimulation of boundary value problems to satisfy field equations throughout some domainof interest. This approach can only yield satisfactory predictions if the constitutive mod-els assumed for every point in the domain are themselves reasonable abstractions of reality.Thereby, the path towards reliable predictions of structural time-effects starts with a predictiveconstitutive model.

Concrete being a heterogeneous material, we can count at least two approaches for the devel-

2

opment of constitutive models. First, if one has enough information about the morphology andthe mechanical behavior of the material, its mesostructure can be represented explicitly and aboundary value problem can be solved down to a resolution at which heterogeneities can not bediscerned any more. Such an approach might be ideal if some limitations did not exist. First,cement-based materials are multiscale materials, meaning that the mechanical behavior of apiece of concrete subjected to a mechanical load is significantly affected by several behaviorsoccurring at different length scales. For this reason, the formulation of a constitutive modelwith finite computational resources is hardly a possibility. Moreover, cement-based materialsare genuinely random systems of which the morphological and small scale constitutive detailsare only known to some degree. Thereby, lack of knowledge is another limiting factor of thisapproach. A second approach is to develop a set of mathematical equations between somestate variables and loading conditions that satisfactorily reproduce the features of the relationbetween these quantities. Although the development of such empirical approaches can a prioriform the basis of a predictive tool, it requires extensive experimental and/or numerical studiesto reach an appropriate calibration. Also, once such a model is developed, it usually is onlyapplicable to the very limited set of conditions under which the calibration was performed.Because cement-based materials are random media subjected to diverse types of solicitations,the resort to empirical methods offers only a small range of applications.

In the last 20 years or so, a different approach has started to emerge. This approach consistsin formulating an abstraction of the mesostructure of a cement-based material parameterizedin terms of meaningful morphological quantities likely to vary from a concrete mix to another.Some estimates of the state variables of the medium are then formulated through some averag-ing methods involving the morphological model assumed. This methodology relies on the ideathat concrete, even when still hydrating, can be modeled as an ensemble of invariant materialphases present in different proportions from one mix design to another and interacting witheach other. This idea was mostly fostered by the work of Bažant (1977). An early use of thisapproach was made by Hua et al. (1997) to assess autogeneous shrinkage in aging cement-based materials. In the last 15 years, this methodology was developed considerably with therise of more accessible nanoindentation facilities used to assess the invariant mechanical prop-erties of the elementary material phases of concrete, see Velez et al. (2001); Constantinidesand Ulm (2004) and others. While almost systematically relying on micromechanics as an ap-plication of the results of Eshelby (1957), these methods have also been extended to accountfor the poroelastic component of the behavior of cement-based materials using results frommicroporomechanics (Dormieux et al., 2006a), see Ulm et al. (2004).

1.3 Research Objective

The resort to nanoindentation techniques along with applications of micromechanics hasproven to be a synergistic method for at least two types of purposes. First, to assess whether

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or not changes of behaviors observed in concrete are triggered by (or correlated with) modi-fications of mechanical behaviors at small length scales of concrete. In this case, one usuallyassesses and compares the small scale mechanical properties of two types of samples knownto exhibit distinct behaviors when subjected two mechanical loads. Instances of such analysesinclude the investigation of heat-treatment (Jennings et al., 2007; Vandamme et al., 2010),calcium leaching (Constantinides and Ulm, 2004) and thermal damage (DeJong and Ulm,2007; Zanjani Zadeh and Bobko, 2013). A second purpose served by this methodology is thedevelopment of versatile models applied to materials with different compositions. An impor-tant example of such models is the one of Bernard et al. (2003) used to predict the early-ageevolution over time of the elastic properties of an arbitrary mix design of concrete.

The objectives of the research presented in this document address each of the two purposesmentioned above. First, we want to illustrate the current state of the art methodology used forindentation analysis as a comparative tool for time effects in cement-based materials. To do so,we consider two distinct concrete materials; one produced with superplasticizer and the otherwith bioplasticizer. While the addition of bioplasticizer is known to enhance the resistance ofconcrete, it is our intent to (i) discover if this improvement is the result of a structural or me-chanical change at the nanoscale of concrete; and (ii) provide a first qualitative measure of theperformance of materials produced with bioplasticizer in terms of relaxation when subjectedto sustained loads. Notably, the resort to nanoindentation enables to survey the long-termrelaxation behavior of calcium silicate hydrates (C–S–H) after few minutes only, while years-long experiments would be required to assess creep after conventional macroscopic methods.The second objective of this research is to understand how the uncertainty of the mechanicaland morphological properties of the invariant material phases considered in micromechanicalmodels affect the prediction of macroscopic state variables of interest to model time effects.If one understands clearly what random parameters of these models are responsible for thegreatest source of uncertainty of model predictions, it can be legitimately decided to providemore efforts to obtain more accurate estimates of the corresponding properties.

1.4 Outline

The outline of the document is as follows. Part I is a comparative study conducted to as-sess whether changes of the behavior of concrete produced with bioplasticizer are triggered ornot by mechanical and/or structural changes at the nanoscale. Part II presents a multiscaleporomechanics-hydration model that could be used as a constitutive model for simulationsintended to assess time effects in concrete structures. The uncertainty of the properties usedas input parameters of the model are propagated up to the level of the predicted macro-scopic effective properties. A sensitivity is also conducted to identify which of these randomparameters affect the most the predictions of the model.

4

Part I

Nanoindentation study of calciumsilicate hydrates in concrete producedwith effective microorganisms-based

bioplasticizer

5

Résumé

Les propriétés mécaniques du béton produit avec un bioplastifiant à base de microor-ganismes efficaces (EM) sont etudiées par nanoindentation statistique, et comparées auxpropriétés mécaniques du béton produit avec un superplastifiant ordinaire (SP). Le recoursà la nanoindentation permet une évaluation du comportement élastique, de la dureté etde la relaxation à long-terme des silicates de calcium hydratés (C–S–H) après seulementquelques minutes. Pour chaque matériau, une analyse de partitionnement de donnéesrévèle différents groupes d’indentations vraisemblablement effectuées sur des C–S–H avecfractions volumiques solides distinctes. Il est trouvé que l’ajout de bioplastifiant à basede produit EM améliore la résistance des C–S–H en augmentant la cohésion et la frictiondes nanograins solides, et réduit le taux absolu de relaxation à long-terme. L’analysestatistique des résultats d’indentation suggère que le bioplastifiant à base de produit EMinhibe la précipitation des C–S–H avec une plus grande fraction volumique solide. Cesobservations corroborent les résultats d’une précédente étude qui attribuait une augmen-tation de l’homogénéité et un raffinement de la structure crystaline des phases de silicate àl’effet d’un agent similaire au bioplastifiant à base de produit EM. Il est aussi montré quel’amélioration des propriétés de résistance du C–S–H coincide avec un gain de résistanceen compression mesuré à l’échelle macroscopique du béton à base de produit EM.

Abstract

The mechanical properties of concrete produced with effective microorganisms (EM)-based bioplasticizer are investigated by means of statistical nanoindentation, and com-pared to the nanomechanical properties of concrete produced with ordinary superplasti-cizer (SP). The resort to nanoindentation enables to survey the elasticity, hardness andlong-term relaxation behavior of calcium silicate hydrates (C–S–H) after few minutes only.For each material, a cluster analysis of the experimental results yields groupings of indentslikely performed on C–S–H with distinct packing densities. It is found that the additionof EM-based bioplasticizer improves the strength of C–S–H by enhancing the cohesionand friction of solid nanograins, and decreases the absolute rate of long-term relaxation.The statistical analysis of indentation results also suggests that EM-based bioplasticizerinhibits the precipitation of C–S–H of higher density. The findings of this work corrob-orate the results of a previous study which attributed an increase of homogeneity anda refinement of the crystalline structure of silicate phases to the effect of a biomodifiersimilar to EM-based bioplasticizer. The improvement of strength properties of C–S–H isalso shown to coincide with a gain of compressive strength measured at the macroscale ofEM-based concrete.

7

Chapter 2

Partial introduction

The development of modern high performance concrete with outstanding mechanical proper-ties relies on an increased packing density of hydration products attainable by reducing thewater-to-cement ratio (w/c) (Tennis and Jennings, 2000; Ulm et al., 2007). The drawbackof decreasing the water-to-cement ratio of a concrete mix is a loss of workability generallycompensated by the addition of expensive superplasticizer (SP) admixtures. Recently, theresort to low-cost bioplasticizers based on effective microorganisms (EM) (see Higa and Wi-didana (1991)) was found to be at least as efficient as SP for improving the workability offresh concrete (Martirena et al., 2012). While EM have also proven to enable healing ofconcrete (Ramachandran et al., 2001; Wu et al., 2012) and strength improvement of mor-tar (Ghosh et al., 2005), no study has yet been performed which investigates the effects ofEM-based admixtures on the nanomechanical properties of hardened concrete. More precisely,the properties of calcium silicate hydrates (C–S–H) are of capital interest. The relevance ofstudying C–S–H lies in the two following points: (i) their crystalline structure and variabilitythroughout the system of hydrates may be affected by concrete admixtures such as EM-basedbioplasticizers (Bolobova and Kondrashchenko, 2000); (ii) differences in their structure andproperties can be responsible of important repercussions on macroscale phenomena such asdrying shrinkage and long term creep (Jennings, 2000; Vandamme and Ulm, 2009). Therefore,the aim of this study is to provide an innovative insight of C–S–H mechanical properties inconcrete produced with EM-based bioplasticizer. The investigation herein presented is focusedon the elastic behavior, strength and long-term relaxation of C–S–H.

Two categories of concrete samples produced either with EM-based bioplasticizer or with or-dinary SP are investigated in order to compare the elastic response, hardness and long-termviscoelastic behavior of C–S–H. The characterization is made by nanoindentation for the fol-lowing reasons: (i) nanoindentation is a high resolution technique which enables to probeseparately the distinct forms of C–S–H in concrete while macroscopic approaches are limitedto the investigation of composite behaviors; (ii) the sharpness of an indenter tip is responsiblefor a high level of stress which shortens the duration of transient creep-relaxation by orders of

9

magnitudes, hence allowing the characterization of a long-term viscoelastic behavior of C–S–Hafter few hundred seconds only (Vandamme and Ulm, 2009, 2013). Noteworthy, the long-termviscoelastic response of C–S–H to nanoindentation has shown to be quantitavely representativeof the long-term creep observed after years of macroscale uniaxial creep (Vandamme and Ulm,2013). Contrarily to previous investigations performed by nanoindentation (Vandamme, 2008;Vandamme and Ulm, 2009; Němeček, 2009; Jones and Grasley, 2011), the viscoelastic prop-erties are surveyed by relaxation (i.e. depth-controlled experiment) as it enables to record amaterial response which is not affected by time-delayed plastic deformation (Vandamme et al.,2012). Also in contrast to precedent studies (Constantinides, 2006; Ulm et al., 2007; Vanzo,2009), the nanoindentation is performed on concrete samples. Thereby, the experimental re-sults are analyzed after a two-step cluster analysis with the objective to isolate the indentsrealized on C–S–H from the ones performed on phases which are out of the scope of thisstudy. Following the current methodology of analysis of indentation results Ulm et al. (2007);Vandamme et al. (2010), the packing density distribution and nanoscale strength propertiesof C–S–H are identified in addition to the elastic stiffness and long-term rate of relaxation.

10

Chapter 3

Materials

The EM-based bioplasticizer used in this study is produced at CIDEM-UCLV (Martirenaet al., 2012). It consists of fallen leaves, rice, waste milk products, yogurt and molasses. Oncethe raw materials are mixed to form a solid substrate, it is left 25 days in a sealed tank forfermentation. The product is then used as inoculate to obtain a liquid fermentation of EM ofwhich the pH is kept between 3.2 and 3.8. The resulting EM-based bioplasticizer is referred toas IH Plus. Meanwhile, the SP used in this study is a naphthalene-based product developedby MAPEI, Italy and referred to as Mapefluid N-200. Each plasticizer is used separatelyfor a distinct sample. The sample preparation is processed after two steps. First, a bulkpreparation consists in casting and curing the material and second, a surface preparation isdone to precondition the sample for nanoindentation.

3.1 Bulk preparation

The samples used for this study consist of self-compacting concrete with mix designs given inTable 3.1. A type I cement (P-35 after Cuban norm NC 54 205:80) produced in Cienfuegos,Cuba is used with Cuban zeolite as a complementary source of fines (Martìnez-Ramìrez et al.,2006). The total amount of fines in the mix is 814 kg. A first sample is prepared with40 L of IH Plus (6% of cement weight) and a second with 6.7 L of Mapelfuid N-200 (2%of cement weight). The amount of water is corrected in the mix with IH Plus in order tocompensate for the excess of plasticizer. The water-to-fines ratio is kept around 0.56 for bothmixes, and the fines to aggregates ratio is 0.72. The samples are cast in cubes of dimensions100 x 100 x 100 mm and cured for one year. In the meantime, various properties of the freshconcrete are measured by Martirena et al. Martirena et al. (2012). At 365 days, some prismsof dimensions 4 x 4 x 1 cm are cut and kept for one week in isopropanol with two solventreplacements before being stored in desiccator for another week.

11

Table 3.1: Mix designs of self-compacting concrete produced either with EM-based bioplasti-cizer or superplasticizer

Sample produced Cement Water Zeolite Sand Aggregate Plasticizerwith [kg] [L] [L] [L] [kg] [L]

Mapefluid N-200 450.0 459.0 364.0 594.6 528.0 6.7IH Plus 450.0 426.0 364.0 594.6 528.0 40.0

3.2 Surface preparation

In order for nanoindentation results to relate to mechanical properties by classical contactanalysis, the surface of the indented material should be flat (Oliver and Pharr, 1992). How-ever, due to a highly heterogeneous pore structure, cement-based materials exhibit significantlyrough sections (Trtik et al., 2008). Thereby, in order to assess the results of surface prepa-ration, Miller et al. (2008) evaluated the surface roughness of finely polished cement pasteat relevant scale for nanoindentation. For this purpose, a correction accounting for both thealignment of the sample and spatial waves of wavelength larger than 8 µm was applied beforecalculation of the surface roughness. A repeatable methodology was hence proposed whichminimizes the surface roughness and ensures the convergence of the results of statistical de-convolution (Miller et al., 2008). Although no evidence was provided that the exposed poresare filled by detritus during surface preparation (Trtik et al., 2008), the issue of pore-fillingremains an undiscussed question. Yet, similar protocols as the one of Miller et al. (2008) havebeen used for nanoindentation studies of cement-based materials (Chen et al., 2010; Davydovet al., 2011; Abuhaikal, 2011; Zanjani Zadeh and Bobko, 2013; Vallée et al., tion).

Before nanoindentation, the samples were cut into pieces of dimensions 1 x 1 x 0.8 cm. Thesurface of each sample was then prepared following the protocol of Vallée (2013) which yieldssurface roughnesses inferior to 20 nm for cement paste at different water-to-cement ratio. Thesurface preparation consists in two steps. First, the sample is glued with mounting wax ona steel disc and mounted on a jig. The jig is left 5 minutes on a polishing machine with an8 inches silicon carbide disc (grain 240 grit). The position of the jig on the lapping wheel issuch that the velocity of the disc underneath the sample remains between 18 cm/s to 25 cm/s.Second, the disc underneath the jig is replaced by a perforated polishing pad with 0.5 mLof 1 µm oil-based diamond suspension. The same lap frequency is kept for 4 hours. Afterpolishing, the surface is gently cleaned with a soft cloth soaked with isopropanol.

12

Chapter 4

Methods

A four-step methodology is adopted to assess and compare the elastic and nanoscale strengthproperties, packing density distribution and long-term relaxation of C–S–H in both samplesproduced either with EM-based bioplasticizer or with SP.

• First, a large number of depth-controlled nanoindentation experiments is performedon both materials under study. Although the indents are preferably realized on bulkcement paste; sand grains, interfacial transition zones (ITZ) and aggregates are alsoprobed unintentionally. The purpose of this step is to collect enough local observationsto provide a statistically accurate picture of the nanomechanical behavior of C–S–H.

• Second, in an attempt to isolate the indents which where performed on C–S–H, allthe indentation results are merged into a common data set irrelevantly of the samplethey were measured on. This enables to increase the number of indents performedon phases which behave similarly in both materials and were sparsely sampled becauseunintentionally indented. The phases of concern are aggregates, sand grains, unhydratedcement grains and, arguably, large crystals of portlandite. A cluster analysis of themerged data set is thus performed with respect to the indentation elastic modulus andhardness. As a result, the indents performed on the material phases which are out ofthe scope of this study are identified and removed from the data set. At this point,mostly results of indentation performed on C–S–H phases packed after distinct densitiesremain.

• Third, the remaining indentation results are split back in two data sets accordinglyto the sample they were measured on. Then, the packing density distribution andstrength properties of C–S–H are assessed for each sample following the same procedureas described by Ulm et al. (2007) and Vandamme et al. (2010). For every indent, theobserved stress relaxation is characterized by a characteristic time of relaxation and anactivation volume which are representative of the viscoelastic behavior described after

13

an energy-activated (Klug and Wittmann, 1974; Wittmann, 1982) rearranging processof elementary particles of C–S–H (Ulm et al., 2000; Jennings, 2004; Vandamme and Ulm,2009).

• Fourth, a cluster analysis of the indentation modulus, hardness and activation volume ofrelaxation is performed to identify the different phases among the remaining data of eachsample. Different C–S–H phases and their distinct mechanical properties are identifiedfor both samples produced either with IH Plus or with Mapefluid N-200.

The homogeneity of the indented microvolumes assumed in nanoindentation studies (Constan-tinides and Ulm, 2007; Ulm et al., 2007; Sorelli et al., 2008; Vandamme et al., 2010) has beenquestioned by Trtik et al. (2009) and Lura et al. (2011). While the extent of this probablelimitation is still being debated (Ulm et al., 2010; Davydov et al., 2011), the resort to nanoin-dentation along with micromechanics was conclusive enough to lead up to a similar stiffnessof C–S–H grains (Constantinides and Ulm, 2007) as what was later assessed by moleculardynamics (Pellenq et al., 2009). A promising approach to circumvent potential issues relatedto the size of the investigated microvolume is the use of peak-force tapping atomic force mi-croscopy (AFM), see Trtik et al. (2012). However, contrarily to statistical nanoindention (Ulmet al., 2007; Bobko et al., 2011; Vandamme and Ulm, 2013), this method neither yet enablesto assess strength properties nor can it measure time-delayed mechanical behaviors. Also, al-though AFM can be used to quickly perform myriads of measurements, it was only used oversmall regions of cement paste so that the full range of hydrates was never investigated withthis sole technique (Trtik et al., 2012). Meanwhile, nanoindentation has already demonstratedsome versatility for the comparative assessment of mechanical properties of hydrates with theintent to determine whether changes of behaviors observed at macroscales were triggered by(or correlated with) modifications of mechanical behaviors at nanoscales. Instances of suchanalyses include the investigation of heat-treatment (Jennings et al., 2007; Vandamme et al.,2010), calcium leaching (Constantinides and Ulm, 2004) and thermal damage (DeJong andUlm, 2007; Zanjani Zadeh and Bobko, 2013). Thereby, nanoindentation is currently the mostsuited technique to compare the stiffness, strength and long-term time-delayed mechanicalbehavior of hydrates in concrete produced either with EM-based bioplasticizer or with SP.

The technical aspects invoked through the four-step methodology adopted in this study aresummarized in the next sections.

4.1 Nanoindentation relaxation analysis

Indentation consists in pressing a reference punch, i.e. indenter, at the surface of a deformablemedium, see Fig. 4.1. By means of a proper analysis of the outputs of an indentation, one canextract information about the mechanical behavior and properties of the indented material.

14

Figure 4.1: Geometry of indentation tests with conical and spherical probes, see Vandamme(2008).

A most common indentation analysis requires some knowledge of the mechanical properties ofthe indenter as well as the histories of the applied load, penetration depth and contact area ofthe indented surface. Here, the indenter is considered infinitely rigid compared to the indentedmaterial.

4.1.1 Self-similarity

Indentation analysis is performed by applying the solution of a quasi-static contact problembetween two solid bodies. The greatest source of difficulty in the analysis comes from the factthat the contact area between the indenter and the specimen is not known a priori. As a meanto simplify the resolution of the associated contact problem, one can perform a self-similarindentation; meaning that the displacement field at any load P2 can be inferred from theknown displacement field at a distinct load P1. Borodich et al. (2003) stated three conditionsthat must be satisfied for an indentation to be self-similar:

1. The constitutive behavior of the indented material is homogeneous with respect to strainsor stresses. This is satisfied if the operator F of the constitutive relation between strains andstresses is a homogeneous function of degree κ:

λκF (ε) = F (λε) (4.1)

where ε is the infinitesimal strain tensor and λ is an arbitrary constant. The constitutiverelationship of non-aging viscoelastic media such as mature cement-based materials satisfiesthis condition. Indeed, the stress state of such media is given by

λσ(t) = λ

∫ t

0

C(t− τ) :dε

dτdτ =

∫ t

0

C(t− τ) :d

dτ(λε) dτ (4.2)

15

Table 4.1: Parameters of the shape functions for the Berkovich and spherical indenters.

Probe type d B θeq R

Berkovich 1 cot θeq 70.32 -Spherical 2 (2R)−1 - 10 µm

where C(t) is a time-dependent stiffness tensor so that the constitutive behavior is homoge-neous of degree one.

2. The shape of the indenter can be described by a homogeneous function of degree d greaterthan or equal to one. Since it is common either to perform indentation with an axisymmetricpunch or, to use a probe with a shape that can reasonably be described by an axisymmet-ric function, it is useful to formulate this condition with respect to a cylindrical system ofcoordinates (r, z, θ):

x · ez = B [x · er] rd for every x at the edge of the probe (4.3)

where B is referred to as a proportionality factor B, see Fig. 4.1. Two different probes whosegeometries can be described or approximated after Eq. (4.3) are commonly used for indentationstudies. Those are:

• The Berkovich indenter which consists in a sharp three sided pyramidal tip used toprobe areas as small as few hundreds square nanometers, i.e. nanoindentation. Thegeometry of this probe induces a high concentration of stresses beneath the contact zonethat results in important plastic deformations. To simplify the contact analysis, thepyramidal shape is represented by an equivalent cone of half-angle θeq (see Table 4.1).

• The spherical indenter of radius R used to reduce the amount of induced plastic defor-mations.

The proportionality factor B and the degree d of the homogeneous shape functions associatedwith these two indenters are given in Table 4.1.

3. The load (or penetration) at the surface of the specimen increases monotonically during in-dentation. The condition of monotonic increase is not satisfied at load (or penetration) releaseso that self-similarity can not be invoked to analyze the unloading phase of an indentationexperiment.

4.1.2 Scaling relations

If self-similarity is satisfied, the load P2 applied at the surface penetrated by the indenter ata depth h2 can be inferred from the applied load P1 for which the indenter penetrates the

16

surface at a depth h1. Therefore, considering that the state of a self-similar indentation isfully characterized by the load P , a penetration depth h, a contact depth hc (see Fig. 4.1)and a projected contact area Ac (see Fig. 4.1), the following scaling relations apply (Borodich,1989):

P1

P2=

(h1

h2

)2+κ(d−1)d

(4.4)

h1

h2=

((Ac)1

(Ac)2

)d/2(4.5)

where κ is equal to one for linear viscoelastic media. For axisymmetric indenters, the contactarea Ac is a disk of radius a so that Eq. (4.5) can be recast in

ad1h1

=ad2h2. (4.6)

Considering that the profile of the indenter is described by Eq. (4.3), it can be shown fromEq. (4.6) that the ratio of contact over penetration depth does not depend on the applied load:

(hc)1

h1=

(hc)2

h2. (4.7)

In other words, the ratio hc/h remains constant during indentation until the sustained pene-tration is released.

4.1.3 Definition of the contact problem with relaxation

For this study, we consider depth-controlled experiments for which the intented material ex-hibits some stress relaxation. The prescribed penetration history is described by a function ofthe form

h(t) = F(t)hmax (4.8)

where hmax is a constant penetration depth sustained during relaxation and F(t) is referred toas a history function with property maxF = 1. The following trapezoidal penetration history(see Fig. 4.2) is considered:

F(t) =

t/τL for 0 ≤ t ≤ τL1 for τL ≤ t ≤ τL + τH

(τL + τH + τU − t)/τU for τL + τH ≤ t ≤ τL + τH + τU

(4.9)

where τL is the duration of the loading phase, τH is the time during which the maximalpenetration depth is sustained and τU is the unloading time of the specimen.

The indented material is considered as a solid half-space Ω(t) with boundary ∂Ω(t) = ∂Ωc(t)∪∂Ωnc(t) in which ∂Ωc(t) is the surface of contact with the probe and ∂Ωnc(t) is the part of theboundary which is not in contact with the indenter. We note n(x, t) and t(x, t) the outwardnormal and tangent vectors at a point x of the boundary ∂Ω(t).

17

Figure 4.2: Penetration depth history of the indenter.

The assumption is made that the deformable medium undergoes infinitesimally small defor-mations only so that the strain tensor is given by

ε(t) =1

2

[∇u(x, t) + t∇u(x, t)

]∀ x ∈ Ω(t), ∀ t (4.10)

in which u is the displacement field at time t. Another assumption is made that every pointwithin the indented solid remains under quasi-static equilibrium:

∇ · σ(x, t) = 0 ∀ x ∈ Ω(t), ∀ t. (4.11)

The boundary points of Ω(t) which are not in contact with the indenter remain traction-freeduring the whole experiment:

σ(x, t) · n(x, t) = 0 ∀ x ∈ ∂Ωnc(t), ∀ t (4.12)

while the contact between the indenter and the deformable solid is assumed friction-free:

t(x, t) · σ(x, t) · n(x, t) = 0 ∀ x ∈ ∂Ωc(t),∀ t. (4.13)

Considering the parametric representation of the contour of the rigid indenter, see Eq. (4.3),the only non-vanishing component of the displacement field is given as follows

u(x, t) · ez = h(t)−B [x · er]d ∀ x ∈ ∂Ωc(t),∀ t (4.14)

onto the moving contact boundary. The global equilibrium of the solid half-space is guaranteedthrough the following equation:

P (t) =

∫

∂Ωc(t)

ez · σ(x, t) · n(x, t) dA ∀ t. (4.15)

4.1.4 Elastic solution of Galin-Sneddon

Analytical approximations to solutions of self-similar indentation boundary value problemsare often built upon the solutions of Galin (1953) and Sneddon (1965). Assuming an isotropic

18

elastic deformable medium, the following relation was established between the load applied atthe surface of a specimen and the penetration depth of the indenter:

P =d

d+ 1

[Γ(d/2 + 1/2)

Γ(d/2 + 1)

]1/d 2h1+1/d

(√πB)

1/dM0 (4.16)

where Γ(x) is the Euler Gamma function1. The indentation elastic modulus M0 is given by

M0 =E0

(1− ν0)2= 4G0

(3K0 +G0

3K0 + 4G0

)(4.17)

where E0 is the elastic modulus, ν0 is the Poisson’s ratio and, K0 and G0 are the elastic bulkand shear moduli. It is also common practice to extract the elastic indentation modulus M0

from the measurement of a contact stiffness through the BASh formula (Bulychev et al., 1975):

S0 ≡dP

dh

∣∣∣∣t=(τL+τH)+

=2√π

√AcM0 (4.18)

where the elastic contact stiffness S0 is the derivative of the applied load with respect to thepenetration depth at the onset of unloading. Still following the solution of Galin (1953) andSneddon (1965), the invariant ratio of contact over penetration depth is given by

Λ ≡ hch

=1√π

Γ(d/2 + 1/2)

Γ(d/2 + 1)=

2/π for d = 1

1/2 for d = 2. (4.19)

4.1.5 Approximate solution for indentation with relaxation

Although the analytical relation of Galin-Sneddon is only valid for elastic solids, it is frequentlyused to analyze results of nanoindentation despite the occurrence of plastic deformations dur-ing penetration of the indenter. Thereby, neglecting first the contribution of plastic defor-mations, Eq. (4.16) is recast in the following convolution integral to account for relaxation(Radok, 1957; Lee and Radok, 1960):

P (t) = 2d

d+ 1

(Λ

B

)1/d ∫ t

0

M(t− τ)d h(τ)1+1/d

dτdτ. (4.20)

where the indentation modulus M(t) becomes a decreasing function of time. Using the previ-ous formulation of penetration history (see Eq. (4.8)), the applied load is recast in

P (t) = 2h1+1/dmax

d

d+ 1

(Λ

B

)1/d ∫ t

0

M(t− τ)d F(τ)1+1/d

dτdτ. (4.21)

Considering the piecewise definition of the controlled penetration, the expression of the appliedload is given as follows for the loading, the holding and the unloading phase:

P (t) =

PL(t) for 0 ≤ t ≤ τLPH(t) for τL ≤ t ≤ τL + τH

PU (t) for τL + τH ≤ t ≤ τL + τH + τU

(4.22)

1The Euler Gamma function is given by Γ(x) =∞∫0

t x−1 exp (−t)dt.

19

where the applied load of the loading phase is

PL(t) = 2h1+1/dmax

(Λ

B

)1/d ∫ t

0

M(t− τ)

τL

(τ

τL

)1/d

dτ, (4.23)

the applied load of the holding phase reads

PH(t) = 2h1+1/dmax

(Λ

B

)1/d ∫ τL

0

M(t− τ)

τL

(τ

τL

)1/d

dτ (4.24)

and, at the onset of unloading, the applied load is

PU (t) = 2h1+1/dmax

(Λ

B

)1/dτL∫

0

M(t− τ)

τL

(τ

τL

)1/d

dτ −

t∫

τL+τH

M(t− τ)

τU

(τT − ττU

)1/d

dτ

(4.25)

where τT is the overall time length of indentation (τL + τH + τU ).As stated by Eq. (4.18), the contact stiffness is the derivative of the applied load with respectto the penetration depth at the onset of unloading. For the given function of penetrationhistory, this becomes

S =dP

dh

∣∣∣∣t=(τL+τH)+

=P (t)

h(t)

∣∣∣∣∣t=(τL+τH)+

=PU (t = (τL + τH)+)

F (t = (τL + τH)+)hmax. (4.26)

For a more accurate evaluation of the elastic indentation modulus, one should take into accountthe contribution of viscous effects on the measurement of the unloading rate PU . Hence, thefollowing expression is obtained by inserting the time derivative of Eq. (4.25) into Eq. (4.26):

S =− PH(t = τL + τH)τUhmax

+

2h1/dmax

(Λ

B

)1/d

τU · limt→ (τL+τH)+

M(t− (τL + τH))

τU

(τT − (τL + τH)

τU

)1/d (4.27)

where M(0+) is the elastic indentation modulus M0. Then, Eq (4.27) is recast in

S = − PH(t = τL + τH) τUhmax

+ 2h1/dmax

(Λ

B

)1/d

M0 (4.28)

where the second term of the right-hand side is the elastic contact stiffness S0 given byEq. (4.18). The discrepancy between S and S0 increases with the inverse of the unload-ing penetration rate and, with the relaxation rate PH at the end of the holding phase. Asshown by Fig. 4.3a, for a given deformable medium loaded at a given penetration rate, thelonger the holding phase, the smaller the relaxation rate reached before release of the sustainedpenetration. Consequently, as shown in Fig. 4.3b, the longer the holding phase and the faster

20

(a) History of force relaxation (b) Force-penetration curve

Figure 4.3: Effect of viscosity on the measurement of contact stiffness.

the unloading, the smaller the contact stiffness. If the relaxation reaches an asymptotic state(PH → 0) or, if the release of the indenter is infinitely fast (τU → 0), the viscous effects onthe measurement of the contact stiffness becomes negligible and Eq. (4.28) is equivalent toEq. (4.18). Finally, the elastic indentation modulus is obtained by

M0 =

√π

2√Ac

(S +

PH(t = τL + τH) τUhmax

)(4.29)

where the contact stiffness S and the relaxation rate PH at the end of the holding phase aremeasured during the indentation experiment.

4.1.6 Correction of the analytical solution

As mentioned previously, the application of the analytical solution of Galin-Sneddon to solvea relaxation indentation problem with plastic deformations occurring during penetration ofthe indenter is an approximation. As a result, one should expect some bias in the indentationanalysis leading to some inaccuracies of the assessed mechanical properties of the indentedmaterial. According to Oliver and Pharr (2004), the most important causes of this bias are:

• The non-consideration of large deformations;

• The negligence of plastic deformations;

• The idealization of pyramidal indenters as axially symmetric probes.

It has been emphasized by Hay et al. (1999) that the assumption of small deformations isresponsible for a mis-prediction of the radial component of the displacement field in compress-ible media. Meanwhile, the negligence of plastic deformations leads up to an overestimation of

21

the elastic indentation modulus when evaluated with the BASh formula. Also, using the ana-lytical axisymmetric solution of Galin-Sneddon is an another approximation of the evaluationof the stress field for indentation with a pyramidal indenter (King, 1987).

A widely used method that consists in applying a correction factor β in Eq. (4.18) has beendeveloped to counteract the biasing effects mentioned above. By doing so, Eq. (4.18) becomes

S0 = β2√π

√AcM0 . (4.30)

King (1987) was the first to emphasize the importance of this factor initially intended tocorrect for the idealization of non-circular probes. Later on, this method was used to correctfor other biasing effects until it finally takes into account all types of physical processes thatmay bias the analytical solution of Galin-Sneddon (Oliver and Pharr, 2004). As a consequenceof Eq. (4.30), the elastic indentation modulus given by Eq. (4.29) is recast in

M0 =

√π

2β√Ac

(S +

PH(t = τL + τH) τUhmax

)(4.31)

where β is taken equal to 1.034 for indentation with a Berkovich indenter and 1 when using aspherical probe King (1987).

4.1.7 Imperfect geometry of the indenter

In practice, the stiffness of the indenter is finite so that indentation after indentation, it deformsand deviates from the geometry described in Eq. (4.3). The consideration of these deviationsis highly relevant as it strongly affects the accuracy of the results obtained by nanoindentation.In order to counteract this effect, Oliver and Pharr (2004) proposed a reformulation of theprojected contact area used in Eqs. (4.29) and (4.31):

Ac =8∑

i=0

Ci (hc)max 2 (1−i)

= C0 (hc)max2 + · · ·+ C8 (hc)max1/128 (4.32)

where Ci’s are constants to be calibrated from results of indentation performed on materialswith known mechanical properties. In case of undeformed conical indenters, Ac reduces toC0 (hc)max2 with C0 = π tan2 θ. For spherical probes of radius R, the contact area isC0 (hc)max2 + C1(hc)max with C0 = −π and C1 = 2πR.

The maximum contact depth is expressed in terms of the maximum sinking depth illustratedin Fig. 4.4:

(hc)max = hmax − (hs)max. (4.33)

Oliver and Pharr (1992) express the maximum sinking depth in terms of the maximum appliedload and the elastic contact stiffness:

(hs)max = εPmaxS0

(4.34)

22

Figure 4.4: Schematic representation of an indentation (adapted from Oliver and Pharr(1992)).

where Pmax is measured at the onset of the holding phase (see Fig. 4.3a), and S0 is free ofviscous effects. The parameter ε is a constant determined from the Galin-Sneddon solution:

ε =d+ 1

d(1− Λ) =

2 (1− 2/π) for d = 1

1 for d = 2(4.35)

where Λ is given by Eq. (4.19).From Eqs. (4.33) and (4.34), the projected contact area at maximum sustained penetration islinked to the maximum applied depth and measured load through the following equation:

Ac =8∑

i=0

Ci

hmax − ε

PmaxS0

2 (1−i)

(4.36)

where assigning a value of 0.75 to ε has proven to be more successful than 2 (1 − 2/π) forBerkovich indenters (Oliver and Pharr, 1992). The elastic contact stiffness of Eq. (4.36) reads

S0 = S +PH(t = τL + τH) τU

hmax(4.37)

where S is measured at the onset of unloading, PH(t = τL+τH) is the relaxation rate measuredat the end of the holding phase and, hmax and τU are prescribed parameters of the indentationexperiment.

4.2 Packing density distribution and strength properties

The hardness H is the average vertical traction component at the surface measured at theonset of the holding phase so that H ≡ P0/Ac where P0 stands for P (t = τL), interchange-ably referred to as Pmax. Both the indentation modulus and hardness of C–S–H phases arerecognized as the composite mechanical signature of a porous material composed of a solid

23

phase and saturated pore space (Ulm et al., 2007; Vandamme et al., 2010). Then, assuminga nanogranular morphology along with the application of a self-consistent homogenizationscheme with a solid percolation threshold at 50% porosity, the indentation modulus relates tothe packing density as follows:

M0(ms, η) = ms(2η − 1) (4.38)

where ms is the indentation modulus of the solid phase and η is the packing density of theindented microvolume. Although Eq. (4.38) is valid for a solid phase Poisson’s ratio of 0.2, ithas a limited sensitivity to discrepancy from this value (Constantinides and Ulm, 2007).

Meanwhile, a cohesive-frictional behavior of the solid phase is assumed so that the indentationhardness relates to the packing density η, the cohesion cs and the friction coefficient αs of thestrength domain of hydrates (Vandamme et al., 2010) as follows:

H(cs, αs, η) = csA[1 +Bαs + (Cαs)

3 + (Dαs)10] [

Π1 + αs(1− η)Π2

](4.39)

where A = 4.7644, B = 2.5934, A = 2.1860 and D = 1.6777. In Eq. (4.39), Π1 is a functionof the packing density given by:

Π1(η) =

√2(2η − 1)− 2η + 1√

2− 1

[1 + a(1− η) + b(1− η)2 + c(1− η)3

](4.40)

where a = −5.3678, b = 12.1933 and c = −10.3071. Similarly, Π2 is a function of the frictioncoefficient and packing density:

Π2(αs, η) =2η − 1

2

[d+ e(1− η) + f(1− η)αs + gα3

s

](4.41)

where d = 6.7374, e = −39.5893, f = 34.3216 and g = −21.2053.

Therefore, an inverse analysis of the indentation results enables to assess the cohesion andfriction coefficient of the solid phase of C–S–H, and the packing density at every indentationlocation within the specimen:

cs, αs, η1, . . . , ηN = argmincs,αs,η1,...,ηN

N∑

i=1

[(M0,i −M(ηi)

M0,i

)2

+

(Hi −H(cs, αs, ηi)

Hi

)2].

(4.42)A number of indents N 2 is considered in Eq. (4.42), where the solid phase of C–S–H hasa fixed indentation modulus ms of 63.5 GPa (Vandamme et al., 2010; Vandamme and Ulm,2013).

4.3 Energy activated relaxation

The measured load relaxation of cement-based materials is assumed to result from the en-ergy activated Klug and Wittmann (1974); Wittmann (1982) rearrangement of nano-meter

24

sized grains of C–S–H (Ulm et al., 2000; Jennings, 2004; Vandamme and Ulm, 2009, 2013).Therefore, in conformity with previous works of Klug and Wittmann (1974); Wittmann (1982);Jennings (2004), rate process theory is invoked to model the time-delayed mechanical responseto indentation.

4.3.1 Simplification to a 1D problem

At any time of an indentation relaxation experiment, the stress field of the indented materialis given by

σ(x, t) =

∫ t

0

C(t− τ) : [ε(x, τ)− εp(x, τ)] dτ ∀ x ∈ Ω(t), ∀ t (4.43)

where εp(t) is the time derivative of the plastic strain field at time t. Here, we consider thatall plastic deformations occur during the penetration of the indenter. We also assume thatthe loading phase is short enough so that no substantial relaxation occurs before the holdingphase. If so, we have:

σ(x, t) ≈ C(t− τL) : [ε(x, τL)− εp(x, τL)] ∀ x ∈ Ω(t), ∀ t s.t. τL ≤ t ≤ τL + τH . (4.44)

At the beginning of the holding phase, the material has not yet relaxed and the force appliedis the largest load observed through the course of the experiment (see Fig. 4.3a):

Pmax ≡ P (τL) =

∫

∂Ωc(τL)

ez · σ(x, τL) · n(x, τL) dA. (4.45)

Once the maximum penetration is sustained, the indented material starts relaxing and theapplied force becomes

PH(t) =

∫

∂Ωc(τL)

ez ·C(t− τL) : [ε(x, τL)− εp(x, τL)]·n(x, τL) dA ∀ t s.t. τL ≤ t ≤ τL+τH .

(4.46)that we recast in

PH(t) =

∫

∂Ωc(τL)

ez · C(t− τL) : S(0) : σ(x, τL) · n(x, τL) dA ∀ t s.t. τL ≤ t ≤ τL + τH

(4.47)where S(0) is the instantaneous elastic compliance of the solid. The indented material beingisotropic, we have

S(0) =1− 2ν0

E0J +

1 + ν0

E0(I− J) (4.48)

in which J = (1/3) 1⊗1 and 1 is the second-order unit tensor. Not only the indented materialis elastic isotropic, but it remains isotropic during relaxation:

C(t) =E(t)

1− 2ν(t)J +

E(t)

1 + ν(t)(I− J) . (4.49)

25

An even stronger assumption is made that the Poisson’s ratio remains constant during relax-ation:

C(t) =E(t)

1− 2ν0J +

E(t)

1 + ν0(I− J) . (4.50)

The statement of constant Poisson’s ratio is commonly made for the investigation of creepmechanisms in cement-based materials (Vandamme, 2008; Vandamme and Ulm, 2009; Jonesand Grasley, 2011). The underlying idea at the origin of this assumption is that the relaxationof the solid is triggered by the relaxation of uniformly distributed microprestresses after akinetic independent of external loadings (Bažant, 1997). As a result, every increment of stressreleased is coaxial with the applied stress state so that the Poisson’s ratio remains constantduring relaxation.

As a result of these assumptions, we have

PH(t) =E(t− τL)

E0Pmax ∀ t s.t. τL ≤ t ≤ τL + τH (4.51)

which, for larger times and short loading phases (t τL) becomes

PH(t) =E(t)

E0Pmax ∀ t s.t. τL ≤ t ≤ τL + τH (4.52)

so that the kinetic of indentation load relaxation can be simplified to a 1D viscous problem.Given the following expression between the elastic indention and Young modulus:

M0 =E0

1− ν20

(4.53)

we haveM(t) = φ(t)M0 (4.54)

where φ(t) ≡ PH(t)/Pmax is defined as a non-dimensional indentation relaxation function.

4.3.2 Rate process theory

Rate process theory was introduced by Eyring (1935) through a general equation governingthe rate of rearrangement of matter for processes that can be idealized as the surmounting ofa potential barrier by activated structural units. While the scope of the initial work of Eyringwas to describe chemical reactions at molecular scale, it was extended to the representationof viscous flows (Eyring, 1936) and applied to the analysis of particulate flows as encounteredin creep and relaxation of soil (Culling, 1983, 1988) and concrete (Klug and Wittmann, 1969,1974). The corresponding rate equations used for activated rheological processes and theparameters related to nanostructural features involved in the mechanisms investigated in thisstudy are presented here.

The general equation for the rate of a process in which matter rearranges by surmounting apotential energy barrier was proposed by Eyring (1935) in the form

κ = θ

(FaFn

)(p

m∗

)(4.55)

26

Displacement

Energy

Figure 4.5: Energy activated process of a flow unit.

where κ is the frequency of a primary process performed by an activated unit. The transmissioncoefficient θ is the probability that a unit having reached the activated state, referred to astransition state (Petersson, 2000), proceeds across the saddle point and completes the processrather than returning to its original configuration. The mean velocity p/m∗ of a unit alongthe axis of motion is given by the ratio of the mean momentum p over the reduced mass m∗.The ratio Fa/Fn is obtained from the partition function of the activated complex per unitlength Fa and the partition function of the normal state Fn. The different features involvedin such rearrangement processes are represented in Fig. 4.5, where Q is the energy barrier tosurmount.

Eyring invokes statistical mechanics to reformulate Eq. (4.55) for the case of thermally acti-vated rearrangement processes with one translational degree of freedom:

κ = θ

(F ∗aFn

)(kB T

h

)exp

(− Q

kB T

)(4.56)

where κ is the rate of translational motion induced by thermal fluctuation. The energy avail-able being distributed among the system with respect to a Maxwell-Boltzmann distribution,the rate of activation is given by the proportion of units that bear enough energy to reachthe transition state times the fundamental frequency kB T in which, kB is the Boltzmann’sconstant and h, the Planck’s constant. The partition function F ∗a differs from Fa in that itis calculated using a zero of energy higher by Q than for Fn and, the partition function forthe degree of freedom normal to the barrier is omitted from F ∗a . By fixing F ∗a /Fn and θ tounity (Eyring, 1936), Eq. (4.56) simplifies to

κ =kB T

hexp

(− Q

kB T

). (4.57)

Following the notions of rate process theory applied to viscous flows (Eyring, 1936), viscosityis considered as the sliding of ordered layers of flow units at a distance λ1 apart. The assumed

27

Figure 4.6: Vacancy migration of flow units as described by the Eyring model applied toviscous flows.

mechanism consists in individual flow units which acquire the activation energy required to slipover the potential barrier to the next equilibrium position along the same plane (see Fig. 4.6).The viscosity of such a system is given by

η = σλ1/∆u (4.58)

where ∆u is the difference in velocity of two layers at a distance λ1 and σ is the appliedshearing stress. The distance from an equilibrium site to another is noted λ and the resultingforce in the direction of motion is given by σλ2λ3, where λ2λ3 is the effective cross-sectionalarea of the moving flow unit. As described in Fig. 4.7, the resulting force leads to a drop ofσ V = σ/2(λλ2λ3) in the potential barrier of activation in direction of the applied stress sothat the rate of forward displacement is given by

κ+ =kB T

hexp

[−(

Q

kB T− σ λλ2λ3

2 kB T

)]. (4.59)

Similarly, the rate of backward translation is

κ− =kB T

hexp

[−(

Q

kB T+σ λλ2λ3

2 kB T

)](4.60)

so that the net frequency of forward displacement is

κ+ − κ− =2 kB T

hexp

(− Q

kB T

)sinh

(V σ

kB T

)(4.61)

where V , the thermodynamic activation volume, is the product of the distance crossed by anactivated flow unit from equilibrium to transition state with the cross-sectional area of thesliding unit. Here, the transition state is located at equal distance from adjacent equilibriumsites so that the activation volume is given by 1/2λ · λ2λ3.

Although Eq. (4.61) does take into account the dispersion of energy in the system, it does notconsider how free space is spread in the system. For a solid medium, not only a particle needsto be provided enough energy to slip over the potential barrier but, a sufficiently large volumeneeds to be free at proximity of a sliding unit in order for it to be able to move at a newequilibrium position. Therefore, the net frequency of activation is multiplied by a structuralfactor X that quantifies the effect of the distribution of room available on the viscosity of

28

Displacement

Energy

(a) No applied forceDisplacement

Energy

(b) Effect of applied force

Figure 4.7: Potential trough of a flow unit with and without shearing applied force.

the system. Spaepen (1977) proposed to quantify this effect by considering X as the volumefraction of potential rearrangement sites among the system and follows Cohen and Turnbull’stheory (Cohen and Turnbull, 1959) to express the probability p(v) of finding a flow unit withadjacent free volume v:

p(v) =γ

vfexp

(−γ vvf

)(4.62)

where γ is a geometrical factor between 1/2 and 1 that stands for overlapping free volumes andvf is the average volume of the available rearrangement sites. In order for a room available toenable the rearrangement of matter, it must be larger than the volume of the activated flowunit v∗. Therefore, X is given by the proportion of free volume greater than v∗:

X =

∫ ∞

v∗p(v) dv =

∫ ∞

v∗

γ

vfexp

(−γ vvf

)dv = exp

(−γ v

∗

vf

)(4.63)

where v∗ is equivalent to λ1λ2λ3. If the free volumes do not overlap, γ is set to unity.

Given that the difference of velocity ∆u between two ordered layers is equivalent to the net rateof activation times the distance λ crossed by a removed flow unit, the expression of viscositygiven by Eq. (4.58) is recast in

η =σ λ1

λX (κ+ − κ−)=τ λ1

λ

[X

(2 kB T

h

)exp

(− Q

kB T

)sinh

(V σ

kB T

)]−1

. (4.64)

Then, the thermally activated strain rate of a viscous medium is given by

ε = X

(λ

λ1

)(2 kB T

h

)exp

(− Q

kB T

)sinh

(V σ

kB T

)(4.65)

in which the thermodynamic significance of the activation parameters Q and V was studiedby Gibbs (1969). Hence, according to Gibbs, the activation energy is a function of the change

29

of strain rate induced by a change in temperature at constant applied stress:

Q = kB T2

(∂ ln ε

∂T

)

τ

. (4.66)

Likewise, the activation volume is a function of the change of strain rate due to a change ofapplied stress at constant temperature:

V = kB T

(∂ ln ε

∂σ

)

T

. (4.67)

Then, as described by Eq. (4.65), the strain rate of any thermally activated rearrangingmedium is governed by five parameters. Namely, those are (i) activation energy, (ii) kineticenergy, (iii) frequency of primary rearrangement, (iv) free volume and (v) applied stress. Somehypotheses are often considered that simplify the formulation of the strain rate. First, if thefree room crossed by an activated flow unit removed to a new equilibrium site is the same sizethan the particle itself, the ratio λ/λ1 vanishes. Second, if the applied stress is high enough sothat the strain work V σ is significantly greater than the kinetic energy represented by kB T ,the formulation of the thermally activated strain rate simplifies in

ε = X

(kB T

h

)exp

(− Q

kB T

)exp

(V σ

kB T

). (4.68)

4.3.3 Zhurkov element

If the scale of the flow units involved in the rearrangement is large enough so that the kineticenergy of each particle is not any more given by statistical mechanics, neither the fundamentalfrequency nor the denominator of the exponential terms of Eq. (4.68) are representative ofthe process undergone by the system. In that case, Culling (Culling, 1983, 1988) proposed analternative description of the strain rate of particulate flows:

ε = X νm exp (−Q/β) exp

(V σ

β

)(4.69)

where νm is the rate of primary displacement of a particle and β is an energy term roughlyanalogous to kB T . In order to improve the simplicity of the next developments, the weightedrate of primary displacement X νm exp (−Q/β) is substituted by ε0 and the amount of energyβ/V available per unit volume of particle is replaced by A, so that Eq. (4.69) is recast in

ε = ε0 exp( σA

). (4.70)

Then, a new dashpot referred to as a Zhurkov body is defined to model the viscous behaviorof energy activated rearranging media (see Fig. 4.8a). This element is characterized by theviscosity

ηQ =σ

ε0exp

(− σA

)(4.71)

where the index Q stands for energy activated processes. The stress sustained by an energyactivated dasphot is given by

σ = ηQ(σ) ε. (4.72)

30

(a) Eqs. (4.70) to (4.72). (b) Eqs. (4.73) to (4.75). (c) See Petrov (1998).

Figure 4.8: Zhurkov dashpot and simple energy activated bodies.

4.3.4 Energy activated Maxwell element

Since the scope of the study is focused on the behavior of viscoelastic media, a compositeelement analogous to the classic Maxwell element is defined so that it involves an energyactivated dashpot in series with an elastic spring (see Fig. 4.8b). The rheological behavior ofthe resulting body is governed by the following equation:

ε =σ

ηQ+

σ

E0= ε0 exp

( σA

)+

σ

E0. (4.73)

The stress released by such an energy activated Maxwell body that supports a constant strainε0 is given by

σ − σ0 = − A

ε0E0ln

[1 +

ε0E0

Aexp

(ε0E0

A

)t

](4.74)

so that the corresponding relaxation function is

φ(t) = 1− A

ε0E0ln

[1 +

ε0E0

Aexp

(ε0E0

A

)t

]. (4.75)

Energy activated dashpots can be used the same way that standard viscous dashpots intro-duced earlier in the section for the construction of composite rheological models, see Petrov(1998) for additional examples.

4.3.5 Application to nanoindentation relaxation analysis

After application of rate process theory for high levels of stress as encountered in solid inden-tation, a 1D viscoelastic strain rate is expressed as follows:

ε = A exp

(V σ

kB T

)exp

(− Q

kB T

)+

σ

E0(4.76)

where V is the activation volume, Q is the activation energy, kB is the Boltzmann constantand A is a frequency factor independent of stress σ and temperature T (Klug and Wittmann,1974; Wittmann, 1982). The activation volume V is arguably recognized as the cross section

31

of a C–S–H basic particle integrated along the motion path it follows to reach equilibrium.However, because rate process theory suggests a Brownian motion which is negligible for C–S–H solid particles (Vandamme and Ulm, 2009), it does not consistently describe the physics ofnanogranular rearrangement in concrete. Nevertheless, rate process theory provides a usefulframework for the quantitative characterization of creep and relaxation in concrete (Klug andWittmann, 1974; Wittmann, 1982; Bažant et al., 1983; Jennings, 2004).

In the case of relaxation experiments, the strain rate of Eq. (4.76) vanishes so that the 1Dstress released over time at the area of contact is given by

σ(t)− σ0 = −kB TV

ln

[1 +

AE0 V

kB Texp

(− Q

kB T

)t

](4.77)

where σ0, the applied stress at the onset of relaxation, is nothing but the contact hardness H.The non-dimensional relaxation function φ(t) = σ(t)/σ0 then reads

φ(t) = 1− kB T

V σ0ln

[1 +

t

τr

](4.78)

where the characteristic time of relaxation τr can easily be expressed in terms of the activationvolume, activation energy and other quantities in Eq. (4.77).

Fitting the observed nanoindentation relaxation with Eq. (4.78) yields the identification ofan activation volume V and a characteristic time τr for each indent. The characteristic timemarks the transition between a transient relaxation phase and the long-term behavior.

The rate of long-term normalized relaxation as derived from Eq. (4.78) yields an expressionwhich is linear with respect to the inverse of time, i.e. φ(t τr) ≈ −kBT/(V σ0t). Therefore,the kinetics of long-term relaxation described by σ0φ(t τr) is governed by the mechanicalwork of relaxation V σ0. For a given instantaneous stress σ0 developed in reaction to sustaineddeformation (or contact hardnessH), the larger the activation volume, the less stress is releasedthrough time.

4.4 Cluster analysis

The samples investigated in this study are assimilated to material mixtures which consist inmixes of phases with distinct mechanical properties. The resort to cluster analysis enables toidentify the statistically most reliable mixture which corresponds to the results obtained by gridnanoindentation (Vanzo, 2009; Abuhaikal, 2011; Krakowiak et al., 2011). A material mixtureconsists in a set of material phases with their respective properties and the probability set ofeach indent to have been performed on either of these phases. As a result of cluster analysis,the volume fractions of the different material phases in the mixture are straightforwardlycalculated.

32

The indentation results are considered as a realization of material mixture in which the me-chanical properties of each phase follow a multivariate Gaussian probability distribution.Given the set of observations X = x1, ...,xN in which N is the number of indents, theprobability density function that an indent xi = M0,i, Hi belongs to the k-th materialphase is noted fk(xi|µk,Σk) in which µk =

(µM0

)k, (µH)kis the mean vector and Σk the

covariance matrix of the corresponding material phase:

fk(xi |µk,Σk) =1

(2π) p/2|Σk|exp

−1

2(xi − µk)T Σ−1

k (xi − µk)

(4.79)

where p is the dimension of the multivariate random variable considered in the analysis. If onlythe indentation modulus and hardness are considered among the indentation results, p equals2. If the activation volume of relaxation is also considered; p equals 3, xi = M0,i, Hi, Vi,and µk and Σk are redefined in consequence.

For a given number G of material phases, the likelihood of a mixture model reads

LM (µk,Σk, τk |X) =

N∏

i=1

G∑

k=1

τk fk(xi |µk,Σk) (4.80)

where τk is the probability that an indent was performed on the k-th material phase. The sta-tistically most reliable mixture is obtained for values of τk, µk and Σk at which the likelihood ismaximized. Here, the cluster analysis is performed by means of the expectation-maximizationalgorithm (Dempster et al., 1977) as implemented by Fraley and Raftery (2006). Therefore, theoptimization of LM with respect to τk, µk and Σk is rather preformed through the equivalentmaximization of the complete-data log-likelihood given by

logLC (µk,Σk, τk, zi |X) =N∑

i=1

G∑

k=1

zik log τkfk(xi |µk,Σk) (4.81)

where zi = (zi1, ..., ziG) is defined to attribute the indent xi to a single material phase in themixture:

zik =

1 if xi belongs to phase k,

0 otherwise(4.82)

so that zi follows a multinomial distribution of ones drawn over G categories with probabilitiesτk.

The expectation-maximization algorithm maximizes Eq. (4.81) through estimates zik, µk, Σk

and τk which are conditioned by X. The converged value of each zik is noted z ∗ik and representsthe conditional probability that the i-th indentation was performed on the k-th material phase.Every indented microvolume i is attributed to one and only one m-th material phase such thatm = k | z ∗ik = maxk z

∗ik. The uncertainty of this classification is quantified by 1− z ∗im for each

indent i. The larger 1− z ∗im, the less reliable is the conception that the i-th indentation wasperformed on a homogeneous microvolume of mechanical properties µk.

33

Because the number of phases in the mixture which composes the sample is not prior in-formation, it is among the objectives of cluster analysis to determine G. Therefore, theexpectation-maximization algorithm is invoked to find the maximum likelihood of differentmixtures with prescribed numbers of phases. For each converged mixture model, the BayesianInformation Criterion (BIC) is computed as follows:

BIC ≡ 2 logLM −G(

(N + 2)(N + 1)

2− 1

G

)log(N) (4.83)

where the larger the value of BIC, the greater the posteriori probability that the data conformsto the mixture model, and the stronger the evidence for the existence of G material phases inthe sample.

34

Chapter 5

Results and discussion

5.1 Nanoindentation relaxation tests

Several grids of 5 x 5 and 10 x 10 nanoindentation relaxation tests were realized on theprepared surfaces of the two samples under study with 4 µm lag in horizontal and verticaldirections. The indentation relaxation tests were performed with a CSM UNHT nanoindenterat the Laboratory of Chemo-Mechanical Characterization of Materials’ Microstructures, LavalUniversity. For each test, a maximum penetration depth of 250 nm was held for 600 s with aBerkovich indenter tip. Both loading and unloading were performed at a constant penetrationrate of 50 nm/s. As it is common practice in nanonindenation studies (Constantinides andUlm, 2007; DeJong and Ulm, 2007; Sorelli et al., 2008; Davydov et al., 2011; Jones andGrasley, 2011), the validity of each test was verified with respect to the requirements ofcontact analysis. The contact load-penetration curve P (h) of the loading phase was hencesystematically investigated so that the indentation results which deviate from the scalingrelation P ∝ h2 were discarded (Constantinides and Ulm, 2007). As a result, a total of 337and 297 indents remain which were respectively performed on the samples produced withMapefluid N-200, and IH Plus.

Some representative results of the relative reaction load and penetration depth during theholding phase of indentation are presented in Figs. 5.1 and 5.2. At first glance, although thespread of relative load is a bit larger for the sample with Mapefluid N-200 (see Fig. 5.1), theobserved relaxation kinetic is rather similar from one sample to another. For both samplesunder study, the relaxation curves are satisfactorily fitted by the logarithmic relation given inEq. (4.78).

35

Figure 5.1: Non-dimensional penetration and load relaxation curves measured on the sampleproduced with Mapefluid N-200.

5.2 Cluster analysis based on indentation modulus andhardness

A cluster analysis was done based on the indentation modulus and hardness of the merged dataset to isolate the indentation results which were measured on C–S–H phases (see Section 4).The results of the BIC computed for different numbers of phases (see Fig. 5.3) show thatthe merged data set is most likely conformed by a mixture model of five phases. A scatter-plot is presented in Fig. 5.3 where every tested microvolume, characterized by an indentationmodulus and a hardness, is attributed to either of the five phases of the optimal mixture. Theconfidence ellipsoids which contain 90% of the observations of each cluster are also drawn. Themeans, standard deviations and correlation coefficient of indentation modulus and hardnesswhich define every bivariate Gaussian distribution of the mixture are given in Table 5.1. Eachvolume fraction in Table 5.1 indicates the proportion of indented microvolumes attributed tothe corresponding cluster.

The cumulated relaxation undergone by the end of the holding phase is also plotted in scatter-plots with the indentation modulus and hardness (see Fig. 5.4). For cluster 5, the relativeamount of stress released (i.e. 1− φ) varies irrelevantly of the values of indentation modulus

36

Figure 5.2: Non-dimensional penetration and load relaxation curves measured on the sampleproduced with IH Plus.

Table 5.1: Summary statistics of the cluster analysis based on the indentation modulus andhardness

Cluster Vol. M0 [GPa] H [GPa] ρM0,H

i frac. µM0 σM0 µH σH1 0.40 14.89 4.93 0.428 0.108 0.6252 0.25 22.80 7.01 0.753 0.167 0.3003 0.13 23.36 4.29 1.328 0.368 0.2204 0.17 37.98 9.37 2.329 0.804 0.0315 0.04 92.56 34.71 7.856 2.348 0.347

and hardness. Meanwhile for clusters 1 to 4, the smaller the indentation modulus and hardness,the larger the relative amount of stress released at the end of the holding phase.

In an attempt to associate actual material phases to the clusters identified in Table 5.1, theaverage mechanical properties of cluster 5 are found to correspond neither solely to unhy-drated cement particles nor solely to sand grains or aggregates. Previous indentation studieshave revealed that the mean elastic modulus and hardness of unhydrated cement particles

37

0 2 4 6 8 10

Number of phases, G [1]

−7500

−7000

−6500

−6000

−5500

−5000

BayesianInformationCriterion,BIC[1]

0 20 40 60 80 100 120 140 160 180

Indentation modulus, M0 [GPa]

0

2

4

6

8

10

12

14

Hardness,H[GPa]

a)

b)

Figure 5.3: Summary of the cluster analysis based on indentation modulus and hardness.(a) Scatter-plot of indentation modulus and hardness with classification of indents and 90%confidence ellipsoids. (b) BIC values for different sizes of mixture.

38

vary in ranges of 125 GPa to 145 GPa and 8 GPa to 10.8 GPa, respectively (Velez et al.,2001; Acker, 2001). Similarly, the mean elastic modulus and hardness of quartz aggregatesmeasured by indentation are 73 GPa and 10 GPa, respectively (Acker, 2001). Then, theaverage indentation properties of cluster 5 lies in between the ranges of values observed forunhydrated cement particles and sand grains. Therefore, cluster 5 likely embeds indentationresults sparsely measured on sand grains and unhydrated cement particles. Cluster 5 also doesnot likely contain data from indents performed on portlandite or C–S–H since the least value ofindentation hardness it exhibits is by far greater than the largest values commonly observedfor C–S–H phases (Constantinides, 2006; Vandamme, 2008). Finally, the small amount ofnon-dimensional stress released and its absence of scaling with the indentation modulus andhardness (see Fig. 5.4) corroborate the assertion that cluster 5 is a set of indentation resultsmeasured on unhydrated cement particles and sand grains

In an effort to associate other material phases to clusters in Table 5.1, the mean indentationmodulus of cluster 4 is found to match with the one of portlandite (Acker, 2001; Constantinidesand Ulm, 2004). The volume fraction of cluster 4 which varies from 17% to 18%, whethercluster 5 is considered or not, is also realistic for portlandite in a cement paste with such ahigh water-to-cement ratio as 0.60. Because the mean hardness of cluster 4 is larger than the1.35 GPa previously measured on large crystals of portlandite (Acker, 2001); cluster 4 is, afortiori, unrepresentative of pure C–S–H phases which exhibit even lower hardness (Constan-tinides, 2006; Vandamme, 2008). Although some mixtures of C–S–H and portlandite havebeen identified that have similar values of indentation modulus and hardness as some indentsof cluster 4, this type of nanocomposite was found to arise as a result of local deficiencies ofspace and water which typically occur in cement pastes with low water-to-cement ratio (Chenet al., 2010; Vandamme and Ulm, 2009; Vandamme et al., 2010). Therefore, if cluster 4 maycontain some results of indentation performed on composite microvolumes of C–S–H, theseare believed to represent only a small proportion of the indents.

Meanwhile, as for C–S–H compared to portlandite, sand grains and unhydrated cement parti-cles; the indentation modulus and hardness of clusters 1 to 3 are smaller, and the normalizedamount of stress released is larger than in clusters 4 and 5. Also, the distributions of posteriorprobabilities that an indent belongs to either of the phases of the mixture are useful to assessthe uncertainty of the classification of tested microvolumes. More particularly, the distributionof z ∗i4 + z ∗i5 provides some information about the uncertainty of the classification of indentationresults among either group of clusters 4 and 5 or 1, 2 and 3. The higher the value of z ∗i4 + z ∗i5,the more reliable is the attribution of the i-th microvolume to either of clusters 4 and 5 whilethe less likely it is to belong to cluster 1, 2 or 3. The closer z ∗i4 + z ∗i5 is to 0.50, the moreuncertain is the attribution of the i-th microvolume to either of clusters 4 and 5 instead ofcluster 1, 2 or 3. The histogram presented in Fig. 5.5 is clearly bimodal with a prominentpeak at each extremity of the range of probabilities spanning from 0 to 1. Also, more than

39

a)

b)

0 20 40 60 80 100 120 140

Indentation modulus, M0 [GPa]

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Relaxationat600s,φ(t=600s)[1]

Cluster 1

Cluster 2

Cluster 3

Cluster 4

Cluster 5

0 2 4 6 8 10 12 14

Hardness, H [GPa]

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Relaxationat600s,φ(t=600s)[1]

Cluster 1

Cluster 2

Cluster 3

Cluster 4

Cluster 5

Figure 5.4: Scatter-plots and classifications of (a) indentation modulus and (b) indentationhardness with non-dimensional relaxation at 600 s.

40

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

z ∗i1 + z ∗

i2 + z ∗i3 [1]

0

100

200

300

400

500

Number

ofindents,N

[1]

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

z ∗i4 + z ∗

i5 [1]

Figure 5.5: Histogram of classification uncertainty for the indents attributed to cluster 4 or 5.

98% of the data in clusters 1 to 3 is at least twice more likely to belong to the phase it wasattributed to than to belong to cluster 4 or 5. Therefore, the attribution of indentation resultsto either of the groups of clusters 1, 2 and 3 or 4 and 5 seems reliable.

In order to isolate the indents likely performed on C–S–H from the others, the indentationresults attributed to cluster 4 and 5 were discarded from the data set. For the followingreasons, a good level of confidence is alloted to the accuracy and relevance of the separationof data attributed to clusters 4 and 5 from the results attributed to clusters 1, 2 and 3. First,clusters 1, 2 and 3 have mechanical properties which are representative of C–S–H in general.Second, clusters 4 and 5 have mechanical properties which are representative of unhydratedcement particles, sand grains and portlandite, but certainly not of pure C–S–H. Finally theattribution of indentation results to either of the groups of clusters 1, 2 and 3 or 4 and 5 isreliable. Hence, from now, only the results which were attributed to clusters 1, 2 and 3 areconsidered in the analysis.

41

5.3 Assessment of packing distributions, strength andrelaxation properties of C–S–H

As the indentation results were split back in two data sets accordingly to the sample they weremeasured on, the packing density distribution and the strength and relaxation properties ofhydrates were investigated for each sample under study. The inverse analysis used to back-calculate the distributions of packing density relies on the assumption that the solid phaseof hydrates solely consists of C–S–H. However, C–S–H is certainly not the sole hydrationproduct present in the tested microvolumes. The error made while applying the methodologyof Section 4.2 to a composite microvolume of C–S–H with nanocrystals of portlandite wasestimated by Vandamme and Ulm (2013). Following the same approach, the largest errorarises for the maximum value of indentation modulus herein equal to 38.85 GPa. The back-calculation of the packing density distribution yields a value of 72.7% for the correspondingindent. Meanwhile, if the microvolume consists of low density C–S–H with 32.7% porositypartially filled with portlandite, a similar application of the self-consistent homogenizationscheme as by Chen and Qiao (2011) yields an equivalent packing density of 78.2%. Hence,the assimilation of C–S–H partially filled with nanocrystals of portlandite to pure C–S–H canlead to an underestimation of the packing density as large as 5.5%. For other composites ofLD C–S–H with nanoinclusions which were more compliant than portlandite, the error madeon the estimation of the packing density would be larger.

Figs. 5.6 to 5.9 present scatter-plots of the indentation modulus, indentation hardness, acti-vation volume and characteristic time of relaxation with the back-calculated packing density.Because the solid phase indentation modulus was fixed (see Table 5.2), the indentation mod-ulus scales similarly with packing density for both samples. A fixed common value of ms wasalready used to assess the packing density distributions of hydrates in samples with variouswater-to-cement ratios, curing conditions, amounts of silica fumes and with or without calcare-ous filler (Vandamme et al., 2010; Vandamme and Ulm, 2013). The solid phase indentationmodulus herein prescribed is also close to the one obtained by molecular simulation (Pellenqet al., 2009). The range of posterior packing densities reveals the heterogeneity of the hy-dration product in both samples under study. The maximal packing density in the sampleproduced with IH Plus is 72.4%, while 12% of the tested microvolumes with Mapefluid N-200have larger packing densities spanning up to 79.1%. Therefore, for a given water-to-binderratio, the addition of IH Plus to concrete seems to inhibit the precipitation of C–S–H withhigh packing density.

For a given packing density, the hardness is larger in the sample with IH Plus than in thesample with Mapefluid N-200. Also, the larger is the packing density, the more importantis the difference of hardness. This difference can be formulated in terms of the solid phasecohesion and friction coefficient obtained by inverse analysis (see Table 5.2). As for a Drucker-

42

a) b)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0

5

10

15

20

25

30

35

40

45

Indentationmodulus,M0[GPa]

Mapefluid N-200

IH Plus

Figure 5.6: Scatter-plots and scaling of packing density with the indentation modulus.

b)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0.0

0.5

1.0

1.5

2.0

2.5

Hardness,H[GPa]

Mapefluid N-200

IH Plus

Figure 5.7: Scatter-plots and scaling of packing density with the indentation hardness.

43

c) d

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Activationvolume,V[nm3]

Mapefluid N-200

IH Plus

Figure 5.8: Scatter-plots and scaling of packing density with the activation volume.

d)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

10−4

10−3

10−2

10−1

100

101

Characteristictimeofrelaxation,τ r[s]

Mapefluid N-200

IH Plus

Figure 5.9: Scatter-plots and scaling of packing density with the characteristic time of relax-ation.

44

Table 5.2: Converged strength properties assessed by inverse analysis of the packing densitydistributions

Mapefluid N-200 IH Plusms / GPa 63.5 63.5cs / GPa 0.39 0.51

αs 0.260 0.499

Prager representation of the strength domain, the larger the cohesion, the stronger the cohesivebonds in between solid particles of C–S–H. Meanwhile, an increase of the friction coefficientindicates an improved sensitivity to pressure of the particle-to-particle contact. The cohesionand friction coefficient of the solid phase in the sample with IH Plus are, respectively, larger bymore than 30% and 90% than the ones obtained for the sample with Mapefluid N-200. As thestrength properties assessed by inverse analysis seem roughly stable for the number of indentsperformed (see Figs. 5.10 and 5.11), the results support the idea that the addition of IH Plusto concrete improves the nanoscale strength properties of C–S–H. This is consistent with theobservation of Bolobova and Kondrashchenko (2000) that the addition of bioplasticizer of thesame nature as EM increases the rate of formation of fine crystalline structure and new fullycrystallized structures of hydrated silicate phases.

The activation volume of the tested microvolumes lies in between 0.07 nm3 and 0.5 nm3 forboth samples under study. Some values of activation volume were previously reported to spanfrom 0.01 nm3 to 40 nm3 for concrete Jennings (2004), and up to 15 nm3 for cement paste(Klug and Wittmann, 1974). However, for a macroscopic strain prescribed on a cement-basedmaterial, the relaxation of C–S–H results in a stress redistribution among unhydrated cement,portlandite and aggregates. Therefore, the rate of stress release measured at macroscale isbiased by the presence of non-viscous phases (Acker and Ulm, 2001), and the activation volumeis overestimated compared to what is measured by indentation of C–S–H. Another reason fordiscrepancies between observations is the dependence of the apparent activation volume tothe level of applied stress (Klug and Wittmann, 1969). As the measured activation volumedecreases for larger stresses, the resort to indentation conduces to smaller activation volumesthan uniaxial relaxation experiments.

Fitting a function of the form V/v = a(η − 0.5)b where v = 1 nm3 yields V/v = 0.0247(η −0.5)−1.049 for the sample with Mapefluid N-200, and V/v = 0.0379(η−0.5)−0.685 for the samplewith IH Plus. For a given stress response σ0 in Eq. (4.78), the larger the activation volumethe less stress is released at long-term. Therefore, because C–S–H has shown to creep less athigh packing density (Vandamme and Ulm, 2009, 2013), the observed scaling of the activationvolume with packing density is counterintuitive (see Fig. 5.8). The reason for this confusionis that the activation volume varies as a decreasing function of the applied stress (Klug and

45

0 50 100 150 200 250

Number of indents, N [1]

0.35

0.40

0.45

0.50

0.55

0.60

0.65Cohesion,c s

[GPa]

Mapefluid N-200

IH Plus

Figure 5.10: Convergence of the cohesion of the solid phase of C–S–H obtained by inverseanalysis with an increasing number of indents.

Wittmann, 1969). Such a dependence is likely due to the exhaustion of the rearrangingprocess of C–S–H solid grains as the availability of free volumes is reduced by increasing theloading stress. For a relaxation indentation experiment, the stress undertaken at the onset ofrelaxation increases with the packing density of C–S–H. Hence, denser C–S–H undergo morecompaction during the loading phase leading up to a more important decrease of the apparentactivation volume. Thereby, a more consistent way to assess the rate of long-term relaxationis the mechanical work V σ0 which, in the name of the Cottrell-Stokes law, remains constantirrelevantly of the applied stress (Klug and Wittmann, 1969).

The range of characteristic times of relaxation identified by fitting Eq. (4.78) over the non-dimensional relaxation curves plotted in Figs 5.1 and 5.2 spans from 5 ·10−4 to 5 s. Therefore,the choice of holding the maximum penetration depth of the indenter during 600 seconds isrelevant for the assessment of a long-time behavior characterized by t τr. In Fig. 5.9, thecharacteristic time of relaxation is found to vary irrelevantly of the packing density, hencesuggesting that the duration of the short-term relaxation behavior during indentation is notgoverned by the nanoscale porosity of C–S–H.

In Figs. 5.12 to 5.14, the measured scatters of indentation modulus, indentation hardness and

46

0 50 100 150 200 250

Number of indents, N [1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Frictioncoeffi

cient,αs[1]

Mapefluid N-200

IH Plus

Figure 5.11: Convergence of the friction coefficient of the solid phase of C–S–H obtained byinverse analysis with an increasing number of indents.

activation volume are presented without consideration of the packing density distributions.The difference of scaling between the two samples under study is preserved for the indentationmodulus with respect to the indentation hardness and activation volume. Hence, the effectsof adding IH-Plus to concrete are also revealed after a simple contact analysis of the inden-tation results. Thereby, the differences of scaling in Figs. 5.6 to 5.8 are certainly not a soleconsequence of the eventual convergence toward a local solution of the inverse problem solvedto assess the packing density distributions.

5.4 Cluster analysis of C–S–H phases based on indentationmodulus, hardness and activation volume

A cluster analysis was performed on the remaining indentation results of each sample to as-sess the heterogeneity of hydrates. Both ensembles of remaining indented microvolumes wereassimilated to finite mixtures of C–S–H phases with distinct packing densities. Each clus-ter analysis was performed with respect to the indentation modulus, indentation hardness,and activation volume of relaxation as these parameters show relevant scaling with packingdensity and with respect to each other. Meanwhile, the characteristic time of relaxation was

47

a)

0 5 10 15 20 25 30 35 40 45

Indentation modulus, M0 [GPa]

0.0

0.5

1.0

1.5

2.0

2.5

Hardness,H[GPa]

Mapefluid N-200

IH Plus

Mapefluid N-200

IH Plus

Figure 5.12: Scatter-plots and scaling of the indentation modulus with the indentation hard-ness.

0 5 10 15 20 25 30 35 40 45

Indentation modulus, M0 [GPa]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Activationvolume,V

nm3

Mapefluid N-200

IH Plus

Mapefluid N-200

IH Plus

b)

Figure 5.13: Scatter-plots and scaling of the indentation modulus with the activation volume.

48

0.0 0.5 1.0 1.5 2.0 2.5

Hardness, H [GPa]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Activationvolume,V

nm3

Mapefluid N-200

IH Plus

Mapefluid N-200

IH Plus

c)

Figure 5.14: Scatter-plot and scaling of the indentation hardness with the activation volume.

Table 5.3: Summary statistics of the cluster analysis of C–S–H phases for the sample withMapefluid N-200

Cluster Vol. M0 [GPa] H [GPa] V [nm3] ρM0,H ρM0,V ρV,Hi frac. µM0 σM0 µH σH µV σV1 0.30 15.19 3.39 0.356 0.060 0.249 0.080 0.688 -0.775 -0.7582 0.42 20.24 3.85 0.531 0.091 0.172 0.040 0.535 -0.438 -0.3633 0.28 29.06 4.87 0.895 0.286 0.133 0.033 -0.011 -0.231 -0.443

not considered as it does not consistently scale with packing density. The means, standarddeviations and correlation coefficients which define every multivariate Gaussian distributionof the assessed mixtures are presented in Tables 5.3 and 5.4 for the samples produced withMapefluid N-200 and IH Plus, respectively. The univariate counterparts of the assessed prob-ability density functions are plotted along with values of BIC for both samples (see Figs. 5.15to 5.18). The results of the BIC computed for different numbers of phases show that theindentation results performed on hydrates are most likely conformed by mixture models ofthree and two phases for the samples with Mapefluid N-200 and IH Plus, respectively. As thedifference of BIC between the most and second-most likely numbers of phases lies in between2 and 6 for both samples, there is positive evidence for the mixtures presented in Tables 5.3and 5.4 (Kass and Raftery, 1995; Fraley and Raftery, 1998).

49

Table 5.4: Summary statistics of the cluster analysis of C–S–H phases for the sample withIH Plus

Cluster Vol. M0 [GPa] H [GPa] V [nm3] ρM0,H ρM0,V ρV,Hi frac. µM0 σM0 µH σH µV σV1 0.45 11.72 3.83 0.450 0.139 0.211 0.076 0.868 -0.535 -0.5682 0.55 20.61 4.82 1.079 0.393 0.136 0.035 0.445 -0.280 -0.252

Mapefluid N-200

Cluster 1 ∼Macropores, ITZ

Cluster 2 ∼ LD CSH

Cluster 3 ∼ HD CSH

∼Macropores

Cluster 2 ∼ LD CSH

IH Plus

Cluster 1

a) d)

b) c

0.0 0.5 1.0 1.5 2.0

Hardness, H [GPa]

0

5

10

15

20

25

30

35

40

Probabilitydensities,τkfk(H)[1]

Figure 5.15: Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. Model distributions of indentation hardness.

In order to associate actual C–S–H phases to the clusters identified in Tables 5.3 and 5.4, thescatters of indentation modulus and mechanical work of relaxation with respect to packingdensity are plotted in Fig. 5.19 to 5.22. The estimated 90% confidence ellipsoids are alsoplotted for each cluster. Similarly, the scatter of indentation hardness with respect to packingdensity is presented for both samples in Fig. 5.23. Every point in Figs. 5.19 to 5.23 representsan indented microvolume of hydrates attributed to either of the identified clusters.

The second cluster exhibits a mean indentation modulus of 20.24 GPa and 20.61 GPa for the

50

Mapefluid N-200

Cluster 1 ∼Macropores, ITZ

Cluster 2 ∼ LD CSH

Cluster 3 ∼ HD CSH

∼Macropores

Cluster 2 ∼ LD CSH

IH Plus

Cluster 1

0 5 10 15 20 25 30 35 40 45

Indentation modulus, M0 [GPa]

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018Probabilitydensities,τkfk(M

0)[1]

Figure 5.16: Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. Model distributions of indentation modulus.

samples with Mapefluid N-200 and IH Plus, respectively, while it has a mean packing densityof 65.9% for both samples. Previous nanoindentation studies (Acker, 2001; Constantinidesand Ulm, 2004, 2007; DeJong and Ulm, 2007) revealed that the mean indentation modulusof LD C–S–H approximately spans from 17.1 GPa to 21.7 GPa for an assumed porosity of37% (Ulm et al., 2004). Also, previous investigations of the packing density of LD C–S–H byinverse analysis of indentation results yielded mean values of packing density as low as 62.0%(Vandamme et al., 2010) for an overall mean of 67.3% (Vandamme and Ulm, 2013). Therefore,the second cluster of each sample likely corresponds to LD C–S–H.

The third cluster of the sample produced with Mapefluid N-200 has a mean packing densityof 72.4%, an indentation modulus of 29.06 GPa and an indentation hardness of 0.895 GPa.For HD C–S–H, previously investigated values of mean indentation modulus lie in between28.3 GPa and 32.2 GPa (Acker, 2001; Constantinides and Ulm, 2004, 2007; DeJong and Ulm,2007) for an average indentation hardness spanning from 0.77 GPa (Constantinides and Ulm,2007) to 1.29 GPa (DeJong and Ulm, 2007) and a packing density of 74%±2% (Vandamme

51

Mapefluid N-200

Cluster 1 ∼Macropores, ITZ

Cluster 2 ∼ LD CSH

Cluster 3 ∼ HD CSH

∼Macropores

Cluster 2 ∼ LD CSH

IH Plus

Cluster 1

0.0 0.1 0.2 0.3 0.4 0.5

Activation volume, V [nm3]

0

50

100

150

200

250

Probabilitydensities,τkfk(V)[1]

Figure 5.17: Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. Model distributions of activation volume.

d)

Mapefluid N-200

IH Plus

Figure 5.18: Partial summary results of the cluster analysis based on indentation modulus,hardness and activation volume. BIC values for different sizes of mixtures.

52

and Ulm, 2013). Hence, the third cluster of the sample with Mapefluid N-200 presents asimilar mechanical profile as HD C–S–H. In accordance with the observations of Section 5.3,there is no such phase as HD C–S–H among the remaining indentation results of the sampleproduced with IH Plus.

The most porous cluster has a mean indentation modulus of 15.19 GPa and 11.72 GPa forthe samples with Mapefluid N-200 and IH Plus, respectively. In previous nanonindentationstudies, a domain of indentation modulus spanning up to 13 GPa was assimilated to mi-crovolumes of hydrates with high porosity referred to as macropores (Constantinides, 2006;Constantinides and Ulm, 2007). It was also stated that the pore size distribution in such re-gions jeopardizes the scale separability assumed for the contact analysis and hence rises someconcerns about the accuracy of the measured indentation modulus (Constantinides, 2006;Constantinides and Ulm, 2007). Nevertheless, the assimilation of small values of indentationmodulus to macropores leads up to consistent, yet biased, estimates of capillary porosity incement paste (Constantinides, 2006; Constantinides and Ulm, 2007). Also, there is no alter-native evidence to explain the measurement of small values of indentation modulus but theindentation of highly porous regions. Therefore, the first cluster of the sample with IH Pluslikely corresponds to macropores. As the mean indentation modulus of the first cluster inthe sample with Mapefluid N-200 is larger than the common values associated to macrop-ores (Constantinides, 2006; Constantinides and Ulm, 2007; Sorelli et al., 2008), it is likelyrepresentative of microvolumes with a higher packing density though yet more porous thanbulk cement paste. A possible explanation for the emergence of such values of indentationmodulus is the indentation of ITZ where spatial constraints leads up to porosity gradients inhydration products (Scrivener et al., 2004; Gao et al., 2013). The first cluster of the samplewith IH Plus is therefore referred to as a mix of macropores with ITZ.

While the scaling of the indentation modulus and hardness was investigated in Section 5.3, themechanical work of relaxation V σ0 is found to scale more consistently with packing densitythan the previously investigated volume of activation (see Fig. 5.8). For both samples, V σ0

increases with the packing density. For a given porosity of hydrates, the mechanical work ofrelaxation is larger in the sample produced with IH Plus and the difference increases alongwith the packing density. Hence, the results support the idea that the addition of IH Plus toconcrete decreases the absolute rate of long-term relaxation of C–S–H.

Noteworthy for the sample with Mapefluid N-200, although the cluster analysis was performedwith no constrain whatsoever, the estimated mixture is compatible with the colloidal modelof Jennings (2000). Indeed, both identified C–S–H phases exhibit an indentation modulus anda packing density which are comparable to the ones of previous studies (Constantinides andUlm, 2007; Vandamme et al., 2010). Also, the estimated volume fractions convert into a massratio of HD over LD C–S–H of 66.7% which is consistent with the predictions of Tennis andJennings (2000). As it was pointed out by Lura et al. (2011), previous deconvolution methods

53

a) b

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0

5

10

15

20

25

30

35

40

45

Indentationmodulus,M0[GPa]

Mapefluid N-200

IH Plus

Cluster 1

Cluster 2

Cluster 3

Figure 5.19: Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationmodulus with the packing density of the sample produced with Maplefluid N-200.

b)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Mechanicalworkofrelaxation,Vσ0[J]

×10−19

Mapefluid N-200

IH Plus

Cluster 1

Cluster 2

Cluster 3

Figure 5.20: Scatter-plots, classifications, confidence ellipsoids and scaling of the activationvolume with the packing density of the sample produced with Maplefluid N-200.

54

c) d

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0

5

10

15

20

25

30

35

40

45

Indentationmodulus,M0[GPa]

Mapefluid N-200

IH Plus

Cluster 1

Cluster 2

Figure 5.21: Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationmodulus with the packing density of the sample produced with IH Plus.

d)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Mechanicalworkofrelaxation,Vσ0[J]

×10−19

Mapefluid N-200

IH Plus

Cluster 1

Cluster 2

Figure 5.22: Scatter-plots, classifications, confidence ellipsoids and scaling of the activationvolume with the packing density of the sample produced with IH Plus.

55

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Packing density, η [1]

0.0

0.5

1.0

1.5

2.0

2.5

Hardness,H[GPa]

Mapefluid N-200

IH Plus

Cluster 1

Cluster 2

Cluster 3

Figure 5.23: Scatter-plots, classifications, confidence ellipsoids and scaling of the indentationhardness with packing density.

(Constantinides and Ulm, 2007; Ulm et al., 2010) would typically not converge toward mixtureswhich confirm current models of C–S–H without a constrain on the number of phases. Theconvergence of the expectation-maximization algorithm toward a global maximum likelihoodis not guaranteed neither (McLachlan and Krishnan, 2008; Wu, 1983; Nettleton, 1999). Also,although not illustrated here, the results of the cluster analysis vary significantly in functionof the number of indents performed. Thereby, the modality of the estimated distribution ofC–S–H is subject to caution. However, the uncertainty related to the identification of C–S–Hphases has little if no impact on the findings of this work. Indeed, irrelevantly of the resultsof deconvolution, the comparative analysis supports the idea that the addition of IH Plusto concrete: (i) improves the strength of C–S–H by enhancing the cohesion and friction ofsolid nanograins, (ii) decreases the absolute rate of long-term relaxation, (iii) inhibits theprecipitation of C–S–H of higher density. The underrepresentation of C–S–H of higher densityin the sample with IH Plus may also be the result of a mistaken attribution of some indentationresults to portlandite in Section 5.2 due to an improved hardness of C–S–H.

56

5.5 Comparison with the macroscopic behavior

It is beyond the scope of this study to provide an accurate model of the macroscopic mechanicalbehavior of concrete produced with EM-based bioplasticizer. However, some relevant insightsof the expected and observed macroscale effects and the way they relate to the observationsgathered at microscale are provided here.

If for a given water-to-cement ratio, the precipitation of HD C–S–H is inhibited by the additionof bioplasticizer, the stiffness of the C–S–H matrix should consequently decrease. Whetheror not the stiffness of cement paste and concrete decreases as well depends on the effect ofbioplasticizer on hydration and, more precisely, on the relative proportions of different hydratesand remaining anhydrous cement phases. However, a multitechnique analysis which enablesto crosscheck the identification of the different material phases should be used for a keenunderstanding of this question. Some instances of this type of approach include the works ofChen and Qiao (2011), Trtik et al. (2012), Vanzo (2009) and Vallée et al. (tion).

The amelioration of the cohesion and friction of solid nanograins must result in an improve-ment of the strength of the C–S–H matrix. Thereby, as failure of cement paste and concretenecessarily involves debonding of the C–S–H matrix, the amelioration of nanoscale strengthproperties should ultimately result in an improvement of strength properties at macroscale.This is actually confirmed by the measurements of compressive strength which were performedat the macroscale of the two samples under study (see Table 5.5). Hence, for a water-to-binderratio of 0.60, the compressive strength of concrete is improved by 10.5% and 9.6% at 28 daysand 365 days, respectively. Similarly, an improvement of 25% of the compressive strength ofcement mortar was observed by other authors after the addition of EM-based mixtures (Ghoshet al., 2005).

The long-term creep and relaxation of concrete is attributed to phenomena which occur atnanoscale in C–S–H Ulm et al. (2000) and were formerly investigated by Vandamme and Ulm(2009). Recently, the long-term creep behavior of C–S–H observed by nanoindentation wasfound to be representative of long-term rates at larger scales (Vandamme and Ulm, 2013; Zhanget al., 2013). Therefore, the decrease of the long-term relaxation rate of C–S–H associatedto the measured increase of mechanical work of relaxation (see Section 5.3) should ultimatelytranslate into a similar effect at macroscale. While nanoindentation enables to anticipate suchbehaviors after very short periods of time, the validation of this finding at macroscale willrequire the experimental exhaustion of short-term relaxation which may require decades oftesting procedures (Ulm et al., 2000; Vandamme and Ulm, 2013).

57

Table 5.5: Macroscopic compressive strength

Mapefluid N-200 IH Plusfc at 28 d / GPa 18.58 20.53fc at 365 d / GPa 25.13 27.49

58

Chapter 6

Partial conclusion

For the first time, the mechanical properties of C–S–H in concrete produced with EM-basedbioplasticizer were studied by a statistical nanoindentation technique. The resort to nanoin-dentation allowed to investigate the long-term viscous behavior of C–S–H after few hundredsseconds while equivalent measurements performed at larger scale would require decade-longuniaxial testing procedures. The nanoindentation tests which were configured for relaxation(i.e. at controlled depth) were performed on concrete produced either with commercial SPor with EM-based bioplasticizer. As a result, the addition of EM-based bioplasticizer to con-crete was found to: (i) improve the strength of C–S–H by enhancing the cohesion and frictionof solid nanograins; (ii) decrease the absolute rate of long-term relaxation; (iii) inhibit theprecipitation of C–S–H of higher density. However, the finding that EM-based bioplasticizerinhibits the precipitation of C–S–H of higher density is the result of assertions solely basedon mechanical considerations. A multitechnique investigation which combines chemical andmechanical information will be needed to thoroughly address the question of the distributionof hydrates.

59

Part II

Uncertainty propagation of amultiscale poromechanics-hydrationmodel for poroelastic properties of

cement paste at early-age

61

Résumé

Ce travail présente des résultats originaux sur la propagation d’incertitude et l’analysede sensibilité d’un modèle multi-échelle poromécanique viellissant pour pâtes de cimentavec ratio d’eau sur ciment entre 0.35 et 0.70. L’approche proposée permet d’obtenirles propriétés poroélastiques requises pour la modélisation du comportoment mécaniquepartiellement saturé des pâtes de ciment viellissantes. Les prédictions du seuil de per-colation et du module de Young non drainé reproduisent convenablement les donnéesexpérimentales. Le développement d’un metamodèle stochastique sur la base du chaospolynomial permet de propager l’incertitude des paramètres cinétiques d’hydratation, dela composition du ciment, des modules élastiques et des paramètres morphologiques dela microstructure. Les résultats présentés montrent que la propagation n’amplifie pasl’incertitude vis-à-vis la simple variabilité des propriétés poroélastiques. Toutefois, lescorrelations caclulées entre ces propriétés révèlent de possibles effets d’amplification dansle cadre des équations poroélastiques d’état. Afin de réduire l’incertitude des prédictionsdu seuil de percolation et des propriétés élastiques, les ingénieurs doivent avoir accès à desmesures plus précises de l’énergie d’activation des aluminates de calcium et, plus tard, dumodule élastique des silicates de calcium hydratés de basse densité.

Abstract

This work presents original results on the uncertainty propagation and sensitivity analysisof a multiscale poromechanics-hydration model for cement pastes of water-to-cement ratiobetween 0.35 and 0.70. Notably, the proposed approach provides poroelastic propertiesrequired to model the behavior of partially saturated aging cement pastes (e.g. autoge-nous shrinkage), and it predicts the percolation threshold and undrained elastic modulusin good agreement with experimental data. The development of a stochastic metamodelusing polynomial chaos expansions allows to propagate the uncertainties of kinetic parame-ters of hydration, cement phase composition, elastic moduli and morphological parametersof the microstructure. The presented results show that the propagation does not magnifythe uncertainty of the single poroelastic properties although, their correlation may am-plify the variability of the estimates obtained from poroelastic state equations. In orderto reduce the uncertainty of the predicted percolation threshold and that of the predictedporoelastic properties at early-age, engineers need to assess more accurately the apparentactivation energy of calcium aluminate and, later on, of the elastic modulus of low densitycalcium-sillicate-hydrate.

63

Chapter 7

Partial introduction

In poroelastic media such as concrete materials (Ulm et al., 2004), the development of internalstresses due to drying and shrinkage and thus, the risk of cracking, significantly depend on theevolution of poroelastic properties (Pichler et al., 2009a; Pichler and Dormieux, 2010a,b). Atearly-age, these properties evolve through time as a result of microstructural changes triggeredby the hydration of anhydrous cement particles and other chemical reactions. Thereby, theaccuracy with which one predicts the percolation threshold and the evolution of the elasticmodulus along with the Biot-Willis parameter and the skeleton Biot modulus influences theaccuracy with which one can assess the risk of cracking of concrete structures at early-age. Asof now, some estimates of the poroelastic properties of cement-based materials can be madeby continuum micromechanics using uncertain representative quantities of the behavior andstructure of concrete at different length scales. It is of interest to understand how these uncer-tainties affect the predictions of the poroelastic properties in order to identify the quantitiesthat should be assessed with more accuracy.

Recently, continuum micromechanics models have been used in many successful assessmentsof the effective mechanical properties of hardened cement-based materials (Constantinides,2002; Bernard et al., 2003; Constantinides and Ulm, 2004; Sorelli et al., 2008). The extensionof these models to take into account the poroelastic behavior of concrete materials (Dormieuxet al., 2004; Ulm et al., 2004; Constantinides, 2006) introduced several applications (Pichleret al., 2007; Lin and Meyer, 2008; Grondin et al., 2010; Stefan et al., 2010). Bernard et al.(2003) were the first to model the evolution of the elastic modulus and Poisson’s ratio ofhydrating concrete materials by means of micromechanics. They proposed a hydration modeland described the aging process by an evolution of the relative volumetric proportions ofelementary phases with invariant mechanical properties. While providing good results formid- to late-stages of hydration, this model lacked of precision at very early-age. Later on,Sanahuja et al. (Sanahuja et al., 2007; Sanahuja, 2008) studied the impact of the shape ofsolid hydrates on the prediction of the solid percolation threshold and the elastic modulusof aging cement pastes. However, portlandite and ettringite were not considered among the

65

hydration products contributing to setting in the model.

The input parameters of these deterministic models can be classified into four categories: theinitial phase quantification, the kinetic parameters of hydration, the invariant elastic propertiesand the microstructural/morphological parameters. This information is uncertain and, as aconsequence, the model responses can be considered as random variables. Berveiller et al.(2009) first propagated uncertainty through a multistep micromechanics model of concrete.They predicted the variability of the elastic modulus and Poisson’s ratio of cement paste bypolynomial chaos expansions. The resulting variances were decomposed into Sobol’ indicesrecovered by post-processing the polynomial expansions for much less calculation than whatis required by Monte-Carlo simulation. Later on, Sudret et al. (2010) used polynomial chaosexpansions as stochastic metamodels in order to assess the full randomness of elastic propertiesat different scales of hardened concrete and perform sensitivity analyses from scale to scale.However, a sensitivity analysis of the poroelastic properties of cement paste over time has notyet been accomplished.

The purpose of this work is threefold. First, we want to provide a deterministic multiscaleporomechanics-hydration model to predict the evolution of the Biot-Willis parameter, theskeleton Biot modulus, the Poisson’s ratio, the elastic modulus and the percolation threshold.Second, we wish to improve the agreement between the prediction of the elastic modulus andexperimental data by considering the effect of the shape of ettringite and portlandite particleson setting. Third, we want to assess the uncertainty of these predictions at different stages ofhydration and identify their greatest contributors among the input parameters. The originalityof this work relies on the application of a non-intrusive method of uncertainty propagation to amicroporomechanics-hydration model for cement pastes in order to assess the contribution ofthe material parameters to the variability of the macroscopic poroelastic properties over time.For the first time, this work presents the second central moment of the Biot-Willis parameterand the Biot skeleton modulus as functions of time.

The uncertainty propagation is performed by polynomial chaos expansion of the random modelresponses Soize and Ghanem (2004). This approach has the advantage to give access tosensitivity information and to allow the computation of large samples of predictions for muchless calculation than required by Monte-Carlo methods.

The remaining of the document is organized as follows. First, the materials studied are pre-sented. Second, some elementary state equations of microporomechanics are derived from thesolution to the generalized problem of Eshelby. Third, a deterministic multiscale poromechanics-hydration model is introduced. Fourth, the polynomial chaos expansion and the post-processingare explained. Fifth, the uncertainties of the input parameters are described. Sixth, the modelis validated, the uncertainty is propagated, a sensitivity analysis is performed, the correlationbetween the poroelastic properties is investigated and the PDF of the elastic modulus is esti-

66

mated at different timesteps.

67

Chapter 8

Materials

This work focuses on three cement pastes with different water-to-cement ratio (w/c) of 0.40,0.50 and 0.60. The cement composition is an important factor that influences the chemistryof hydration and the properties of cement paste. The cement composition of this study istaken from the experimental study of Boumiz et al. (Boumiz et al., 1996, 2000) on the elasticproperties at early-age of cement-based materials. Two approaches can be used to describe thecement composition: the oxide composition and the chemical phase composition. The majoroxides composition of this cement is presented in Table 8.1.

To model hydration, one needs to assess the phase composition. Cements are made of fourmajor grinded clinker phases that react differently with water: tricalcium silicate, dicalciumsilicate, tricalcium aluminate (aluminate) and tetracalcium aluminoferrite (ferrite). Accord-ing to cement chemistry those are refered to as C3S, C2S, C3A and C4AF where C = CaO,A = Al2O3, F = Fe2O3, S = SO3 and H = H2O so that C3S = 3CaO · SiO2 and so on. Anamount of gypsum ((CSH2) about 5% of the total mass of the system is usually added tothese phases by cement producers.

Several methods can be used to determine the phase composition using the stoichiometry ofthose compounds and the respective amounts of the former oxides. The approach adoptedin this study is a widely used method developped by Bogue (1929). It consists in a linear

Table 8.1: Major oxides composition of cement PCCB9402 (Boumiz et al., 1996)

Oxyde Mass fraction [%]CaO 66.23SiO2 21.68Fe2O3 0.29Al2O3 4.17SO3 3.54

69

system of equations based on a quantitative oxide composition (see Table 8.1) under certainassumptions:

• the main phases of the cement are C3S, C2S, C3A and C4AF, to which are added somegypsum and free lime;

• all the Fe2O3 present in the system occurs as C4AF;

• the remaining amount of Al2O3 occurs as C3A;

• the CaO occurs either as C3S, C2S, free lime or gypsum.

The mathematical formulation of the Bogue calculation is well documented and can be foundin the book of Taylor (1990). We assume an amount of free lime equal to 1g/100g of cement.The amount of calcium oxide that occurs as gypsum is fixed to 70% the mass of SO3.

70

Chapter 9

From the general inclusion problem ofEshelby to microporomechanics

We introduce here the generalized inclusion problem of Eshelby as a fundamental solutioninvoked later on for the derivation of a multiscale microporomechanics model. The solution ofthis problem is used to develop common estimates of macroscopic average states and mechan-ical properties of heterogeneous solid media. Finally, the resulting homogenization schemesare applied to the derivation of common average state equations of microporomechanics.

9.1 Generalized inclusion problem of Eshelby

We consider an elastic matrix V0 subjected to some remote uniform strain ε∞ so that it admitsthe displacement field u(x) = ε∞ · x ∀ x s.t. ||x|| → ∞ as a boundary condition on ∂V∞. Atthe center of the matrix lies an ellipsoidal inhomogeneity Vr perfectly bounded by V0 at theboundary ∂Vr with outward unit normal nr0:

Vr ≡ x s.t. ||x · Z|| < 1 (9.1)

where, for now, Z =∑nd

d=1 1/λd zd⊗zd with λd referring to the principal radius of the ellipsoid

of dimension nd in direction zd. Both the matrix and the inhomogeneity undergo some uniformeigenstrain noted µ0 and µr, respectively. The system of interest is presented in Fig. 9.1.

The resulting strain field, which is heterogeneous, is given by:

ε(x) =

S0 : σ(x) + µ0 ∀ x ∈ V0

Sr : σ(x) + µr ∀ x ∈ Vr(9.2)

where S0 and Sr are the elastic compliance tensors of the matrix and the inhomogeneity,respectively. Our intent is to express the strain field within the inhomogeneity as a lineartransformation of the prescribed eigenstrains and remote strain field. To do so, we recast the

71

Figure 9.1: 2D heterogeneous representation of the Eshelby equivalent eigenstrain problemwith inelastic deformations.

strain field in the following form:

ε(x) = S0 : σ(x) + µ0 + ∆µ(x) ∀ x ∈ V (9.3)

with

∆µ(x) =

0 ∀ x ∈ V0

µr + µeq,r − µ0 ∀ x ∈ Vr. (9.4)

Note that Eqs. (9.2) to (9.4) imply(Sr − S0

): σ(x) = µeq,r ∀ x ∈ Vr so that the stress and

strain field are assumed uniform within the inhomogeneity. Now, if we isolate the stress inEq. (9.3):

σ(x) = C0 :[ε(x)− µ0 −∆µ(x)

]∀ x ∈ V, (9.5)

the statement of quasi-static equilibrium takes the following form:

∇ ·[C0 : ε(x)

]−∇ ·

[C0 : ∆µ(x)

]= 0 ∀ x ∈ V . (9.6)

Thus, given that ∆µ(x) is a step function from Vr to V0, we have:

∇ ·[C0 : ∇u(x)

]+ δ (∂Vr)C0 : ∆µr · nr0 = 0 ∀ x ∈ V, (9.7)

where δ (∂Vr) is the Dirac distribution of ∂Vr and ∆µr = µr + µeq,r − µ0.

Thereby, the elastic problem of the infinite medium with an inhomogeneity as described inFig. 9.1 can be equivalently expressed as the problem of a homogeneous medium with fictitious

72

body force distributed along ∂Vr, see Fig 9.2:

∇ ·[C0 : ∇u(x)

]+ feq(x) = 0 ∀ x ∈ V, (9.8)

wherefeq(x) = −∇ ·

[C0 : ∆µ(x)

]= δ (∂Vr)C0 : ∆µr · nr0 . (9.9)

Now that the inhomogeneity is accounted for by the forcing term feq, the displacement andstrain fields in V can be solved using superposition. If no body force is applied, the resultingdisplacement is such that strain is uniform in the whole domain and is given by ε∞. The totalstrain field is then recast as follows:

ε(x) = ε∞ + εd,∞(x) (9.10)

where εd,∞(x) is derived from the particular solution ud,∞(x) of the Navier equation, i.e.Eq. (9.8).

Then we introduce the Fourier transforms ud,∞(ξ) and ∆µ(ξ):

ud,∞(ξ) =1

(2π)3

∫

R3

ud,∞(x) exp(−ix · ξ

)dVx and ∆µ(ξ) =

1

(2π)3

∫

R3

∆µ(x) exp(−ix · ξ

)dVx

(9.11)of which the inverse transforms are obtained by:

ud,∞(x) =

∫

R3

ud,∞(ξ) exp(ix · ξ

)dVξ and ∆µ(x) =

∫

R3

∆µ(ξ) exp(ix · ξ

)dVξ . (9.12)

Then, when substituting Eq. (9.12) in Eq. (9.6), we obtain:

∇·C0 : ∇

[∫

R3

ud,∞(ξ) exp(ix · ξ

)dVξ

]−∇·

C0 :

[∫

R3

∆µ(ξ) exp(ix · ξ

)dVξ

]= 0 ∀ x ∈ V

(9.13)where differentiation is operated with respect to x so that, given the minor symmetry of C0,we have:

−∫

R3

[ξ · C0 · ξ

]· ud,∞(ξ) + i

[ξ · C0 : ∆µ(ξ)

]exp

(ix · ξ

)dVξ = 0 ∀ x ∈ V . (9.14)

Thus the Navier equation has the following particular solution in terms of Fourier transforms:

ud,∞(ξ) = −i(ξ · C0 · ξ

)−1 ·[ξ · C0 : ∆µ(ξ)

]. (9.15)

Then, using Eq. (9.12), we obtain:

ud,∞(x) = −i∫

R3

(ξ · C0 · ξ

)−1 ·[ξ · C0 : ∆µ(ξ)

]exp

(ix · ξ

)dVξ (9.16)

73

Figure 9.2: 2D homogeneous representation of the Eshelby equivalent eigenstrain problemwith inelastic deformations.

which can be recast in:

ud,∞(x) = −i∫

R3

(ξ · C0 · ξ

)−1 ·[ξ · C0 :

1

(2π)3

∫

R3

∆µ(x′) exp(−ix′ · ξ

)dVx′

]exp

(ix · ξ

)dVξ

(9.17)or, similarly:

ud,∞(x) = − i

(2π)3

∫

R3

∫

R3

(ξ · C0 · ξ

)−1 ⊗ ξ : C0 : ∆µ(x′) exp[i(x− x′

)· ξ]

dVx′dVξ

(9.18)where use was made again of the minor symmetry of C0. The corresponding strain field is:

εd,∞(x) =1

(2π)3

∫

R3

∫

R3

ξs⊗(ξ · C0 · ξ

)−1 ⊗ ξ : C0 : ∆µ(x′) exp[i(x− x′

)· ξ]

dVx′dVξ .

(9.19)Then, given the definition of ∆µ(x), see Eq. (9.4), the particular solution reduces to:

εd,∞(x) =1

(2π)3

∫

R3

∫

Vr

ξs⊗(ξ · C0 · ξ

)−1 ⊗ ξ : C0 : ∆µr exp[i(x− x′

)· ξ]

dVx′dVξ

(9.20)which is valid only for x in Vr, and is equivalently given by:

εd,∞(x) =

[1

(2π)3

∫

R3

ξs⊗(ξ · C0 · ξ

)−1 s⊗ ξ

∫

Vr

exp[i(x− x′

)· ξ]

dVx′

dVξ

]: C0 : ∆µr .

(9.21)The second symmetrized tensor product in the above expression is introduced thanks to theminor symmetry of C0 with which the term in square brackets is contracted. As a result, it

74

is obvious that the strain within the domain of the inhomogeneity depends on the stiffnessof the matrix as well as on the geometry of Vr. Eq. (9.21) can be further simplified usingRadon transformation operations, see for instance Li and Wang (2008), such that the followingexpression is reached:

εd,∞(x) = P(Z,C0) : C0 : ∆µr ∀ x ∈ Vr (9.22)

where the polarization tensor P is given by the following surface integral, see Laws (1977):

P(Z,C0) =detZ

4π

∫

||ξ||=1

ξs⊗(ξ · C0 · ξ

)−1 s⊗ ξ

||Z · ξ||3 dSξ (9.23)

so that, indeed, strain is uniform within the inhomogeneity. The tensor P exhibits bothminor and major symmetry so that it has 21 independent components in R3. Moreover, thepolarization in Vr is size-independent, i.e. P(αZ,C0) = P(Z,C0). Also, the strain withinVr clearly does not depend on the mechanical properties of the inhomogeneity. The interiorEshelby tensor is then defined as follows:

S = S(Z,C0) ≡ P(Z,C0) : C0 (9.24)

such that, for a homogeneous remote displacement boundary condition, the strain within theinhomogeneity relates linearly to ∆µr through the interior Eshelby tensor S. In contrast to thepolarization tensor, S has only minor symmetry so that a maximum of 36 of its componentscan be independent from each other in R3.

Now, the stress field σ is isolated in Eq. (9.2) and the resulting expression is set equal toEq. (9.5):

Cr : [ε(x)− µr] = C0 :[ε(x)− µ0 −∆µr

]∀ x ∈ Vr . (9.25)

The equation above can be recast in:

Cr : ε(x)− Cr : µr = C0 : ε(x)− C0 : µ0 − C0 : S−1 : εd,∞(x) ∀ x ∈ Vr (9.26)

or, equivalently:

Cr : ε(x)− Cr : µr = C0 : ε(x)− C0 : µ0 − C0 : S−1 : [ε(x)− ε∞] ∀ x ∈ Vr (9.27)

so that:

ε(x) =(Cr − C0

)−1:[Cr : µr − C0 : µ0 − C0 : S−1 : ε(x) + C0 : S−1 : ε∞

]∀ x ∈ Vr .

(9.28)By factorizing the expression above with respect to ε(x), we obtain:[I +

(Cr − C0

)−1: P−1

]: ε(x) =

(Cr − C0

)−1:(Cr : µr − C0 : µ0 + P−1 : ε∞

)∀ x ∈ Vr

(9.29)

75

that we recast in:

ε(x) =[I +

(Cr − C0

)−1: P−1

]−1:(Cr − C0

)−1: P−1 :

[P :

(Cr : µr − C0 : µ0

)+ ε∞

]∀ x ∈ Vr

(9.30)and in:

ε(x) =[P :

(Cr − C0

)+ I]−1

:[P :

(Cr : µr − C0 : µ0

)+ ε∞

]∀ x ∈ Vr . (9.31)

Finally, the strain within the inhomogeneity is given by:

ε(x) = A∞ :[ε∞ + P :

(Cr : µr − C0 : µ0

)]∀ x ∈ Vr (9.32)

with the strain concentration tensor A∞ defined as:

A∞ ≡[P :

(Cr − C0

)+ I]−1

. (9.33)

As assumed and verified earlier, the strain is uniform within the inhomogeneity so that inte-gration of (9.32) lead up to:

〈ε〉r = A∞ :[ε∞ + P :

(Cr : µr − C0 : µ0

)]∀ x ∈ Vr (9.34)

where 〈ε〉r = 1||Vr||

∫Vr ε(x)dV is a volume average calculated over Vr. Note that Eq. (9.32)

was equivalently given as follows in Zaoui (2002):

ε(x) = A∞ :[ε∞ − P :

(πr − π0

)]∀ x ∈ Vr (9.35)

where πr = −Cr : µr and π0 = −C0 : µ0 are eigenstresses which arise when inelastic defor-mations are kinematically constrained within the inhomogeneity and the matrix, respectively.In case of vanishing inelastic deformation, i.e. µr = µ0 = 0, Eq. (9.32) yields the famousresult of Eshelby (1957).

9.2 Homogenization scheme of Mori and Tanaka

9.2.1 Mean field theory assumptions

In Section 9.1, some elements of solution of the boundary-value problem of interest were givenfor the case of a single inhomogeneity lying in an infinitely large domain. For all practicalpurposes, a solution is also needed for the case of multiple inhomogeneities embedded in amatrix of finite dimensions.

Given the additive decomposition of the strain field considered in Eq. (9.10), the uniforminfinitely remote boundary condition is satisfied if

lim||x||→∞

εd,∞(x) = 0 . (9.36)

76

Also, it was demonstrated that εd,∞(x) is constant within the inhomogeneity. Although theclosed form of this disturbance strain is not resolved here for within the surrounding matrix,we know that εd,∞(x) does not vanish for some arbitrary points in V0 which are far away from∂V∞, see Li and Wang (2008). Thereby, for a finite domain of arbitrary boundary ∂V, anadditional strain field component εim(x) must be added as follows:

ε(x) = ε∞ + εd,∞(x) + εim(x) (9.37)

so that an equivalent uniform boundary condition is met at the limit of the finite domain:

u(x) = ε∞ · x ∀ x ∈ ∂V . (9.38)

Meanwhile, the image strain εim(x) should be such that u(x) remains a solution to the Eshelbyequivalent Navier equation stated in Eqs. (9.8) and (9.9).

Solving for εim(x) is complicated and beyond the scope of this work. Instead, we resort tomean field theory as proposed by Mori and Tanaka (1973). To do so, we consider a domainpopulated by multiple inhomogeneities so that the strain field at any point within V can berecast in the form:

ε(x) = 〈ε〉0 + εd(x) (9.39)

where 〈ε〉0 = 1||V0||

∫V0 ε(x)dV is referred to as the mean field. The mean field includes both

boundary effects and effects of interactions between inhomogeneities. The disturbance εd(x)

from the mean field is unknown.

We then consider the averaged finite domain subjected to a uniform strain 〈εold〉0 and intowhich an inhomogeneity is added so that the average strain becomes

〈εnew〉0 = 〈εold〉0 + 〈εd,∞(x) + εim(x)〉0 ∀ x ∈ V0 (9.40)

where 〈εnew〉0 is the averaged strain in the matrix after addition of the inhomogeneity. Ifthe number of inclusions within the averaged matrix is large, the addition of a single inhome-geneity may not affect the mean strain field in the matrix so that 〈εd,∞(x)〉 is negligible.Similarly, Mori and Tanaka argue that if the number of inhomogeneities is large, the effect ofthe additional image strain onto the mean field strain is also negligible so that:

〈εnew〉0 = 〈εold〉0 = 〈ε〉0 ∀ x ∈ V0 . (9.41)

Within the inhomogeneity, the average strain field may be written as

〈ε〉r = 〈ε〉0 + 〈εd,∞(x) + εim(x)〉r (9.42)

where the effect of the image strain is also negligible so that we end up having

〈ε〉r = 〈ε〉0 + 〈εd,∞〉r . (9.43)

77

Eq. (9.43) is consistent with the former additive decomposition given in Eq. (9.10) only if

ε∞ = 〈ε〉0 . (9.44)

Thereby, the uniform strain field ε∞ at the boundary of the domain is an auxiliary quantity;it is not truly an independent variable of the problem. Thus, following these mean fieldassumptions, Eq. (9.35) is recast as follows:

〈ε〉r = Ar,∞ :[〈ε〉0 − Pr :

(πr − π0

)]∀ x ∈ Vr (9.45)

where Vr is the domain of an inhomogeneity for r = 1, . . . , n, with n being the number offamilies of inhomogeneities in the medium. Similarly, a strain concentration tensor

Ar,∞ ≡[Pr :

(Cr − C0

)+ I]−1

. (9.46)

is prescribed for each familiy of inhomogeneities with a polarization tensor Pr given by

Pr ≡ P(Zr,C0) =detZr

4π

∫

||ξ||=1

ξs⊗(ξ · C0 · ξ

)−1 s⊗ ξ

||Zr · ξ||3dSξ (9.47)

Zr is the shape tensor of all inhomogeneities from the r-th family.

9.2.2 Homogenized stiffness tensor

For practical reasons, it is important to understand how a prescribed mean strain 〈ε〉 andsome eigenstresses π0 and πr affect the average stress 〈σ〉 within the domain. In other words,we are interested in having a macroscopic constitutive equation for the system of interest.For sure, part of this equation will involve a homogenized stiffness tensor Chom which linearlyrelates 〈ε〉 to 〈σ〉.

In order to solve for the homogenized stiffness tensor, we first average the strain field over thedomain V:

〈ε〉 = f0〈ε〉0 +

n∑

r=1

fr〈ε〉r (9.48)

where f0 and fr are volume fractions of the matrix and families of inhomogeneities respectivelygiven by 1

||V||∫V0 dV and 1

||V||∫Vr dV . The average strain field 〈ε〉r is then substituted in the

equation above by Eq. (9.45) so that we have:

〈ε〉 = f0〈ε〉0 +

n∑

r=1

frAr,∞ :[〈ε〉0 − Pr :

(πr − π0

)](9.49)

which can be recast in

〈ε〉 =

[f0I +

n∑

r=1

frAr,∞]

: 〈ε〉0 −n∑

r=1

frAr,∞ : Pr :(πr − π0

). (9.50)

78

The average strain field within the surrounding matrix for some prescribed mean strain andeigenstresses is then given by:

〈ε〉0 = A : 〈ε〉+ A :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]

(9.51)

where the tensor A is introduced to simplify further developments:

A =

[f0I +

n∑

r=1

frAr,∞]−1

. (9.52)

Note that, equivalently, we have

f0A +

n∑

r=1

frAr,∞ : A = I. (9.53)

Then, by introducing Eq. (9.51) in Eq. (9.45), we find that the average strain within a familyk of inhomogeneities is given by:

〈ε〉k = Ak : 〈ε〉+ Ak :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]− Ak,∞ : Pk : (πk − π0) (9.54)

where Ak is referred to as the weighted concentration tensor of the k-th family of inhomo-geneities:

Ak = Ak,∞ : A = Ak,∞ :

[f0I +

n∑

r=1

frAr,∞]−1

. (9.55)

Once the average strain is resolved in every subset of the domain, these expressions can beintroduced into the following equation for stress averaging over the domain:

〈σ〉 = f0〈σ〉0 +n∑

k=1

fk〈σ〉k (9.56)

where every local stress field consists of elastic and inelastic contributions expressed as follows:

〈σ〉 = f0

[C0 : 〈ε〉0 + π0

]+

n∑

k=1

fk

[Ck : 〈ε〉k + πk

]. (9.57)

Then, by introducing Eqs. (9.51) and (9.54) in the equation above, we obtain:

〈σ〉 = f0

[C0 :

A : 〈ε〉+ A :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]

+ π0

]

+

n∑

k=1

fk

[Ck :

Ak : 〈ε〉+ Ak :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]− Ak,∞ : Pk : (πk − π0)

+ πk

]

(9.58)

79

which can be recast in:

〈σ〉 =

[f0C0 +

n∑

r=1

frCr : Ar,∞]

: A : 〈ε〉

+

[f0C0 +

n∑

r=1

frCr : Ar,∞]

: A :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]

+ f0π0 +

n∑

r=1

frπr −

n∑

r=1

frCr : Ar,∞ : Pr :(πr − π0

)

(9.59)

and in:

〈σ〉 =Chom : 〈ε〉+ Chom :

[n∑

r=1

frAr,∞ : Pr :(πr − π0

)]

+ f0π0 +

n∑

r=1

frπr

−n∑

r=1

frCr : Ar,∞ : Pr :(πr − π0

)(9.60)

where the homogenized stiffness tensor is given by:

Chom =

[f0C0 +

n∑

r=1

frCr : Ar,∞]

: A =

[f0C0 +

n∑

r=1

frCr : Ar,∞]

:

[f0I +

n∑

r=1

frAr,∞]−1

.

(9.61)Finally, Eq. (9.60) can be recast as follows:

〈σ〉 = Chom : 〈ε〉+n∑

r=1

fr(Chom − Cr) : Ar,∞ : Pr :(πr − π0

)+ f0π

0 +n∑

r=1

frπr . (9.62)

Note that Eq. (9.62) can be further simplified by introducing influence tensors which explicitlyrender the effect of eigenstrains within the constitutive equation, see Pichler (2010).

9.2.3 Levin’s theorem

From Eq. (9.62), it is clear that the average stress of the system depends linearly on theprescribed average strain 〈ε〉 and the set of uniform eigenstresses π0 and πr. Thereby, theaverage stress response admits the following decomposition:

〈σ〉 = 〈σ′〉+ 〈σ′′〉 (9.63)

where σ′(x) is the local stress field of the system subjected to a compatible strain field ε′(x)

with volume average 〈ε′〉 = 〈ε〉 and zero eigenstresses π0 = πr = 0. On the other hand, σ′′(x)

is the local stress field of the system subjected to a compatible strain field ε′′(x) with volumeaverage 〈ε′′〉 = 0 and arbitrary uniform eigenstresses π0 and πr.

As long as both σ′(x) and σ′′(x) result from uniform boundary conditions and satisfy localequilibrium within the domain, the Hill lemma (see Dormieux (2005)) applies as follows:

〈σ′′(x) : ε′(x)〉 = 〈σ′′〉 : 〈ε′〉 . (9.64)

80

For the loading conditions described above, we have:

〈σ′′(x) : ε′(x)〉 = 〈σ′′〉 : 〈ε〉 (9.65)

that we recast in:〈[C(x) : ε′′(x) + π′′(x)

]: ε′(x)〉 = 〈σ′′〉 : 〈ε〉 . (9.66)

Considering the absence of eigenstresses for the first loading case, we have ε′(x) = S(x) : σ′(x)

so that:〈ε′′(x) : σ′(x)〉+ 〈π′′(x) : ε′(x)〉 = 〈σ′′〉 : 〈ε〉 . (9.67)

Now, a second application of the Hill lemma yields the following expression:

〈ε′′(x) : σ′(x)〉 = 〈ε′′〉 : 〈σ′〉 (9.68)

where it was mentioned before that 〈ε′′〉 = 0 so that Eq. (9.67) simplifies to:

〈π′′(x) : ε′(x)〉 = 〈σ′′〉 : 〈ε〉 . (9.69)

Then knowing that π′′(x) is piecewise uniform, the equation above is recast in:

f0π0 : 〈ε′〉0 +

n∑

r=1

frπr : 〈ε′〉r = 〈σ′′〉 : 〈ε〉 (9.70)

in which we introduce relevant parts of Eqs. (9.51) and (9.54) so that:

f0π0 : A : 〈ε〉+

n∑

r=1

frπr : Ar : 〈ε〉 = 〈σ′′〉 : 〈ε〉 (9.71)

and thus:

〈σ′′〉 = f0π0 : A +

n∑

r=1

frπr : Ar . (9.72)

Meanwhile, we know from Eq. (9.62) that the mean stress is as follows for the first loadingcase:

〈σ′〉 = Chom : 〈ε〉 (9.73)

so that introduction of Eqs. (9.72) and (9.73) in Eq. (9.63) leads up to:

〈σ〉 = Chom : 〈ε〉+ f0π0 : A +

n∑

r=1

frπr : Ar . (9.74)

which is referred to as the Levin’s theorem, see Dormieux (2005).

81

9.3 Self-consistent homogenization scheme

In the previous Section, we considered the case of heterogeneous media which consist of a solidmatrix with embedded inhomogeneities. As a result, the homogenization scheme of Mori andTanaka is not appropriate for the case in which the morphology of the medium is such thatno material phase can clearly be identified as a matrix in which the other phases lie. For suchcases, we rather consider a self-consistent approach meaning that the matrix V0 is referred toas a fictitious reference medium of stiffness Chom subjected to some average stress, strain andeigenstress states 〈σ〉, 〈ε〉 and πhom, respectively. Consequently, the application of mean fieldassumptions yields an auxiliary remote strain states ε∞ = 〈ε〉. The volume fractions of the nfamilies of inhomogeneities constituting the medium are such that

n∑

r=1

fr = 1. (9.75)

Similarly to Eq. (9.45), we obtain

〈ε〉r = Ar,∞ :[〈ε〉 − Pr :

(πr − πhom

)]∀ x ∈ Vr (9.76)

where the strain concentration tensor is

Ar,∞ =[Pr :

(Cr − Chom

)+ I]−1

. (9.77)

The macroscopic average strain relates as follows to the local averages

〈ε〉 =n∑

r=1

fr〈ε〉r (9.78)

so that we have

〈ε〉 =

n∑

r=1

frAr,∞ :[〈ε〉 − Pr :

(πr − πhom

)](9.79)

which must also hold when no eigenstresses are prescribed. Thereby, because the strain con-centration tensors are independent of the prescribed set of eigenstresses, we have

I =n∑

r=1

frAr,∞ (9.80)

which leads up to the following implicit expression for the effective homogenized stiffness:

I =

n∑

r=1

fr

[Pr :

(Cr − Chom

)+ I]−1

. (9.81)

The application of Levin’s theorem for the self-consistent scheme leads up to the followingexpression of the macroscopic average stress:

〈σ〉 =Chom : 〈ε〉+n∑

r=1

frπr : Ar,∞ (9.82)

82

in which we note the absence of the effective eigenstress πhom. We solve for the averageeigenstress by reformulating Eq. (9.79) using Eq. (9.80). We obtain

πhom =

[n∑

r=1

frAr,∞ : Pr]−1

:

(n∑

k=1

fkAk,∞ : Pk : πk

). (9.83)

9.4 Applications to microporomechanics

As seen in the previous Sections, the solution to the generalized inclusion problem of Eshelbycan be applied to derive estimates of the effective macroscopic state of a heterogeneous mediumsubjected to some prescribed average strain (or stress) and eigenstrains (or eigenstresses). Anapplication of interest is the case in which one of the phases of the heterogeneous mediumconsists of saturated uniformly pressurized pores. We refer to the treatment of this problem bymeans of the results presented in the previous Sections as microporomechanics, see Dormieuxet al. (2006a). Here, we present elementary results of microporomechanics for the followingsystems: homogeneous porous matrices, granular porous materials with homogeneous skele-ton,and granular porous materials with heterogeneous skeleton.

9.4.1 Case of a homogeneous porous matrix

The homogenized porous matrix consists of two phases: a solid matrix Vs of stiffness Cs anda compliant porous network subjected to a uniform compressive pressure p, i.e. an eigenstressπp = −p1. The volume fraction of the porous phase is noted φ.

Appropriate substitutions in Eq. (9.46) leads up to the following expression of the strainconcentration tensor on the porous network:

Ap,∞ = [I− Pp : Cs]−1 = [I− Sp]−1 (9.84)

auxh that〈ε〉p = Ap,∞ : [〈ε〉p + Pp : 1p] (9.85)

in which the interior Eshelby tensor is given by Sp = Pp : Cs with the following polarizationtensor

Pp ≡ P(Zp,Cs) =detZp

4π

∫

||ξ||=1

ξs⊗(ξ · Cs · ξ

)−1 s⊗ ξ

||Zp · ξ||3dSξ (9.86)

where Zp is the shape tensor of the pores. Consequently, Eq. (9.52) becomes

A = [(1− φ)I + φAp,∞]−1 =[(1− φ)I + φ (I− Sp)−1

]−1. (9.87)

The effective stiffness of the porous matrix is

Chom = (1− φ)Cs : A. (9.88)

83

Note that Chom is the elastic stiffness of the medium when no pore pressure is applied. Therebyit is the drained elastic stiffness of the medium.

The expression for the averaged stress obtained by application of the Levin’s theorem (seeEq. (9.74)) is

〈σ〉 = Chom : 〈ε〉 − φp1 : Ap,∞ : Ap (9.89)

where we considered that no eigenstress is directly supported by the solid matrix, i.e. πs = 0.

Another quantity of interest to describe the state of the porous medium is the change ofporosity observed after application of a given pressure p and strain state 〈ε〉. To obtain arelation between these quantities, we express the average strain within the pores as followsusing Eq. (9.54):

〈ε〉p = Ap,∞ : A : 〈ε〉+ [I− φAp,∞ : A] : Ap,∞ : Pp : 1p. (9.90)

The porosity change φ− φ0 is nothing but the volume change of the porous network given by

φ− φ0 ≡ φ1 : 〈ε〉p = φ1 : Ap,∞ : A : 〈ε〉+ φ1 : [I− φAp,∞ : A] : Ap,∞ : Po : 1p. (9.91)

Introducing the specific form of the strain concentration tensor of the porous network withinthe above equations leads up to the following expression of the drained effective stiffness:

Chom = (1− φ)Cs : A = (1− φ)Cs :[(1− φ)I + φ(I− Sp)−1

]−1. (9.92)

We also obtain the following expression for the average macroscopic stress state:

〈σ〉 = Chom : 〈ε〉 −Bp (9.93)

where the Biot tensor B is given by

B = φ1 : Ap,∞ : A = φ1 : [I− Sp]−1 :[(1− φ)I + φ (I− Sp)−1

]−1(9.94)

with[I− Sp]−1 :

[(1− φ)I + φ (I− Sp)−1

]−1= [I− (1− φ)Sp]−1 (9.95)

so thatB = φ1 : [I− (1− φ)Sp]−1 . (9.96)

The Biot tensor transforms linearly the applied pore pressure into a contribution to the macro-scopic average stress of the porous medium.

Similarly, the expression for the change of porosity becomes

φ− φ0 = B : 〈ε〉+p

N(9.97)

where N is the skeleton Biot modulus given by1

N= φ1 : [I− φAp,∞ : A] : Ap,∞ : Pp : 1 (9.98)

so that1

N= φ(1−B) : (I− Sp)−1 : Sp : Ss : 1. (9.99)

84

9.4.2 Case of granular porous materials with homogeneous skeleton

The case of granular porous materials is the one of solid media for which the morphologyand spatial arrangement of the pores can not be reasonably assimilated to isolated inclusionswithin a solid matrix. These systems are more appropriately addressed using a self-consistenthomogenization scheme. First, we consider the case for which the porous medium consists ofa single solid phase Vs of stiffness Cs and a compliant porous network subjected to a uniformcompressive pressure p, i.e. an eigenstress π = −p1. The volume fraction of the porous phaseis noted φ.

Appropriate substitutions in Eq. (9.77) leads up to the following strain concentration tensors:

Ap,∞ =[I− Pp : Chom

]−1(9.100)

andAs,∞ =

[I + Pp : (Cs − Chom)

]−1(9.101)

which are such that

〈ε〉p = Ap,∞ :[〈ε〉+ Pp :

(πhom + 1p

)]and 〈ε〉s = As,∞ : [〈ε〉+ Pp : πhom] (9.102)

where the homogenized eigenstress is given as follows from Eq. (9.83):

πhom = −φ[φAp,∞ : Pp + (1− φ)As,∞ : Ps]−1 : Ap,∞ : Pp : 1p. (9.103)

The Eshelby tensors are given by Sr = Pr : Chom with the following corresponding polarizationtensors:

Pr ≡ P(Zr,Chom) =detZr

4π

∫

||ξ||=1

ξs⊗(ξ · Chom · ξ

)−1 s⊗ ξ

||Zr · ξ||3dSξ (9.104)

where Zr is the shape tensor of the inhomogeneities of the r-th family. The following im-plicit expression is then obtained for the effective stiffness of the drained porous medium (seeEq. (9.81)):

I = φ[I− Pp : Chom

]−1+ (1− φ)

[I + Ps :

(Cs − Chom

)]−1. (9.105)

Following the result of Levin’s theorem (see Eq. (9.82)), we have

〈σ〉 = Chom : 〈ε〉 − φ1 : Ap,∞p (9.106)

where we assume no eigenstress is directly supported by the solid. Similarly as before,thisexpression is recast in

〈σ〉 = Chom : 〈ε〉 −Bp (9.107)

such that the Biot tensor is

B = φ1 : Ap,∞ = φ1 :[I− Pp : Chom

]−1= φ1 : [I− Sp]−1. (9.108)

85

The average strain of the porous network is given as follows using Eq. (9.76):

〈ε〉p = Ap,∞ :[〈ε〉+ Pp :

(I− φ[φAp,∞ : Pp + (1− φ)As,∞ : Ps]−1 : Ap,∞ : Pp

): 1p

](9.109)

so that the change of porosity becomes

φ− φ0 = φ1 : Ap,∞ :[〈ε〉+ Pp :

(I− φ[φAp,∞ : Pp + (1− φ)As,∞ : Ps]−1 : Ap,∞ : Pp

): 1p

]

(9.110)that we also recast in

φ− φ0 = B : 〈ε〉+p

N(9.111)

where the skeleton Biot modulus is

1

N= φ1 : Ap,∞ : Pp :

(I− φ[φAp,∞ : Pp + (1− φ)As,∞ : Ps]−1 : Ap,∞ : Pp

): 1. (9.112)

Another way to derive an expression for the skeleton Biot modulus is to decompose and solvefor the change of porosity through the application of Levin’s theorem, see Ghabezloo (2011).This leads up to

1

N= (1− φ)1 : Ss : [1− 1 : As,∞] (9.113)

that we easily change as follows using Eq. (9.80):

1

N= φ1 : [Ap,∞ − I] : Ss : 1 (9.114)

so that eventually, we have1

N= (B− φ1) : Ss : 1. (9.115)

9.4.3 Case of granular porous materials with heterogeneous skeleton

The case of granular porous media with heterogeneous skeletons is a generalization of theprevious case to skeletons with different solid phases. Here, we consider a porous mediumconstituted of n solid phases Vs, each with a distinct stiffness Cs and a volume fraction fs. Inaddition to these, the medium has a compliant porous network Vp subjected to an eigenstress−p1 and with volume fraction

φ = 1−n∑

s=1

fs. (9.116)

Similarly as for granular media with a homogeneous skeleton, the strain concentration tensorson the porous network and each solid phase are given by Eqs. (9.100) and (9.101), respectively.

The application of Levin’s theorem still yields the same expression:

〈σ〉 = Chom : 〈ε〉 − φ1 : Ap,∞p (9.117)

86

where it is assumed that no eigenstress is directly supported by any of the solid phases. UsingEq. (9.80), this can be recast in

〈σ〉 = Chom : 〈ε〉 − φ1 :

[I−

n∑

s=1

fsAs,∞]p (9.118)

so that we have again〈σ〉 = Chom : 〈ε〉 −Bp (9.119)

in which

B = φ1 : Ap,∞ = 1−n∑

s=1

fs1 : As,∞ (9.120)

while the effective drained stiffness is solution of

I =

(1−

n∑

s=1

fs

)[I− Pp : Chom

]−1+

n∑

s=1

fs

[I + Ps :

(Cs − Chom

)]−1. (9.121)

Finally, the skeleton Biot modulus is given by

1

N=

n∑

s=1

fs1 : Ss : [1− 1 : As,∞]. (9.122)

87

Chapter 10

A multiscale poromechanics-hydrationmodel

The multiscale poromechanics-hydration model developed in this chapter consists of two un-coupled models: a hydration model and a multiscale poromechanics model. The hydrationmodel allows to define the evolution of the volumetric fractions of the invariant materialphases contained in the aging cement paste. The multiscale poromechanics model is obtainedby recursive application of the average state equations obtained in the previous chapter; it isused to upscale the poroelastic properties based on an idealized morphological scheme of themicrostructure.

10.1 Hydration model

The hydration model adopted in this work was proposed by Bernard et al. (2003) and improvedby Pichler et al. (2007). It consists of stoichiometric and kinetic equations used to assess theevolution of volume fractions of hydration products and reactants through time.

10.1.1 Stoichiometry

Independently from the rate of reaction, the amounts of reactants and products involved incement hydration can be assessed by stoichiometric relations. Those reactions occur betweenthe anhydrous compounds referred to as tricalcium silicate (C3S), dicalcium silicate C2S,tricalcium aluminate C3A, tetracalcium aluminoferrite C4AF, and gypsum CSH2) with waterto form hydration products. Some of the hydration products also react further on. Tennisand Jennings (2000) proposed stoichiometric relations to describe this process:

2C3S + 10.6H→ C3.4S2H8 + 2.6CH (10.1)

2C2S + 8.6H→ C3.4S2H8 + 0.6CH (10.2)

89

C3A + 3CSH2 + 26H→ C6AS3H32 (10.3)

2C3A + C6AS3H32 + 4H→ 3C4ASH12 (10.4)

C4AF + 2CH + 10H→ 2C3(A,F)H6 (10.5)

Eq. (10.1) and Eq. (10.2) describe the formation of calcium-silicate-hydrates (C3.4S2H8) andportlandite (CH). Eq. (10.3) allows to quantify the amount of ettringite (C6S3H32), also notedAFt, formed by hydration of gypsum and aluminate. Ettringite can react further with alumi-nate to form some monosulfoaluminate (C4ASH12) also noted AFm. The hydration of ferriteleads to the precipitation of hydrogarnet (C3(A,F)H6).

10.1.2 Kinetic

The mass exchanges described by stoichiometry occur at different rates depending on theclinker phase involved in the reaction. The advancement of the hydration process of a singleclinker phase is expressed by means of a hydration degree:

ξX(t) ≡ 1− mX(t)

mX,0

(10.6)

Where the hydration degree ξX refers to the relative amount of reactant consumed mX com-pared to the initial amount mX,0 present in the system. The overall hydration degree ξ isobtained by summing the hydration degrees of all the anhydrous phases weighted by theirrespective initial weight fraction. The kinetics of hydration of the four major clinker phasesis described by relationships that link the reaction rate dξX/dt to the affinity A(ξX) referredto as the driving force of the hydration reaction. In a first order approach, we disregardthe chemomechanical couplings which are of secondary importance for normal conditions oftemperature and pressure. The normalized affinity is expressed as follows:

A(ξX) = τX,0 exp

[Ea,XR

(1

T0

− 1

T

)]dξXdt

(10.7)

Where Ea,X is the apparent activation energy for the hydration of a phase X among the majorclinker compounds and R is the universal gas constant. The characteristic time τX,0 is definedat a reference temperature T0 and depends on the hydration process. Three main reactionprocesses can be considered to describe the hydration of a cement mixture: dissolution, growthand nucleation and diffusion.

Dissolution is the first reaction to occur. It can be described with a normal affinity A(ξX)

of 1 and a characteristic time τX,0 equal to tX,0/ξX,0. The end of dissolution is marked by athreshold degree of hydration ξX,0 for each of the main clinker phases.

Growth and nucleation follow the dissolution. This process is well described by the phaseevolution model of Avrami (1939). The corresponding normalized affinity is:

AA(ξX) =1− 〈ξX − ξX,0〉

[− ln (1− 〈ξX − ξX,0〉)](1/κ)−1(10.8)

90

where the operator 〈·〉 is such that 〈x〉 = 1/2 (x+ |x|). The characteristic time τX,0 related tothis kinetic law is equal to 1/(κXkX) where κX is the reaction order and kX , a constant rate.

Diffusion occurs as a late hydration process beyond a threshold degree ξ∗X defined for everymain clinker phase. The normalized affinity associated with this process is:

AD(ξX) =(1− ξX)

23

(1− ξ∗X)13 − (1− ξX)

13

(10.9)

After the work of Fujii and Kondo (1974), the characteristic time τX,0 of this kinetic mech-anism can be expressed by R2/(3DX) where R is the initial mean radius of the anhydrouscement particles and DX , the coefficient of diffusion of dissolved ions through existing layersof hydration products towards the remaining anhydrous phase X.

Because the hydration of aluminate follows two stoichiometric reactions, an order of priorityneeds to be specified between Eq. (10.3) and Eq. (10.4). Aluminate reacts primarily withgypsum to form ettringite as described by Eq. (10.3). Once all the gypsum is consumed, itcan react with ettringite to form monosulfoaluminate with respect to Eq. (10.4).

It is assumed that the diffusion process leads to the precipitation of calcium-silicate-hydrates(C-S-H) that are denser than those associated with dissolution and nucleation (Bernard et al.,2003). Hence, a distinction is made between the C-S-H precipitated either during the firsthydration processes from the ones generated through diffusion. With respect to the nomen-clature introduced by Jennings (2000), the first ones are referred to as low density (LD) C-S-Hand the latter as high density (HD) C-S-H.

10.1.3 Volume fractions

The volumes VX of the main clinker phases can be computed as functions of time equal tomX,0ξX(t)/ρX where the molar massesMX and densities ρX are given by Tennis and JenningsTennis and Jennings (2000). The hydration degrees ξX(t) are obtained from the solution toEq. (10.7) with respect to the appropriate expression of the normalized affinity of each kineticprocess. It is assumed that the sample is hydrated with an infinite supply of water so thatthe volumes of hydration products can be computed as functions of time governed by theremaining amounts of major clinker phases:

VP (t) =MP

MX

ρX

ρP

mX,0nP/XξX(t) (10.10)

where the index P refers to the hydration product and nP/X is the number of moles of productgenerated by hydration of one mole of anhydrous cement phase X. The stoichiometric ratiosnP/X are straightforwardly obtained from Eqs. (10.1)-(10.5). The volume fractions of reactantsfX(t) and hydration products fP (t) are obtained by normalization of the respective volumesVX and VP with the total volume at time t.

91

10.2 Multiscale poromechanics model

The purpose of the multiscale poromechanics model is to provide us with the Biot-Willisparameter, the skeleton Biot modulus, the Poisson’s ratio, the elastic modulus and the perco-lation threshold of cement paste as functions of volume fractions which are obtained from thehydration model.

The approach adopted is inspired from underlying works in microporomechanics (Dormieuxet al., 2004; Ulm et al., 2004; Constantinides, 2006) and model applications to early-agecement-based materials (Bernard et al., 2003; Sanahuja et al., 2007; Sanahuja, 2008). However,the proposed model assumes different hypotheses with respect to the work of Sanahuja et al.(2007), such as:

(i) The nanoscale elementary C-S-H particles are spherical. As demonstrated by Sanahujaet al. (2008), the shape of particles of polycrystals with packing density greater than 60% (asLD and HD C-S-H) is a second order parameter;

(ii) There are larger numbers of anhydrous phases and hydration products. One of the ob-jectives of this study is to investigate the sensitivity of the predicted macroscopic states tothe uncertainty of quantities related to these phases which were disregarded in the work ofSanahuja et al. (2008).

(iii) Spherical particles of HD C-S-H are embedded into a matrix of LD C-S-H. Although theobserved morphology of C-S-H at the scale of several hundreds of nanometers is often referredto as fibrillar, it varies over time (Jennings et al., 1981) and depends on drying conditions thatprovoke the rearrangement of the elementary C-S-H particles (Fonseca and Jennings, 2010).As it is difficult to define a precise morphology that will correspond to the shape of LD andHD C-S-H in any conditions, the same representation that Bernard et al. (2003) is adopted(see Fig. 10.1);

(iv) The inclusions of portlandite and ettringite are not spherical. It is known that crystals ofportlandite and ettringite exhibit strongly aspherical shapes and contribute to the setting ofcement paste (Taylor, 1990);

(v) A fixed porosity of LD C-S-H.

The model is developed as follows. First, the microstructure of cement paste is represented interms of elementary phases of invariant mechanical properties. A representative volume ele-ment (RVE) (Zaoui, 2002) that contains qualitative and morphological information is drawnfor every characteristic length scale of the material. These elements are significantly smallerthan the size of the structure and larger than the inhomogeneities they contain. Second,the microscopic mechanical responses of the cement paste subjected to prescribed macro-scopic boundary conditions are expressed by means of localization tensors. Third, the general

92

macroscopic response of the system is expressed in terms of state equations obtained by ho-mogenization of the local reactions. The investigated poroelastic properties are recovered fromthese resulting equations.

10.2.1 Microstructure representation

Converging efforts of experimental characterization of mechanical properties at nanoscale(Constantinides, 2002; Constantinides and Ulm, 2004) and modeling of calcium-silicate-hydrates(Jennings, 2000, 2008) have led to a multiscale representation of heterogeneity in cement-basedmaterials. A microstructure of cement paste can be described over three length scales (Ulmet al., 2004). As presented in Fig. 10.1, those are, from the coarsest to the finest: the cementpaste itself (level II), the aging C-S-H (level I) and the scale of preferential invariant densitiesof hydrates (level0). The qualitative phase composition of each scale and the correspondingessential morphological information are communicated below.

Figure 10.1: Multiscale representation of the microstructure of cement paste, adapted fromConstantinides (2002).

93

Level II is representative of heterogeneities of characteristic lengths smaller than 10−4 m. Thephases at this scale are the anhydrous cement particles, hydration products and pores. Theanhydrous phases (C3S, C2S, C3A, C4AF, and CSH2) are not very sharp (Taylor, 1990) andare usually considered as spherical inclusions Bernard et al. (2003); Sanahuja et al. (2007). Themain non-porous elementary hydration products are portlandite, AFt, AFm and hydrogarnet.The shape of AFm and hydrogarnet is not well defined and could even change during hydration(Taylor, 1990). Due to lack of information and their small volume fractions, they are heresimply represented by spherical inclusions Stora et al. (2006); Pichler et al. (2007). Portlanditecan be present in great amounts in form of platy crystals which are here represented byoblate ellipsoids (Stora et al., 2006). AFt crystals exhibit very sharp rod-like shapes andcan be modeled by prolate ellipsoids (Stora et al., 2006). The pores are part of the capillaryporosity and remain filled of water, they are considered as spherical inclusions. Calcium-silicate-hydrates are present as a sole homogeneous aging phase. At this scale, C-S-H particlescan either be considered as needles (Pichler et al., 2008, 2009b) or as spherical inclusions(Bernard et al., 2003; Constantinides and Ulm, 2004). This is a long-standing and still openscientific debate. Here, the C-S-H particles are considered spherical.

When hydration starts, the mixture contains a greater amount of liquid than solids and thepaste behaves like a viscous fluid. As long as enough hydration product is precipitated toensure a contact between solid particles, the paste sets and exhibits an increasing stiffness asa function of time. As stated by Bernard et al. (2003), this is the scale of solid percolation(Garboczi, 1993) for which Sanahuja et al. (2007) have demonstrated the strong influence ofthe shape of inclusions on the estimated threshold.

Level I is representative of heterogeneities of characteristic length smaller than 10−6 m. Thisis the smallest scale that can be characterized by current nanoindentation techniques dueto roughness limitations (Constantinides et al., 2003). According to the colloidal model ofcalcium-silicate-hydrates proposed by Jennings (Jennings, 2000, 2008), these phases are thehydration products that constitute the actual aging C-S-H: low density and high density C-S-H,also refered to as LD and HD C-S-H or outer and inner products (Tapli, 1959). The invarianceof the properties of these phases was the major finding of Constantinides (2006) confirmed byVandamme (2008). Early-hydration is characterized by nucleation and growth of LD C-S-H.HD C-S-H precipitates later, within smaller regions confined by low density product. Hence,the high density calcium-silicate-hydrates are represented by spherical inclusions embeddedinto a matrix of low density hydration product.

Level 0 is representative of heterogeneities of characteristic length about 10−9 m. Two RVE’sare drawn in order to describe the two types of calcium-silicate-hydrates. According to thecolloidal model of Jennings (Jennings, 2000, 2008), LD and HD C-S-H are consituted of thesame spherical C-S-H solid particles. These particles, referred to as globules (with the su-perscript g`), are packed at different densities with respect to each of the C-S-H phases. The

94

corresponding gel porosities φLD and φHD were estimated to 37.3% and 23.7% and remain filledwith water for a great range of relative humidities (Jennings, 2000; Ulm et al., 2004).

Three position vectors xk with k = 0, 1, 2 are defined over the RVEs drawn at levels 0, I andII, respectively. A remote uniform strain boundary condition is applied on the RVE at level IIso that the displacement u(x2) is equal to 〈ε〉 · x2 for ‖x2‖ → ∞ where 〈ε〉 is the macroscopicaverage strain tensor. At level I, the displacement u(x1) is equal to 〈ε〉LD · x1 for ‖x1‖ → ∞where 〈ε〉LD is the average strain in the LD C-S-H matrix. At levels 0, the displacementsu(x0) are equal to 〈ε〉Y · x0 for ‖x0‖ → ∞ where 〈ε〉Y stands for the average strain in the LDC-S-H (Y=LD), or the HD C-S-H (Y=HD). A common pressure p (i.e. eigenstress −p1) isapplied in the two interconnected porosities at levels 0 and II.

10.2.2 Localization

At level I, the poroelastic behavior of the LD and HD C-S-H phases is described by meansof the usual equations of microporomechanics (Ulm et al., 2004; Dormieux et al., 2006b;Constantinides, 2006) also derived in the previous chapter. The mean stresses 〈σ〉Y inducedby the mean strain 〈ε〉Y and the pore pressure p are related as follows:

〈σ〉Y = CY : 〈ε〉Y −BY p (10.11)

where Y stands either for LD or HD C-S-H. The stiffness tensors CY are known for the twotypes of hydrates. The contribution of the pore pressure to the stress states of these RVEs isquantified by the Biot tensors BY given by

BY = 1 :[I− (1− φY )〈A(x0)〉g`,∞

](10.12)

where φY refers to the porosities of LD and HD C-S-H. Contrarily to the previous chapter,a strain concentration (or localization) tensor Ar,∞ is noted 〈A(xk)〉r,∞ to indicate the levelk in which the corresponding phase Vr lies while emphasizing that we only consider esti-mates of the average concentration tensors. As seen in the previous chapter, the localizationtensor 〈A(x0)〉g`,∞ would concentrate the totality of the strain 〈ε〉V on the globules of thecorresponding phase Y if the pore pressure was equal to zero.

The second poromechanical state equation expresses the change of porosity as follows:

(φ− φ0)Y

= BY : 〈ε〉V +1

NY

p (10.13)

where NY ’s are skeleton Biot moduli:

1

NY

= (1− φY ) 1 : Sg` :(1− 1 : 〈A(x0)〉g`,∞

). (10.14)

The index Y stands for LD and HD C-S-H and the compliance tensor of the globules Sg` isknown.

95

The strains 〈ε〉Y developed at scale 0 are due to the combined effect of the pore pressure pand the mean strain 〈ε〉CSH applied over the aging C-S-H at scale I. Thanks to linear elasticity,this problem can be decomposed into superimposable loading cases. First, the strain 〈ε〉CSH isapplied and the mean stresses 〈σ′〉Y and mean strains 〈ε′〉Y occur at level 0. Second, the porepressure p is applied and the mean stresses 〈σ′′〉Y and mean strains 〈ε′′〉Y are developed. Theapplication of Levin’s theorem (Levin, 1967; Laws, 1973; Dormieux et al., 2006b) provides thefollowing relation:

∑

Y =LD,HD

fY 〈σ′′〉Y : 〈ε′〉Y =

[ ∑

Y =LD,HD

fY 〈σ′′〉Y]

: 〈ε〉CSH (10.15)

where the strains 〈ε′〉Y developed under the first loading case are of the form 〈A(x1)〉Y,∞ :

〈ε〉CSH. The relations between stress and strain developed under the second loading case aredetermined with respect to Eq. (10.11) and the above equation is recast in

p∑

Y =LD,HD

fY BY :(〈A(x1)〉Y,∞ − I

)=

∑

Y =LD,HD

fYCY : 〈ε′′〉Y :(〈A(x1)〉Y,∞ − I

). (10.16)

A second use of Levin’s theorem leads to the following equation:∑

Y =LD,HD

fYCY : 〈A(x1)〉Y,∞ : 〈ε〉CSH : 〈ε′′〉Y = 0. (10.17)

An expression for 〈ε′′〉Y is obtained from Eqs. (10.16) and (10.17) and the total strains 〈ε〉Yare recovered by summing the strains developed under the first and second loading cases:

〈ε〉Y = 〈A(x1)〉Y,∞ : 〈ε〉CSH + SY : BY :(〈A(x1)〉Y,∞ − I

)p (10.18)

where the compliance tensors SY are the inverses of the stiffness tensors CY .

At level II, the strain acting over the aging C-S-H is due to the pore pressure p and themacroscopic strain 〈ε〉 applied over the cement paste. Similarly, the problem is decomposedinto superimposable loading cases. First, the strain〈ε〉 is applied and a mean stress 〈σ′〉CSH

and mean strain 〈ε′〉CSH occur at level I. Second, the pore pressure p is applied and the meanstress 〈σ′′〉CSH and mean strain 〈ε′′〉CSH are developed. A first application of Levin’s theoremgives the following relation:

−ϕ2p1+ fCSH〈σ′′〉CSH +

∑

s∈As

fs〈σ′′〉s

: 〈ε〉 = −ϕ2p1 : 〈A(x2)〉p,∞ : 〈ε〉

+ fCSH

[CCSH : 〈ε′′〉CSH −BCSHp

]: 〈A(x2)〉CSH,∞ : 〈ε〉

+∑

s∈As

fsCs : 〈ε′′〉s : 〈A(x2)〉s,∞ : 〈ε〉

(10.19)

96

where As is the set of non-porous solid phases Vs of the RVE at level II, ϕ2 is the porosityof level II and the Biot tensor BCSH is defined to simplify the formulation (Constantinides,2006):

BCSH = f∗LD〈A(x1)〉LD,∞ : BLD + f∗HD〈A(x1)〉HD,∞ : BHD (10.20)

where f∗LD = fLD/(fLD + fHD) and f∗HD = fHD/(fLD + fHD) so that f∗LD + f∗HD = 1.

The stiffness tensor CCSH of the aging C-S-H is:

CCSH = f∗LDCLD : 〈A(x1)〉LD,∞ + f∗HDCHD : 〈A(x1)〉HD,∞. (10.21)

Because the volume average of the localization tensors over a RVE is equal to the fourth orderunity tensor, the concentration tensor of the porosity of level II can be recast in:

〈A(x2)〉p,∞ =1

ϕ2

I− fCSH〈A(x2)〉CSH,∞ −

∑

s∈As

fs〈A(x2)〉s,∞ . (10.22)

A second application of Levin’s theorem leads to the following equation:∑

s∈As

fsCs : 〈A(x2)〉s,∞ : 〈ε〉 : 〈ε′′〉s + fCSHCCSH : 〈A(x2)〉CSH,∞ : 〈ε〉 : 〈ε′′〉CSH = 0. (10.23)

The total strain developed over the aging C-S-H can then be recovered from Eqs. (10.19),(10.22) and (10.23).

〈ε〉CSH = 〈A(x2)〉CSH,∞ : 〈ε〉+ SCSH : (1−BCSH) :(〈A(x2)〉CSH,∞ − I

)p. (10.24)

The mean strain applied over every non-porous phase of scale II reduces to:

〈ε〉s = 〈A(x2)〉s,∞ : 〈ε〉+ Ss : 1 :(〈A(x2)〉s,∞ − I

)p. (10.25)

The mean strain applied over the capillary porosity is then equal to:

〈ε〉p = − 1

ϕ2

fCSH〈ε〉CSH +

∑

s∈As

fs〈ε〉s

(10.26)

The localizer tensor for an ellipsoidal inclusion in a medium subjected to uniform boundaryconitions assessed from the solution of Eshelby (1957). The assumption is made that, foreach phase i of the level k, the inclusions are randomly oriented in the RVE with respect to auniform distribution. Then, the strain concentration tensor 〈A(xk)〉r,∞ is obtained from thefollowing space average over Euler angles (Sanahuja, 2008):

〈A(xk)〉r,∞ =

∫ 2π

φ=0

∫ π

θ=0

[I + Pr(θ, φ) : (C0 − Ci)

]−1 sin θ

4πdθdφ. (10.27)

97

The Hill polarization tensor Pi(θ, φ) depends on the shape of the inclusions of Ωi, their orien-tation and the stiffness tensor C0 of the reference medium. Every polarization tensor is relatedto an Eshelby tensor Seshi by the relation Seshi = Pi : C0. The form of the Eshelby tensor iswidely documented in textbooks on micromechanics (Nemat-Nasser and Hori, 1993). Theonly missing parameter is the aspect ratio defined as the ratio of the length of the symmetryaxis and the diameter in the symmetry plane. For oblates, spheres and prolates it is respec-tively smaller than 1, equal to 1 and greater than 1. The double integration of Eq. (10.27) isperformed with respect to the approximation of Stroud (Stroud, 1971; Pichler et al., 2009b).

A reference phase Ω0 is defined as a function of both the morphology the homogenizationscheme adopted for each RVE. Two homogenization schemes are considered: the self consistentscheme and the Mori-Tanaka scheme (Mori and Tanaka, 1973). The Mori-Tanaka scheme istypical for media with a strong matrix-inclusion morphology and considers the embeddingphase as the reference medium. The self consistent scheme is typical for polydisperse granularmedia and considers the resulting homogenized medium as a reference and involves an implicitformulation of the auxiliary problem. According to Bernard et al. (2003), the self-consistentscheme is appropriate to model the solid percolation of the RVE drawn at level II. This schemeis also appropriate to model the nanogranular nature of the colloidal representation of LD andHD C-S-H at scale 0 (Constantinides and Ulm, 2007). The Mori-Tanaka scheme is adopted totake into account the strong matrix-inclusion morphology of the aging C-S-H. The LD C-S-His then refered to as the reference medium of scale I.

10.2.3 Homogenization

The macroscopic resulting stress tensor 〈σ〉 can be obtained by volumetric average of thelocalized resulting stresses:

〈σ〉 =∑

s∈As

fsCs : 〈ε〉s + fCSHCCSH : 〈ε〉CSH − (fCSHBCSH + ϕ21) p. (10.28)

The state equation (10.28) can be recast in:

〈σ〉 = C : 〈ε〉 −Bp (10.29)

where C is the homogenized stiffness tensor:

C =∑

s∈As

fsCs : 〈A(x2)〉s,∞ + fCSHCCSH : 〈A(x2)〉CSH,∞. (10.30)

The effective Biot tensor of the cement paste, B, is expressed by:

B =1 :

I−

∑

s∈As

fs〈A(x2)〉s,∞ − fCSH〈A(x2)〉CSH,∞

+ fCSH〈A(x2)〉CSH,∞ : BCSH. (10.31)

98

The total change in porosity is formulated as follows:

φ− φ0 = ϕ2tr〈ε〉p + fCSH [f∗LD (φ− φ0)LD

+ f∗HD (φ− φ0)HD

] (10.32)

where the first term represents the change in the capillary porosity (level II) and the secondone represents the total change in the gel porosity (level 0). This state equation can be recastin:

φ− φ0 = B : 〈ε〉+1

Np (10.33)

where N is the skeleton Biot modulus. It exhibits two components, for the two scales wherethe pore pressure p is applied:

1

N=

1

N0

+1

N2

. (10.34)

The contribution of the pressure applied in the gel porosity on the total change in porosity isquantified by 1/N0:

1

N0

= fCSH (1−BCSH) :[SCSH : BCSH :

(〈A(x2)〉CSH,∞ − I

)]+ fCSH

(f∗LD

NLD

+f∗HD

NHD

). (10.35)

Likewise, the contribution of the pressure applied in the capillary pores on the total changein porosity is quantified by 1/N2:

1

N2

=fCSH (BCSH − 1) :[(SCSH : 1) :

(〈A(x2)〉CSH,∞ − I

)]

+

∑

s∈As

fs (BCSH − 1) :[(Ss : 1) :

(I− 〈A(x2)〉s,∞

)] .

(10.36)

In the absence of fluid mass exchange, the undrained homogenized stiffness tensor is expressedas follows (Dormieux et al., 2006b):

Cu = C +MB⊗B (10.37)

where the Biot modulus, M , is a function of the skeleton Biot modulus N and the totalporosity ϕ0:

1

M=

1

N+

ϕ0

kf`.. (10.38)

By considering the pore solution as water, the fluid bulk modulus kf` can be estimated at2.3 GPa. The total porosity of the cement paste ϕ0 contains capillary pores and gel pores sothat ϕ0 = ϕ2 + fLDφLD + fHDφHD = fH + fLDφLD + fHDφHD because the capillary pores remainsaturated.

All the invariant elementary phases considered in this model and the resulting homogenizedcement paste are considered to be isotropic. Then, the stiffness tensor C can be recast in

99

the form 3kJ + 2gK where the fourth order tensors J and K are the spherical and deviatoricprojectors given by 1/31 ⊗ 1 and I − J. Similarly, the Biot tensor B is isotropic so that itcan be recast in b1 where b is referred to as the Biot-Willis parameter. The bulk and shearmoduli, respectively k and g, can be related to the elastic modulus E and the Poisson’s ratioν as follows:

E =9kg

3k + g; ν =

3k − 2g

6k + 2g(10.39)

The solid percolation threshold t0 of the cement paste is such that

t0 = arg maxt ∈ R+

dg

dt(t) (10.40)

where g is the shear modulus of the cement paste.

100

Chapter 11

Polynomial chaos expansion andpost-processing

Now that the multiscale poromechanics-hydration model of has been presented, the random-ness of the input parameters is assumed and the resulting uncertainty of the predicted poroe-lastic properties is investigated by a spectral non-intrusive approach (Soize and Ghanem, 2004;Berveiller et al., 2006). The model responses are represented by polynomial chaos expansionsand the resulting metamodel is post-processed in order to extract statistical moments, performa global sensitivity analysis and estimate some probability density distribution of the outputsof the model (Sudret, B., 2007).

11.1 Polynomial chaos representation

The deterministic multiscale poromechanics-hydration model is notedM. It has M randominput parameters Xi, i = 1, ...,M gathered in a vector X with prescribed probability densityfunction (PDF) fX(x). The model predicts poroelastic properties at different time steps. Allthese predictions are represented by random variables gathered in Yj , j = 1, ..., N. Thisrelation is expressed as follows:

Y ≡M (X) (11.1)

where X and Y are the random vectors of input parameters and model predictions. Theirrealizations are noted x and y.

11.1.1 Construction of the basis

It is assumed that M is square integrable with respect to the probability measure P(dx) =

fX(x)dx, meaning that every model response has a finite variance. Thereby, each outputvariable can be represented by a polynomial chaos expansion of the following form (Soize and

101

Ghanem, 2004):Yj =M(j) (X) ≡

∑

α∈NM

a(j)α φα (X) (11.2)

where α denotes all the possible multi-indices (α1, ..., αM ) with αi ∈ N, a(j)α are unknown

deterministic coefficients and φα(x) are multivariate basis functions orthonormal with respectto the joint probability density function fX(x) of the input parameters. The orthonormalityis verified by the following inner product of the Hilbert space H = L2(RM ,R,P(dx)):

⟨φα(x), φβ(x)

⟩H≡∫

RM

φα(x)φβ(x)fX(x)dx = δαβ (11.3)

where δαβ = 1 if α = β and 0 otherwise.

In the case of independent input parameters, the multivariate basis functions of Eq. (11.2) areexpressed in terms of tensor products of univariate functions (Sudret, B., 2007):

φα(x) ≡M∏

i=1

π(i)αi

(xi). (11.4)

Eq. (11.3) is verified by defining every univariate Hilbertian basis π(i)αi (xi) as the normalized

function of a classical orthogonal polynomial. The choice of the suitable family of polynomialsis dictated by the PDF of the input parameter Xi. In the sequel the input parameters of themultiscale poromechanics-hydration model are described either by lognormal (lnN ) or uniform(U) distributions. In the former case, lognormal variables are transformed into Gaussianvariables using the classical exponentiation and the associated Hermite polynomials (thatare othogonal with respect to the Gaussian measure) are used. In the latter case, so-calledLegendre polynomials are used.

π(i)αi

(xi) =

(2αi + 1)−1/2Leαi(xi) if Xi ∼ U

(αi!)−1/2Heαi(xi) if lnXi ∼ N

(11.5)

where Leαi and Heαi refer to the Legendre and Hermite polynomials of degree αi, respectively.

Thereby, the multiscale poromechanics-hydration model M(X) is replaced by a polynomialexpansion of the form of Eq. (11.2). For practical applications, one needs to determine thecoefficients a(j)

α associated with each model prediction Yj .

11.1.2 Practical implementation

For practical implementation, the infinite polynomial expansion of Eq. (11.2) is truncated andthe model responses are approximated. This is done by limiting the expansion to multivariatepolynomials of total degree less or equal to a fixed degree p. The total degree q of anymultivariate basis polynomial φα(x) is given by:

q = |α| ≡M∑

i=1

αi. (11.6)

102

The size P of the corresponding finite set of coefficients a(j)α , q = |α| ≤ p, j = 1, ..., N is:

P = N

(M + p

p

)= N

(M + p)!

M !p!(11.7)

where N is the number components of the model response.

Each model response can then be approximated by a series of the following form:

Yj =M(j) (X) ≈∑

|α|≤p

a(j)α φα (X) =

P/N−1∑

k=0

a(j)k φk (X) . (11.8)

The multi-indices α of the last right-hand side are changed into k indices in order to improve theunderstanding and simplify the forthcoming formulations. However, the amount of informationcontained by the multi-indices α is greater than what the integer indices k provide. Hence, atrack is kept between the multi-indices α and the coefficients a(j)

k . Each coefficient a(j)k is part

of the following set:

Aa =a

(j)k , k = 0, ..., P/N − 1, j = 1, ..., N

. (11.9)

The coefficients of the truncated expansion can be determined by solving the following least-squares minimization problem:

Aa = arg mina

(j)k ∈ R

N∑

j=1

ns∑

n=1

M(j)

(x(n)

)−P/N−1∑

k=0

a(j)k φk

(x(n)

)

2

(11.10)

where x(n) is a realization of the random input vector of parametersX among the experimentaldesign X = x(n), n = 1, ..., ns. The size of X needs to be at least as big as P/N so that asolution exists for Eq. (11.10). It is usually recommended to take ∼ 2−3P/N (Berveiller et al.,2006; Blatman, 2009). The experimental design can be generated by Monte-Carlo simulationor Latin-Hypercube-Sampling (LHS).

11.2 Post-processing

Once the finite set of coefficients of Eq. (11.9) is correctly estimated, the truncated seriesexpansion of Eq. (11.8) can be used as a surrogate of the multiscale poromechanics-hydrationmodel in order to provide random predictions. Moreover, some quantitative information can beobtained by post-processing these coefficients without any need to simulate additional modelresponses. Indeed, the first statistical moments and Sobol’ indices of the random outputs arestraightforwardly computed that way. Eq. (11.8) can also be used together with Monte Carlosimulation to assess the PDF of the poroelastic properties using kernel smoothing techniques(Wand and Jones, 1995).

103

11.2.1 First statistical moments

The mean average value of every model predictionM(j)(X) is the coefficient a(j)α of order zero

Sudret, B. (2007):µYj ≡ E[Yj ] = a

(j)0 (11.11)

where it is assumed that the index k equals 0 for |α| = 0.

The variance of each model response can be obtained from the following expression:

σ2Yj ≡ V[Yj ] =

P/N−1∑

i=1

(a

(j)i

)2. (11.12)

11.2.2 Sobol’ decomposition

The univariate polynomials defined at Eq. (11.5) are independent of xi if αi is equal to zero.Thereby, it is possible to identify the input parameters the multivariate polynomials are notfunctions of, and to reformulate the expansion of Eq. (11.2) by taking this information intoaccount. An arbitrary set Ii1,...,is that contains all the multi-indices α for which αi, i =

i1, ..., is are the only indices greater than zero is defined for this purpose:

Ii1,...,is =

α :

αi > 0 ∀ i = 1, ...,M, i ∈ (i1, ..., is)

αi = 0 ∀ i = 1, ...,M, i /∈ (i1, ..., is)

. (11.13)

As presented by Sudret (Sudret, 2008; Blatman and Sudret, 2010a), it is possible to gatherthe terms of Eq. (11.2) with respect to the input parameters they depend on:

M(j)(x) = a(j)0 +

M∑

i=1

∑

α∈Ii

a(j)α φα(xi)

+∑

1≤i1<i2≤M

∑

α∈Ii1,i2

a(j)α φα(xi1 , xi2) + ...

+∑

1≤i1<...<is≤M

∑

α∈Ii1,...,is

a(j)α φα(xi1 , ..., xis) + ...

+∑

α∈I1,...,M

a(j)α φα(x1, ..., xM )

(11.14)

where a(j)0 refers to the mean of Yj (see Eq. (11.11)). The summands of the above formula-

tion constitute the unique Sobol’ decomposition Sobol’ (1993) of Yj . Each summand can beexpressed as follows Sudret (2008):

M(j)i1,...,is

(xi1 , ..., xis) =∑

α∈Ii1,...,is

a(j)α φα(xi1 , ..., xis) (11.15)

whereM(j)i1,...,is

is the part of the model response Yj that depends only on the input parametersxi, i = i1, ..., is. Consequently, the part of the total variance that depends only on these

104

parameters is:

V(j)i1,...,is

(xi1 , ..., xis) =∑

α∈Ii1,...,is

(a(j)α

)2(11.16)

The relative contribution of any combination of random input parameters to the variance ofYj is expressed by the following polynomial chaos-based Sobol’ indices (Sudret, 2008):

SU(j)i1,...,is

= V(j)i1,...,is

(xi1 , ..., xis)/σ2Yj (11.17)

where σ2Yj

is given by Eq. (11.12) for a truncated polynomial expansion and a finite set ofcoefficients given by Eq. (11.9).

The Sobol’ indices constitute the quantitative information used for the sensitivity analysis ofmodel predictions to the uncertainty of the input parameters.

11.2.3 Probability density function

The polynomial chaos expansion in Eq. (11.8) may also be considered as a surrogate modelof the original model M. In order to estimate and plot the probability density function ofany scalar random response a large sample set of points is drawn according to the input jointprobability density function fX(x), say y(k)

j , k = 1, ..., ns (e.g. ns = 10, 000 - 100, 000).Then the PDF ofM(j)(X) may be estimated by kernel smoothing:

fYj (yj) =1

nsh

ns∑

k=1

K

(yj − y(k)

j

h

)(11.18)

where fYj (yj) is the estimator of the PDF of Yj , ns is the sample size, K(·) is a kernelfunction and h is the bandwidth parameter. The kernel is a positive function defined suchthat

∫K(yj)dyj = 1. The most usual kernels are Gaussian and Epanechnikov functions (Wand

and Jones, 1995). For a given kernel function, the quality of the estimation depends on thesmoothness controlled by the band width parameter. Good results can be obtained from theempirical Silverman rule for the calculation of the bandwidth:

h (K) =

[8√πR (K)

3µ2 (K)2

]1/5

σn−1/5s (11.19)

where σ is the estimator of the standard deviation of Yj based on the sample y(k)j , k =

1, ..., ns. This formulation is obtained by the minimization of the asymptotic mean integratedsquare error for densities not far from normal. R(K) and µ2(K) are obtained from:

R (K) =

∫K(y)2dy ; µ2 (K) =

∫y2K(y)2dy (11.20)

105

Chapter 12

Model input parameters

The multiscale poromechanics-hydration model presented in this study requires the specifica-tion of four kinds of input parameters: the initial phase composition, the kinetic parametersof hydration, the invariant elastic properties and the morphological parameters of the mi-crostructure. Most of this information has either been directly characterized with uncertaintyor assessed by inverse modeling and calibration. The aim of this section is to present themodels of uncertainty considered for those input parameters.

12.1 Phase composition

The initial quantitative phase composition of the cement mixture (see Table 12.1) is determinedby application of the Bogue method briefly described in Chapter 8.

The inaccuracy of the mass fractions predicted by Bogue calculation has been discussed byseveral authors (Taylor, 1990; Aldridge, 1982; Idorn, 1983; Gutteridge, 1983; Aldridge, 1983).The first reason for this discrepancy with the pure composition is that this method assumesthat the clinker reaches equilibrium during cooling while it is very unlikely to happen in cementmanufacture (Taylor, 1990). The non-consideration of the significant amounts of substituent

Table 12.1: Quantitative phase composition

Anhydrous Mass fraction, mX [1]phase, X Mean PDFC3S 0.622 U (0.568,0.676)C2S 0.152 U (0.126,0.178)C3A 0.106 U (0.097,0.115)C4AF 0.009 U (0.008,0.010)CSH2 0.074 U (0.068,0.080)

107

Table 12.2: Kinetic parameters

Phase, w/c τX,0 κX [1] ξX,0 DX [cm2/h] ξ∗XX [1] [h] Mean SD PDF [1] Mean SD PDF [1]

C3S 0.4 12.7 1.79 0.18 ln N 1.05× 10−10 0.13× 10−10 ln N0.5 11.9 1.72 0.17 ln N 0.02 2.64× 10−10 0.32× 10−10 ln N 0.600.6 11.2 1.69 0.17 ln N 6.42× 10−10 0.77× 10−10 ln N

C2S 0.4 66.1 1.03 0.10 ln N0.5 60.9 0.96 0.10 ln N 0.00 6.64× 10−13 0.80× 10−13 ln N 0.600.6 59.8 0.90 0.09 ln N

C3A 0.4 53.5 1.07 0.11 ln N0.5 49.2 1.00 0.10 ln N 0.04 2.64× 10−10 0.32× 10−10 ln N 0.600.6 42.6 0.93 0.09 ln N

C4AF 0.4 24.2 2.37 0.24 ln N 1.05× 10−10 0.13× 10−10 ln N0.5 21.4 2.30 0.23 ln N 0.40 2.64× 10−10 0.32× 10−10 ln N 0.600.6 17.9 2.23 0.22 ln N 6.42× 10−10 0.77× 10−10 ln N

ions present in the anhydrous phases and the attribution of the whole amount of major oxides(see Table 8.1) solely to the main clinker phases also contribute to estimation errors (Taylor,1990). Without experimental data available, in this first approach, we reasonably assume thoseinput parameters as uniform random variables. The mean of each mass fraction is taken equalto the estimation obtained from Chapter 8 and a coefficient of variation of 5% is considered.This corresponds to selecting a range [−0.05

√3µ•, 0.05

√3µ•] around each mean value µ•.

12.2 Kinetic parameters

The input parameters of the hydration model described at Chapter 10.1 are summarizedin Tables 12.2 and 12.3. The characteristic time τX,0, the reaction order κX , the diffusioncoefficient DX and the hydration degrees ξX,0 and ξ∗X that mark the advancement stages ofhydration at the transition between kinetic processes (see Chapter 10) are listed in Table 12.2for C3S, C2S, C3A and C4AF. All those parameters were presented by Bernard et al. Bernardet al. (2003) and most of them vary as functions of the water-to-cement ratio (w/c). Followingto Berliner et al. Berliner et al. (1998), the reaction order κX . The diffusion coefficientDX are considered as random input parameters while the characteristic time τX,0 and thehydration degrees are deterministic. The characteristic times are modeled by logarithmicPDF with coefficients of variation about 10% and the diffusion coefficients are also consideredas lognormal random input parameters with coefficients of variation about 12%.

The apparent activation energies presented in Table 12.3 are obtained from Bernard et al.Bernard et al. (2003). Without precise quantitative information about their uncertainty,the authors consider these input parameters as uniform random variables with coefficients of

108

Table 12.3: Apparent activation energies

Anhydrous Ea,X/R [K]phase, X Mean PDFC3S 4500 U (4110,4890)C2S 2500 U (2285,2715)C3A 5500 U (5025,5975)C4AF 4200 U (3835,4565)

variation of 5%.

12.3 Elastic parameters

The input parameters of the multiscale poromechanics model presented in Chapter 10.2 aresummarized in Table 12.4. Those are the elastic moduli E and the Poisson’s ratios ν ofthe elementary material phases presented in Fig. 13.1. Most of the elastic moduli have beendetermined from measurements obtained by nanoindentation while considering fixed values ofPoisson’s ratios. These Young moduli are defined as lognormal random input parameters withcoefficients of variation between 5 and 20%, depending on the values reported by the authors(see Table 12.4). A coefficient of variation of 10% is considered to fill the lack of quantitativeinformation about uncertainty for gypsum, hydrogarnet, AFm and AFt. The Poisson’s ratiosare considered as deterministic input parameters.

12.4 Microstructure parameters

The morphological input parameters of microstructure are the aspect ratios rX of inclusionsand the gel porosities φX of LD and HD C-S-H. Most of the inclusions of the multiscalemodel are considered spherical with a deterministic aspect ratio of 1. Ettringite crystals aremodeled by prolate ellipsoids with an aspect ratio rAFt of 20 in order to take their sharpnessinto account. Portlandite inclusions are represented by oblate ellipsoids and their aspect ratiorCH is set to 0.25 (Stora et al., 2006). Both of these input parameters are defined as uniformrandom variables with a coefficient of variation of 10%. The porosities φLD and φHD arerespectively set to 37.3 and 24.7% with respect to the estimations of Jennings (2000) and Ulmet al. (2004). Because the density of early-age calcium-silicate-hydrates is much less certainthan the density of latter hydration product (Jennings et al., 2007), the porosity of LD C-S-His considered as a random input parameter while φHD is kept deterministic. A lognormal PDFis considered for φLD with a coefficient of variation of 10%.

109

Table 12.4: Elastic parameters

Compound Nominal E [GPa] ν Ref.formula Mean SD PDF

Tricalcium C3S 135.0 7.0 ln N 0.3 Acker (2001), Velez et al. (2001)silicateDicalcium C2S 130.0 20.0 ln N 0.3 Acker (2001), Velez et al. (2001)silicateTricalcium C3A 145.0 10.0 ln N 0.3 Acker (2001), Velez et al. (2001)aluminateTetracalcium C4AF 125.0 25.0 ln N 0.3 Acker (2001), Velez et al. (2001)aluminoferriteGypsum CSH2 45.7 4.6 ln N 0.33 Choy et al. (XXXX),

Bhalla et al. (XXXX)Portlandite CH 38.0 5.0 ln N 0.305 Constantinides and Ulm (2004)Hydrogarnet C3(A,F)H6 22.4 2.2 ln N 0.25 Kamali et al. (2004)AFm C4ASH12 42.3 4.2 ln N 0.324 Kamali et al. (2004)AFt C6AS3H32 22.4 2.2 ln N 0.25 Kamali et al. (2004)LD C-S-H C3.4S2H8 21.7 2.2 ln N 0.24 Constantinides and Ulm (2004)HD C-S-H C3.4S2H8 29.4 2.4 ln N 0.24 Constantinides and Ulm (2004)

110

Chapter 13

Results and discussion

The multiscale poromechanics-hydration model proposed in this document allows one to pre-dict the solid percolation threshold of a cement paste as a function of the water-to-cement ratiofor 0.35 ≤ w/c ≤ 0.70. It also provides estimates of the Biot-Willis parameter, the skeletonBiot modulus and the drained and undrained elastic moduli and Poisson’s ratios as functionsof time. The aim of this Chapter is to validate this deterministic model, to present the resultsof uncertainty propagation from the input parameters, to identify the greatest contributors tothe uncertainty of model predictions and to estimate the PDF of the drained elastic modulusat different time steps.

13.1 Model validation

The predictive capabilities of the deterministic model are evaluated with respect to the evo-lution of the undrained elastic modulus and Poisson’s ratio compared to experimental dataobtained by Boumiz et al. (Boumiz et al., 1996, 2000) on cement pastes similar to the materialsconsidered in this study.

The evolution of the volume fractions required by the multiscale poromechanics model is pre-sented in Fig. 13.1 for a water-to-cement ratio of 0.50 at a curing temperature T of 25oC. Thevolume fractions of hydration products (fLD,fHD,fCH,fAFt and fC3AH6

) increase as the amountsof reactants (fC3S,fC2S,fC3A,fC4AF,fCSH2

and fH) decrease. The late reaction of ettringite toform AFm starts at an overall hydration degree ξ of 0.68 and is not complete; a remainingamount of AFt is predicted at the end of hydration.

The prediction of the undrained elastic modulus Eu is presented in Fig. 13.2 as functions ofthe overall hydration degree and time. A good agreement is observed between those estimatesand the experimental data of Boumiz et al. (1996) obtained by accoustic wave measurements.The root mean square error computed between experiments and predictions are 1.22, 0.43 and0.39 GPa for water-to-cement ratios of 0.40, 0.50 and 0.60. Graphically, the solid percolation

111

0.0 0.2 0.4 0.6 0.8 1.0Overall hydration degree, [1]

0.0

0.2

0.4

0.6

0.8

1.0Volumefractions,f X

[1]

fC3S

fC2S

fC3A

fC4AF

fCSH2

fAFt

fAFm

fC3AH6

fLDCSH

fHDCSH

fCH

fH

Figure 13.1: Model predictions of the volume fractions for w/c=0.50.

thresholds can be interpreted as the times t0 or hydration degrees where the curves of undrainedelastic moduli intercept the horizontal axes. The greatest discrepancy with experimental datais observed for the mix of water-to-cement ratio of 0.40.

The predictions of the Poisson’s ratios for the three mix designs of the study are presented inFig. 13.3 and compared to experimental data from Boumiz et al. (2000) at a water-to-cementratio of 0.40. The model predicts a slightly faster decrease of the Poisson’s ratio than suggestedby the experimental data. However the relation νu(t) has a shape that is representative of theobservations and that tends towards the same value than what was measured on the hardenedcement paste.

The multiscale poromechanics-hydration model is validated for its capacity to predict theevolution at early-age of the elastic properties of cement pastes with water-to-cement ratiosbetween 0.35 and 0.70. The upper-limit value of w/c is imposed by the validity domain of thekinetic parameters calibrated by Berliner et al. (1998) and the lower value is prescribed by thelimitation of the self-consistent homogenization scheme adopted at level II (see Section 10.2.2)to model percolation. Indeed, the multiscale poromechanics-hydration model predicts initialstiffnesses significantly greater than zero for mix designs of water-to-cement ratios lower than0.35.

112

0.0 0.1 0.2 0.3 0.4 0.5 0.6Overall hydration degree, [1]

0

5

10

15

20

Undrained

elasticmodulus,Eu[G

Pa]

w/c=0.40

w/c=0.50

w/c=0.60

0 24 48 72 96t [h]

0

10

20

30

Eu[GPa]

Figure 13.2: Model predictions of the undrained elastic modulus and experimental data fromBoumiz et al. (1996).

13.2 Uncertainty propagation

A set of nt time steps is prescribed for the three mix designs of this study (w/c = 0.40, 0.50, 0.60).For each mix, the solid percolation threshold t0 is computed and the drained elastic modu-lus E, the Biot-Willis parameter b and the skeleton Biot modulus N are simulated at everytime step. There are 3nt + 1 model responses calculated by mix and 31 input parameters. Adegree p = 2 is prescribed for the truncature of the polynomial expansion (see Eq. (11.8)) sothat the number of unknown coefficients is 528 per model response. According to precedentworks (Berveiller, 2005; Berveiller et al., 2006; Blatman and Sudret, 2010b), these coefficientscan be accurately determined by regression from experimental designs of sizes 2 to 3 timesgreater than the number of unknowns. An experimental design that contains 2,000 real-izations by model response is drawn by Latin Hypercube Sampling (LHS). The calculatoinsare carried out using the open source software for uncertainty quantification Open TURNS(http://www.openturns.org/).

Means and variances are computed with respect to Eqs. (11.11) and (11.12). The evolution ofthe means plus or minus a standard deviation and the coefficients of variation of the poroelasticproperties (E, b, N) are represented as functions of time in Fig. 13.7. The uncertainties of

113

0 12 24 36Time, t [h]

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

Undrained

Poisson

′ sratio,ν u

[1]

w/c=0.40

w/c=0.50

w/c=0.60

0 0.2 0.4 0.6ξ [1]

0.2

0.3

0.4

0.5

0.6

ν u[1]

Figure 13.3: Model predictions of the undrained Poisson’s ratio and experimental data forw/c=0.40 from Boumiz et al. Boumiz et al. (2000).

the model predictions are described as follows:

Drained elastic modulus, E:A very high uncertainty is observed during the first 12 hours, when percolation is more likelyto happen. The lower the water-to-cement ratio, the faster is the decrease of the coefficient ofvariation towards a limit of 5% independent of the mix design. In comparison to the uncer-tainty models of input parameters (see Section 12), the propagation of randomness throughscales up to the macroscopic elastic modulus tends to diminish the uncertainty.

Biot-Willis parameter, b:The initial uncertainty is negligible but it increases quickly during the first 12 hours of hydra-tion. The limit coefficients of variation lie between 1.5 and 4% and are inversely proportionalto the water-to-cement ratio. The propagation through scales does not show any magnificationof the uncertainty.

Skeleton Biot modulus, N :The coefficients of variation of the skeleton Biot modulus vary between 5 and 10%. Theuncertainty of the mix design with a water-to-cement ratio of 0.40 exhibits a significant increaseduring the first 24 hours.

114

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.600 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

Figure 13.4: Prediction of the uncertainty of poroelastic properties of cement paste and pair-wise correlations.

Solid percolation threshold, t0:The mean predictions of the solid percolation threshold are 1.01, 2.06 and 2.60 h respectivelyfor water-to-cement ratios of 0.40, 0.50 and 0.60. The coefficients of variation are 2.5, 6.3and 9.6%. The greater the initial amount of water, the greater is the uncertainty on settingpredictions.

Fig. 13.7 also presents the evolution of pairwise correlations among the macroscopic poroelasticproperties as a function of time. The elastic modulus E and the Biot-Willis parameter b arenegatively correlated. Considering that an increase of pore pressure leads up to a decrease oftensile strain at fixed average stress (see Eq. 10.29), this correlation is likely to amplify theuncertainty of the computed stress acting over an isotropic element of cement paste subjectedto tensile strains. In the case of a prescribed pore pressure with stress-free deformations(e.g. autogenous shrinkage), this correlation may be responsible of a magnification of theuncertainty of the computed volume changes. Similarly, beyond 12 hours of hydration, thejoint uncertainty of the skeleton Biot modulus N and the Biot-Willis parameter b amplifiesthe uncertainty of the porosity change predicted by Eq. (10.33).

The pairwise correlations of the poroelastic properties with the solid percolation threshold arepresented in Fig. 13.8 as a function of time. The more the hydration process is advanced, the

115

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.600 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

Figure 13.5: Prediction of the uncertainty of poroelastic properties of cement paste and pair-wise correlations.

less significant is the correlation between the setting time and those macroscopic properties.This is consistent with the fact that different micromechanics-hydration models (Bernardet al. (2003); Sanahuja et al. (2007); Pichler et al. (2009b)) can predict significantly differentbehaviors during the first hours of hydration while providing equally good estimates of matureelastic properties. The correlations of the Biot-Willis parameter tensor and the skeleton Biotmodulus with t0 decrease faster than ρE,t0(t).

13.3 Global sensitivity analysis

The results of the global sensitivity analysis are respectively presented in Figs. 13.9, 13.10 and13.11 for the percolation threshold, the drained elastic modulus and the Biot-Willis parame-ter. For the sake of clarity, the 31 input parameters were gathered into four categories (seeSection 12) and allocated as follows: 5 initial amounts of anhydrous cement phases, 12 kineticparameters, 11 elastic moduli and 3 morphological parameters of microstructure.

The contribution of each category consists in the sum of the first order Sobol’ indices (seeEq. (11.18)) of the input parameters from the same category. The total sum of these indicesis almost 1 for every model response. This means that there are negligible interaction effectsbetween these variables. Hence, only the univariate contributions are considered in the sequel.

116

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.600 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

Figure 13.6: Prediction of the uncertainty of poroelastic properties of cement paste and pair-wise correlations.

The greatest contributors to the uncertainty of the percolation threshold are the kinetic pa-rameters. Independently of any category, the variability of the apparent activation energy ofaluminate Ea,C3A/R, the aspect ratio of ettringite rAFt, and the aspect ratio of portlandite rCH

is responsible of more than 94% of the uncertainty of the setting time for any mix design. Thelower the water-to-cement ratio, the lower is the influence of the kinetic parameters and thebigger is the contribution of the shape of inclusions. However, the total contribution of themicrostructure parameters does not go over 19.6%. The quantitative phase composition has anegligible influence and the elastic moduli of the elementary material phases do not contributeat all.

The most important contribution to the variability of the elastic modulus during the first 12hours (see Fig. 13.10) comes from the kinetic parameters and, more precisely, the apparentactivation energy of aluminate. The part of variance due to the elastic moduli of the elementarymaterial phases increases with time until the end of hydration. After 18 hours, the elasticparameters govern and ELD is the greatest source of uncertainty. After 36 hours, the secondgreatest contributor is the elastic modulus of portlandite, and at 96h, approximately 60%of the variance is due to ELD. The microstructure parameters and the quantitative phasecomposition have negligible effects. The sensitivity of the elastic modulus of cement pastedoes not vary much as a function of the water-to-cement ratio.

117

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

0 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.600 24 48 72 960

5

10

15

20

25

E[GPa]

0 24 48 72 96t [h]

10-210-1100101

CV[1]

0 24 48 72 96-1

0

1

E,b[1]

-1

0

1

E,N[1]

-1

0

1

N,b[1]

0 24 48 72 96Time, t [h]

80

100

120

140

160

180

200

220

240

260

N[GPa]

0 24 48 72 96t [h]

10-2

10-1

100

CV[1]

0 24 48 72 96Time, t [h]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b[1]

0 24 48 72 96t [h]

10-410-310-210-1

CV[1]

w/c=0.40 w/c=0.50 w/c=0.60

Figure 13.7: Prediction of the uncertainty of poroelastic properties of cement paste and pair-wise correlations.

The Biot-Willis parameter b exhibits the same decreasing sensitivity to the apparent activationenergy Ea,C3A/R (Fig. 13.11). However, the rate of the increasing part of variance due to theelastic moduli is slower. After 12 hours, the quantitative phase composition starts to influencethe uncertainty and this contribution grows until it lies between 14 and 22%, depending onthe water-to-cement ratio.

13.4 Probability density function

The stochastic metamodel (see Eq. (11.8)) is used to generate samples of 7,000 realizations ofthe elastic modulus at different time steps. The PDF of these reponse quantities is estimated byGaussian kernel smoothing (see Section 11.2.3) and represented in Fig. 13.12. Then, extremevalues of the elastic modulus can be obtained from those distributions. For instance, the 5%quantile is represented in Fig. 13.12.

118

-1

0

1

E,t0[1] w/c=0.40 w/c=0.50 w/c=0.60

-1

0

1

b,t 0[1]

0 24 48 72 96Time, t [h]

-1

0

1

N,t0[1]

Figure 13.8: Pairwise correlations of the poroelastic properties with the percolation threshold.

77.1%

19.6%

3.3%

80.5%

17.2%

2.3%

85.9%

12.5%

1.5%

w/c=0.40 w/c=0.50 w/c=0.60

Elastic parameters Kinetic parameters

Microstructure parameters Phase composition

Figure 13.9: Sensitivity analysis of the percolation threshold - First Sobol’ indices.

119

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Sobolindex

,Si[1]

6 h0.0

0.2

0.4

0.6

0.8

1.0

12 h 18 h 24 h 36 h 48 h 72 h 96 h

w/c=0.40

w/c=0.50

w/c=0.60

Time, t [h]

Elastic parameters Kinetic parameters

Microstructure parameters Phase composition

Figure 13.10: Sensitivity analysis of the drained elastic moduls - First order Sobol’ indices.

120

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Sobolindex

,Si[1]

6 h0.0

0.2

0.4

0.6

0.8

1.0

12 h 18 h 24 h 36 h 48 h 72 h 96 h

w/c=0.40

w/c=0.50

w/c=0.60

Time, t [h]

Elastic parameters Kinetic parameters

Microstructure parameters Phase composition

Figure 13.11: Sensitivity analysis of the Biot-Willis parameter - First order Sobol’ indices.

121

t[h]

1224364860728496 E [GPa]5 10 15 20

p(E

)[1]

0.1

0.2

0.3

0.4

0.5

Figure 13.12: Smoothed probability density function of the drained elastic modulus as afunction of time.

122

Chapter 14

Partial conclusion

A multiscale poromechanics-hydration model was proposed to estimate the Biot-Willis param-eter, the skeleton Biot modulus and the drained and undrained elastic moduli and Poisson’sratio of hydrating cement paste as a function of time. The shape of the inclusions of ettringiteand portlandite in cement paste was considered to provide a better modeling of the evolutionof the elastic properties during the first hours of hydration. These properties can then be usedin macroscopic state equations to model the poromechanical behavior of partially saturatedcement paste.

The model was validated for cement pastes with water-to-cement ratios between 0.35 and0.70. We employed a powerful probabilistic approach based on polynomial chaos expansionto propagate the uncertainty of the phase composition, the kinetic parameters of hydration,the elastic moduli and the morphological parameters through different length scales. Thepolynomial expansion allows to predict variability and identifying its greatest contributorsamong the uncertain input parameters. The results show that the propagation does notmagnify the uncertainty for the single poroelastic properties although, their correlation mayamplify the variability of the estimates obtained from the macroscopic state equations. Inorder to reduce the uncertainty of the predicted percolation threshold and the poroelasticproperties at early-age, engineers should attempt to decrease the uncertainty of the apparentactivation energy of calcium aluminate. Later on, the variability of the poroelastic propertiescan be reduced by improving the accuracy of the elastic modulus of LD C-S-H.

123

Conclusion

First, the mechanical properties of C–S–H in concrete produced with EM-based bioplasticizerwere studied by a statistical nanoindentation technique. The resort to nanoindentation allowedto investigate the long-term viscous behavior of C–S–H after few hundreds seconds whileequivalent measurements performed at larger scale would require decade-long uniaxial testingprocedures. The nanoindentation tests which were configured for relaxation (i.e. at controlleddepth) were performed on concrete produced either with commercial SP or with EM-basedbioplasticizer. As a result, the addition of EM-based bioplasticizer to concrete was found to:(i) improve the strength of C–S–H by enhancing the cohesion and friction of solid nanograins;(ii) decrease the absolute rate of long-term relaxation; (iii) inhibit the precipitation of C–S–Hof higher density. However, the finding that EM-based bioplasticizer inhibits the precipitationof C–S–H of higher density is the result of assertions solely based on mechanical considerations.A multitechnique investigation which combines chemical and mechanical information will beneeded to thoroughly address the question of the distribution of hydrates.

Second, a multiscale poromechanics-hydration model was proposed to estimate the Biot-Willisparameter, the skeleton Biot modulus and the drained and undrained elastic moduli andPoisson’s ratio of hydrating cement paste as a function of time. The shape of the inclusionsof ettringite and portlandite in cement paste was considered to provide a better modeling ofthe evolution of the elastic properties during the first hours of hydration. These propertiescan then be used in macroscopic state equations to model the poromechanical behavior ofpartially saturated cement paste.

Third, the model proposed was validated for cement pastes with water-to-cement ratios be-tween 0.35 and 0.70. We employed a powerful probabilistic approach based on polynomialchaos expansion to propagate the uncertainty of the phase composition, the kinetic parame-ters of hydration, the elastic moduli and the morphological parameters through different lengthscales. The polynomial expansion allows to predict variability and to identify the greatest con-tributors among the uncertain input parameters. The results show that the propagation doesnot magnify the uncertainty for the single poroelastic properties although, their correlationmay amplify the variability of the estimates obtained from the macroscopic state equations.In order to reduce the uncertainty of the predicted percolation threshold and the poroelastic

125

properties at early-age, engineers should attempt to decrease the uncertainty of the apparentactivation energy of calcium aluminate. Later on, the variability of the poroelastic propertiescan be reduced by improving the accuracy of the elastic modulus of LD C-S-H.

126

Appendix A

Matrix representation

The tensorial equations presented in this work are most of the time implemented using equiv-alent matrix representations. Matrix representations can only be equivalently used if thetensors of interest exhibit some specific symmetries. When resorting to matrices, a choice isdone concerning the basis in terms of which the components of the tensors are formulated.Unless specified, all tensorial components are expressed with respect to the same orthogonalbasis eii=1,3.

A.1 Matrix representation of tensors

Most of the second-order tensors encountered in this work are symmetric. Thus, for a tensorα, we have αijei ⊗ ej = αjiei ⊗ ej . Symmetric second-order tensors can be defined by meansof 6 independent components α11, α22, α33, α23, α31 and α12. The three other componentsare α32 = α23, α13 = α31 and α21 = α12. Such second-order tensors are represented in termsof column vectors as follows:

α ≡

α11

α22

α33√2α12√2α23√2α31

.

Similarly, all the fourth-order tensors encountered in this work exhibit at least a minor sym-metry. This means that for a tensor A, we have Aijklei⊗ ej ⊗ ek⊗ el = Ajiklei⊗ ej ⊗ ek⊗ el =

Ajilkei ⊗ ej ⊗ ek ⊗ el = Aijlkei ⊗ ej ⊗ ek ⊗ el. These tensors have 36 independent components

127

so that they can be recast in 6× 6 matrices as follows:

[A] ≡

A1111 A1122 A1133

√2A1112

√2A1123

√2A1131

A2211 A2222 A2233

√2A2212

√2A2223

√2A2231

A3311 A3322 A3333

√2A3312

√2A3323

√2A3331√

2A1211

√2A1222

√2A1233 2A1212 2A1223 2A1231√

2A2311

√2A2322

√2A2333 2A2312 2A2323 2A2331√

2A3111

√2A3122

√2A3133 2A3112 2A3123 2A3131

Some fourth-order tensors also exhibit a major symmetry, so that Aijklei ⊗ ej ⊗ ek ⊗ el =

Aklijei ⊗ ej ⊗ ek ⊗ el and so on. In this case, the matrix representation [A] is symmetric, andonly 21 components of A are independent from each other.

A.2 Matrix representation of tensorial operations

The purpose of introducing matrix representations is to facilitate the implementation of ten-sorial equations. Thereby, for some given symmetric second-order tensor α and fourth-ordertensor A with minor symmetry, the components of the second-order tensor resulting from thecontracted product α : A are obtained from the following matrix operation:

α : A = [A]T α .

Obviously, due to the minor symmetry of A, α : A is symmetric so that the vector obtainedfrom the equation above satisfies the matrix representation of symmetric second-order tensorsintroduced previously. Similarly, the operation A : α is implemented as follows in terms ofthe product of a matrix and a vector:

A : α = [A]α .

The double contraction of two fourth-order tensors is expressed as follows in terms of a matrixoperation:

[A : B] = [A][B]

where, once again, the minor symmetry of A and B is such that A : B also has minor symmetry.The matrix obtained from the equation above thus satisfies the representation of fourth-ordertensors with minor symmetry introduced earlier.

A.3 Transformation of matrix representations

For some cases encountered in this work, the components of a fourth-order tensor A withminor symmetry are given with respect to a local basis noted eri i=1,3. It is then of particularinterest to find the components of A with respect to the reference basis eii=1,3. If the local

128

basis can be rotated into the reference basis using the transformation law ei = Qαimeαm, the

matrix representation of A in terms of eii=1,3 is obtained as follows from the local matrixform [A]eri :

[A] = [QrQr]T [A]eri [QrQr]

where [QrQr] is a matrix given by:

[QαQα] ≡

Qα11Qα11 Qα21Q

α21 Qα31Q

α31

√2Qα21Q

α31

√2Qα11Q

α31

√2Qα11Q

α21

Qα12Qα12 Qα22Q

α22 Qα32Q

α32

√2Qα22Q

α32

√2Qα12Q

α32

√2Qα12Q

α22

Qα13Qα13 Qα23Q

α23 Qα33Q

α33

√2Qα23Q

α33

√2Qα13Q

α33

√2Qα13Q

α23√

2Qα12Qα13

√2Qα22Q

α23

√2Qα32Q

α33 Qα22Q

α33 +Qα32Q

α23 Qα12Q

α33 +Qα32Q

α13 Qα12Q

α23 +Qα22Q

α13√

2Qα11Qα13

√2Qα21Q

α23

√2Qα31Q

α33 Qα21Q

α33 +Qα31Q

α23 Qα11Q

α33 +Qα31Q

α13 Qα11Q

α23 +Qα21Q

α13√

2Qα11Qα12

√2Qα21Q

α22

√2Qα31Q

α32 Qα21Q

α32 +Qα31Q

α22 Qα11Q

α32 +Qα31Q

α12 Qα11Q

α22 +Qα21Q

α12

with Qαij = ei · eαj independently of the basis, i.e. Qαijei ⊗ ej = Qαijeαi ⊗ eαj .

129

Bibliography

Abuhaikal, M. (2011). Nano-chemomechanical assessment of rice husk ash cement by wave-length dispersive spectroscopy and nanoindentation. Master’s thesis, Massachusetts Insti-tute of Technology.

ACI Committee 209 (1972). SP-27: Prediction of Creep, Shrinkage, and Temperature Effectsin Concrete Structures. American Concrete Institute, Farmington Hills, MI, pages 51–93.

ACI Committee 209 (2008). 209.2R-08: Guide for Modeling and Calculating Shrinkage andCreep in Hardened Concrete. American Concrete Institute, Farmington Hills, MI, page 48pp.

Acker, P. (2001). Micromechanical analysis of creep and shrinkage mechanisms. Creep,Shrinkage and Durability Mechanics of Concrete and Other Quasi-Brittle Materials. Editedby F.J. Ulm, Z.P. Bazant and F.-H. Wittman, pages 15–25.

Acker, P. and Ulm, F.-J. (2001). Creep and shrinkage of concrete: physical origins and practicalmeasurements. Nuclear Engineering and Design, 203(2-3):143–158.

Aldridge, L. (1982). Accuracy and precision of phase analysis in portland cement by Bogue,microscopic and X-ray diffraction methods. Cement and Concrete Research, 12(3):381–398.

Aldridge, L. (1983). A reply to the discussions by G.M. Idorn and W.A. Gutteridge of “Accu-racy and precision of phase analysis in Portland cement by Bogue, microscopic and X-raydiffraction methods”. Cement and Concrete Research, 13(6):897–898.

Avrami, M. (1939). Kinetics of phase change. I. Journal of Chemical Physics, 7:1103–1112.

Bažant, Z. (1977). Viscoelasticity of solidifying porous material–concrete. Journal ofEngineering Mechanics, 103(6).

Bažant, Z. (1997). Microprestress-Solidification Theory for Concrete Creep I: Aging andDrying Effects. Journal of Engineering Mechanics, 123(11):1188–1194.

Bažant, Z., Li, G., Yu, Q., Klein, G., and Křítek, V. (2008). Explanation of excessive long-time deflections of collapsed record-span box girder bridge in palau. Proc., 8th Int. Conf.

131

on Creep, Shrinkage and Durability of Concrete and Concrete Structures, T. Tanabe et al.,eds., The Maeda Engineering Foundation, Ise-Shima, Japan, pages 1–31.

Bažant, Z., Tsubaki, T., and Celep, Z. (1983). Singular history integral for creep rate ofconcrete. J. Eng. Mech., 109(3):866–884.

Bažant, Z., Yu, Q., Li, G., Klein, G., and Křítek, V. (2010). Excessive deflections of record-span prestressed box girder: Lessons learned from the collapse of the Koror-BabeldaobBridge in Palau. Concrete International, 32(6):44–52.

Berliner, R., Popovici, M., Herwig, K., Berliner, M., Jennings, H., and Thomas, J. (1998).Quasielastic neutron scattering study of the effect of water-to-cement ratio on the hydrationkinetics of tricalcium silicate. Cement and Concrete Research, 28(2):231–243.

Bernard, O., Ulm, F.-J., and Lemarchand, E. (2003). A multiscale micromechanics-hydrationmodel for the early-age elastic properties of cement-based materials. Cement and ConcreteResearch, 33(9):1293–1309.

Berveiller, M. (2005). Éléments finis stochastiques: approches intrusive et non intrusive pour des analyses de fiabilité.PhD thesis, Université Blaise Pascal - Clermont II.

Berveiller, M., Le Pape, Y., Sanahuja, J., and Giorla, A. (2009). Sensitivity analysis and un-certainty propagation in multiscale modeling of concrete. 4th Biot Conference on Porome-chanics.

Berveiller, M., Sudret, B., and Lemaire, M. (2006). Stochastic finite element : a non intrusiveapproach by regression. Eur. J. Comput. Mech., 15:81–92.

Bhalla, A., Cook, W., Hearmon, R., Jerphagnon, J., Kurtz, S., Liu, S., Nelson, D., and Oudar,J.-L. (XXXX). Landolt-Bornstein: Numerical data and functionnal relationships in scienceand technology. Group III. Crystal and solid state physics. Vol. 11. Elastic, piezoelectric,pyroelectric, piezooptic, electroptic constants and nonlinear dielectric susceptibilities of crys-tals, Supplement to Volume III/11, Springer-Verlag Berlin and Heidelberg GmbH & Co. K,1984.

Blatman, G. (2009). Adaptive sparse polynomial chaos expansions for uncertainty propagationand sensitivity analysis. PhD thesis, Université Blaise Pascal - Clermont II.

Blatman, G. and Sudret, B. (2010a). Efficient computation of global sensitivity indices usingsparse polynomial chaos expansions. Reliab. Eng. Sys. Safety, 95:1216–1229.

Blatman, G. and Sudret, B. (2010b). Use of sparse polynomial chaos expansions in adaptivestochastic finite element analysis. Prob. Eng. Mech., 25:183–197.

132

Bobko, C. P., Gathier, B., Ortega, J. A., Ulm, F.-J., Borges, L., and Abousleiman, Y. N.(2011). The nanogranular origin of friction and cohesion in shale—a strength homogeniza-tion approach to interpretation of nanoindentation results. Int. J. Numer. Anal. Meth.Geomech., 35(17):1854–1876.

Bogue, R. H. (1929). Calculation of the compounds in portland cement. Industrial &Engineering Chemistry Analytical Edition 1, 4:192–197.

Bolobova, A. and Kondrashchenko, V. (2000). Use of yeast fermentation waste as a biomodifierof concrete (review). 36(3):205–214–.

Borodich, F. M. (1989). Hertz contact problems for an anisotropic physically nonlinear elasticmedium. Strength of Materials, 21(12):1668–1676.

Borodich, F. M., Keer, L. M., and Korach, C. S. (2003). Analytical study of fundamentalnanoindentation test relations for indenters of non-ideal shapes. Nanotechnology, 14(7):803–.

Boumiz, A., Sorrentino, D., Vernet, C., and Tenoudji, F. C. (2000). Modelling the developmentof the elastic moduli as a function of the hydration degree of cement pastes and mortars.2nd International Rilem Workshop On Hydration and Setting: Why Does Cement Set? AnInterdisciplinary Approach, 13:295–316.

Boumiz, A., Vernet, C., and Tenoudji, F. (1996). Mechanical properties of cement pastes andmortars at early ages: Evolution with time and degree of hydration. Advanced CementBased Materials, 3(3-4):94–106.

Bulychev, S., Alekhin, V., Shorshorov, M., Ternovskii, A., and Shnyrev, G. (1975). Determin-ing young’s modulus from the indentor penetration diagram. Ind. Lab., 41(9):1409–1412.

Burgoyne, C. and Scantlebury, R. (2006). Why did Palau bridge collapse? The StructuralEngineer, 84(11):30–37.

Chen, J. J., Sorelli, L., Vandamme, M., Ulm, F.-J., and Chanvillard, G. (2010). A CoupledNanoindentation/SEM-EDS Study on Low Water/Cement Ratio Portland Cement Paste:Evidence for C–S–H/Ca(OH)2 Nanocomposites. Journal of the American Ceramic Society,93(5):1484–1493.

Chen, Y. and Qiao, P. (2011). Crack growth resistance of hybrid fiber-reinforced cementmatrix composites. J. Aerosp. Eng., 24(2):154–161.

Choy, M., Cook, W., Hearmon, R., Jaffe, J., Jerphagon, J., Kurtz, S., Liu, S., and Nelson, D.(XXXX). Landolt-Bornstein: Numerical data and functionnal relationships in science andtechnology. Group III. Crystal and solid state physics. Vol. 11. Elastic, piezoelectric, pyro-electric, piezooptic, electroptic constants and nonlinear dielectric susceptibilities of crystals,

133

Revised and Extended Edition of Vols III/1 and III/2, Springer-Verlag-Berlin and Heidel-berg GmbH & Co. K, 1979.

Cohen, M. H. and Turnbull, D. (1959). Molecular transport in liquids and glasses. J. Chem.Phys., 31(5):1164–1169.

Constantinides, G. (2002). The elastic properties of calcium leached cement pastes and mor-tars: A multi-scale investigation. Master’s thesis, Massachusetts Institute of Technology.

Constantinides, G. (2006). Invariant Mechanical Properties of Calcium-Silicate-Hydrates. PhDthesis, Massachusetts Institute of Technology.

Constantinides, G., Ulm, F., and Van Vliet, K. (2003). On the use of nanoindentation forcementitious materials. Materials and Structures, 36(257):191–196.

Constantinides, G. and Ulm, F.-J. (2004). The effect of two types of C-S-H on the elasticityof cement-based materials: Results from nanoindentation and micromechanical modeling.Cement and Concrete Research, 34(1):67–80.

Constantinides, G. and Ulm, F.-J. (2007). The nanogranular nature of C–S–H. Journal of theMechanics and Physics of Solids, 55(1):64–90.

Culling, W. (1983). Rate process theory in geomorphic soil creep. Rainfall simulation, runoffand soil erosion, pages 191–214.

Culling, W. E. H. (1988). A unified theory of particulate flows in geomorphic settings. EarthSurf. Process. Landforms, 13(5):431–440.

Davydov, D., Jirásek, M., and Kopecký, L. (2011). Critical aspects of nano-indentation tech-nique in application to hardened cement paste. Cement and Concrete Research, 41(1):20–29.

DeJong, M. J. and Ulm, F.-J. (2007). The nanogranular behavior of C-S-H at elevated tem-peratures (up to 700 °C). Cement and Concrete Research, 37(1):1–12.

Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete datavia the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological),39(1):1–38.

Dormieux, L. (2005). Poroelasticity and strength of fully or partially saturated porous mate-rials. In Applied micromechanics of porous materials. CISM Courses and Lectures editedby L. Dormieux and F.-J. Ulm.

Dormieux, L., Kondo, D., and Ulm, F.-J. (2006a). A micromechanical analysis of damagepropagation in fluid-saturated cracked media. Comptes Rendus Mécanique, 334(7):440–446.

134

Dormieux, L., Kondo, D., and Ulm, F.-J. (2006b). Microporomechanics. Wiley.

Dormieux, L., Lemarchand, E., Kondo, D., and Fairbairn, E. (2004). Elements of poro-micromechanics applied to concrete. Materials and Structures, 37(1):31–42.

Eshelby, J. D. (1957). The determination of the elas-tic field of an ellipsoidal inclusion, and related problems.Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences,241(1226):376–396.

European Concrete Committee (Comité européen du béton, CEB) (1972). RecommandationsInternationales pour le Calcul et l’Exécution des Ouvrages en Béton. , page 291 pp.

Eyring, H. (1935). The activated complex in chemical reactions. J. Chem. Phys., 3(2):107–115.

Eyring, H. (1936). Viscosity, plasticity, and diffusion as examples of absolute reaction rates.J. Chem. Phys., 4(4):283–291.

Fonseca, P. and Jennings, H. (2010). The effect of drying on early-age morphology of C–S–Has observed in environmental SEM. Cement and Concrete Research, 40(12):1673–1680.

Fraley, C. and Raftery, A. (2006). Mclust version 3 for r: Normal mixture modeling andmodel-based clustering. Technical report, Technical report.

Fraley, C. and Raftery, A. E. (1998). How many clusters? which clustering method? answersvia model-based cluster analysis. The Computer Journal, 41(8):578–588.

Fujii, K. and Kondo, W. (1974). Kinetics of the hydration of tricalcium silicate. Journal ofthe American Ceramic Society, 57(11):492–497.

Galin, L. (1953). Contact Problems in the Theory of Elasticity, 1953 (Gostekhizdat, Moscow).English translation.

Gao, Y., De Schutter, G., Ye, G., Huang, H., Tan, Z., and Wu, K. (2013). Porosity charac-terization of itz in cementitious composites: Concentric expansion and overflow criterion.Construction and Building Materials, 38(0):1051–1057.

Garboczi, E. (1993). Computational materials science of cement-based materials. Materialsand Structures, 26(4):191–195.

Ghabezloo, S. (2011). Effect of the variations of clinker composition on the poroelastic prop-erties of hardened class g cement paste. Cement and Concrete Research, 41(8):920–922.

Ghosh, P., Mandal, S., Chattopadhyay, B., and Pal, S. (2005). Use of microorganism toimprove the strength of cement mortar. Cement and Concrete Research, 35(10):1980–1983.

135

Gibbs, G. (1969). Thermodynamic analysis of dislocation glide controlled by dispersed localobstacles. Materials Science and Engineering, 4(6):313–328.

Grondin, F., Bouasker, M., Mounanga, P., Khelidj, A., and Perronnet, A. (2010). Physico-chemical deformations of solidifying cementitious systems: multiscale modelling. Materialsand Structures, 43(1):151–165.

Gutteridge, W. (1983). A discussion of the paper “Accuracy and precision of phase analyses inPortland cement by Bogue, microscopic and X-ray diffraction methods”, by L.P. Aldridge.Cement and Concrete Research, 13(6):896.

Hay, J., Bolshakov, A., and Pharr, G. (1999). A critical examination of the fundamental rela-tions used in the analysis of nanoindentation data. Journal of materials research, 14(6):2296–2305.

Higa, T. and Wididana, G. (1991). The concept and theories of effective microorganisms.Proceedings of the first international conference on Kyusei nature farming. US Departmentof Agriculture, Washington, DC, USA, pages 118–124.

Hua, C., Ehrlacher, A., and Acker, P. (1997). Analyses and models of the autogenous shrinkageof hardening cement paste ii. modelling at scale of hydrating grains. Cement and ConcreteResearch, 27(2):245–258.

Idorn, G. (1983). A discussion of the paper “Accuracy and precision of phase analyses inPortland cement by Bogue, microscopic and X-ray diffraction methods”, by L.P. Aldridge.Cement and Concrete Research, 13(6):895.

Jennings, H. M. (2000). A model for the microstructure of calcium silicate hydrate in cementpaste. Cement and Concrete Research, 30(1):101–116.

Jennings, H. M. (2004). Colloid model of C–S–H and implications to the problem of creepand shrinkage. Materials and Structures, 37(1):59–70.

Jennings, H. M. (2008). Refinements to colloid model of C-S-H in cement: CM-II. Cementand Concrete Research, 38(3):275–289.

Jennings, H. M., Dalgeish, B. J., and Pratt, P. L. (1981). Morphological development ofhydrating tricalcium silicate as examined by electron-microscopy techniques. Journal of theAmerican Ceramic Society, 64(10):567–572.

Jennings, H. M., Thomas, J. J., Gevrenov, J. S., Constantinides, G., and Ulm, F.-J. (2007).A multi-technique investigation of the nanoporosity of cement paste. Cement and ConcreteResearch, 37(3):329–336.

136

Jones, C. A. and Grasley, Z. C. (2011). Short-term creep of cement paste during nanoinden-tation. Cement and Concrete Composites, 33(1):12–18.

Kamali, S., Moranville, M., Garboczi, E., Prené, S., and Gérard, B. (2004). Hydrate dissolutioninfluence on the young modulus of cement paste. FRAMCOS V: Linking scales, fromnanostructure to infrastructures, pages 631–638.

Kass, R. E. and Raftery, A. E. (1995). Bayes factors. Journal of the American StatisticalAssociation, 90(430):773–795.

King, R. (1987). Elastic analysis of some punch problems for a layered medium. InternationalJournal of Solids and Structures, 23(12):1657–1664.

Klug, P. and Wittmann, F. (1969). Activation energy of creep of hardened cement paste.Materials and Structures, 2(1):11–16.

Klug, P. and Wittmann, F. (1974). Activation energy and activation volume of creep ofhardened cement paste. Materials Science and Engineering, 15(1):63–66.

Krakowiak, K. J., Lourenço, P. B., and Ulm, F. J. (2011). Multitechnique investigation ofextruded clay brick microstructure. Journal of the American Ceramic Society, 94(9):3012–3022.

Laws, N. (1973). On the thermostatics of composite materials. Journal of the Mechanics andPhysics of Solids, 21(1):9–17.

Laws, N. (1977). The determination of stress and strain concentrations at an ellipsoidalinclusion in an anisotropic material. Journal of Elasticity, 7(1):91–97.

Lee, E. and Radok, J. (1960). The contact problem for viscoelastic bodies. J. Appl. Mech.,27(3):438–444.

Levin, V. (1967). Thermal expansion coefficient of heterogeneous materials. Mekh. Tverd.Tela, 2:83–94.

Li, S. and Wang, G. (2008). Introduction to micromechanics and nanomechanics. Worldscientific.

Lin, F. and Meyer, C. (2008). Modeling shrinkage of portland cement paste.ACI Materials Journal, 105(3):302–311.

Lura, P., Trtik, P., and Münch, B. (2011). Validity of recent approaches for statistical nanoin-dentation of cement pastes. Cement and Concrete Composites, 33(4):457–465.

Martìnez-Ramìrez, S., Blanco-Varela, M., Ereña, I., and Gener, M. (2006). Pozzolanic re-activity of zeolitic rocks from two different cuban deposits: Characterization of reactionproducts. Applied Clay Science, 32(1–2):40–52.

137

Martirena, J., Gonzalez, D., Alvarado, Y., Abreu, M., and Machado, I. (2012). Bioplasticizeras fluidizer in cementitious materials. Construction and Building Materials.

McDonald, B., Saraf, V., and Ross, B. (2004). A spectacular collapse: Koror-Babeldaob(Palau) balanced cantilever prestressed, post-tensioned bridge. Indian Concrete Journal,77(3):955–962.

McLachlan, G. and Krishnan, T. (2008). The EM Algorithm and Extensions (Wiley Series inProbability and Statistics). Wiley-Interscience.

Miller, M., Bobko, C., Vandamme, M., and Ulm, F.-J. (2008). Surface roughness criteria forcement paste nanoindentation. Cement and Concrete Research, 38(4):467–476.

Mori, T. and Tanaka, K. (1973). Average stress in matrix and average elastic energy ofmaterials with misfitting inclusions. Acta Metallurgica, 21(5):571–574.

Nemat-Nasser, S. and Hori, M. (1993). Micromechanics: Overall properties of heterogeneousmaterials, volume 47. Applied Mathematics and Mechanics.

Nettleton, D. (1999). Convergence properties of the em algorithm in constrained parameterspaces. Can J Statistics, 27(3):639–648.

Němeček, J. (2009). Creep effects in nanoindentation of hydrated phases of cement pastes.Materials Characterization, 60(9):1028–1034.

Oliver, W. and Pharr, G. (1992). Improved technique for determining hardness and elasticmodulus using load and displacement sensing indentation experiments. Journal of materialsresearch, 7(6):1564–1583.

Oliver, W. and Pharr, G. (2004). Measurement of hardness and elastic modulus by instru-mented indentation: Advances in understanding and refinements to methodology. Journalof Materials Research, 19(01):3–20.

Pellenq, R. J.-M., Kushima, A., Shahsavari, R., Van Vliet, K. J., Buehler, M. J., Yip, S.,and Ulm, F.-J. (2009). A realistic molecular model of cement hydrates. Proceedings of theNational Academy of Sciences, 106(38):16102–16107.

Petersson, G. A. (2000). Perspective on “the activated complex in chemical reactions”.Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica ChimicaActa), 103(3):190–195.

Petrov, M. (1998). Rheological properties of materials from the point of view of physicalkinetics. Journal of Applied Mechanics and Technical Physics, 39(1):104–112.

138

Pichler, B. (2010). Estimation of influence tensors for eigenstressed multiphase elastic mediawith nonaligned inclusion phases of arbitrary ellipsoidal shape. Journal of EngineeringMechanics, 136(8):1043–.

Pichler, B., Cariou, S., and Dormieux, L. (2009a). Damage evolution in an undergroundgallery induced by drying. International Journal For Multiscale Computational Engineering,7(2):65–89.

Pichler, B. and Dormieux, L. (2010a). Cracking risk of partially saturated porous media—PartI: Microporoelasticity model. Int. J. Numer. Anal. Meth. Geomech., 34(2):135–157.

Pichler, B. and Dormieux, L. (2010b). Cracking risk of partially saturated porous media—PartII: Application to drying shrinkage. Int. J. Numer. Anal. Meth. Geomech., 34(2):159–186.

Pichler, B., Hellmich, C., and Eberhardsteiner, J. (2009b). Spherical and acicular represen-tation of hydrates in a micromechanical model for cement paste: prediction of early-ageelasticity and strength. Acta Mechanica, 203(3-4):137–162.

Pichler, B., Scheiner, S., and Hellmich, C. (2008). From micron-sized needle-shaped hydratesto meter-sized shotcrete tunnel shells: micromechanical upscaling of stiffness and strengthof hydrating shotcrete. Acta Geotechnica, 3(4):273–294.

Pichler, C., Lackner, R., and Mang, H. A. (2007). A multiscale micromechanics model forthe autogenous-shrinkage deformation of early-age cement-based materials. EngineeringFracture Mechanics, 74(1-2):34–58.

Pilz, M. (1997). The collapse of the KB Bridge in 1996. Master’s thesis, Imperial CollegeLondon.

Radok, J. (1957). Visco-elastic stress analysis. Quart. Appl. Math., 15:198–202.

Ramachandran, S., Ramakrishnan, V., and Bang, S. (2001). Remediation of concrete usingmicro-organisms. ACI Materials Journal-American Concrete Institute, 98(1):3–9.

Sanahuja, J. (2008). Impact de la morphologie structurale sur les performances mécaniquesdes matériaux de construction: Application au plâtre et à la pâte de ciment. PhD thesis,École Nationale des Ponts et Chaussées.

Sanahuja, J., Dormieux, L., and Chanvillard, G. (2007). Modelling elasticity of a hydratingcement paste. Cement and Concrete Research, 37(10):1427–1439.

Sanahuja, J., Dormieux, L., and Chanvillard, G. (2008). A reply to the discussion “does C–S–Hparticle shape matter?” F.-J. Ulm and H. Jennings of the paper “modelling elasticity of ahydrating cement paste”, CCR 37 (2007). Cement and Concrete Research, 38(8-9):1130–1134.

139

Scrivener, K., Crumbie, A., and Laugesen, P. (2004). The interfacial transition zone (itz)between cement paste and aggregate in concrete. 12(4):411–421–.

Sneddon, I. N. (1965). The relation between load and penetration in the axisymmetric Boussi-nesq problem for a punch of arbitrary profile. International Journal of Engineering Science,3(1):47–57.

Sobol’, I. (1993). Sensitivity estimates for nonlinear mathematical models. Mathematicalmodeling and computational experiment, 1:407–414.

Soize, C. and Ghanem, R. (2004). Physical systems with random uncertainties: Chaos repre-sentations with arbitrary probability measure. SIAM J. Sci. Comput., 26(2):395–410.

Sorelli, L., Constantinides, G., Ulm, F.-J., and Toutlemonde, F. (2008). The nano-mechanicalsignature of ultra high performance concrete by statistical nanoindentation techniques.Cement and Concrete Research, 38(12):1447–1456.

Spaepen, F. (1977). A microscopic mechanism for steady state inhomogeneous flow in metallicglasses. Acta Metallurgica, 25(4):407–415.

Stefan, L., Benboudjema, F., Torrenti, J. M., and Bissonnette, B. (2010). Prediction of elasticproperties of cement pastes at early ages. Computational Materials Science, 47(3):775–784.

Stora, E., He, Q.-C., and Bary, B. (2006). Influence of inclusion shapes on the effective linearelastic properties of hardened cement pastes. Cement and Concrete Research, 36(7):1330–1344.

Stroud, A. (1971). Approximate calculation of multiple integrals. Prentice-Hall.

Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. ReliabilityEngineering & System Safety, 93(7):964–979.

Sudret, B., Yalamas, T., Noret, E., and Willaume, P. (2010). Sensitivity analysis of nestedmultiphysics models using polynomial chaos expansions. Safety, Reliability and Risk ofStructures, Infrastructures and Engineering Systems – Furuta, Frangopol & Shinozuka (eds),pages 3883–3890.

Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models – Contributions to structural reliability and stochastic spectral methods.Habilitation à diriger des recherches, Université Blaise Pascal, Clermont-Ferrand, France.

Tapli, J. (1959). A method for following the hydration reaction in portland cement paste.Australian Journal of Applied Science, 10:329–345.

Taylor, H. (1990). Cement chemistry. Academic Press.

140

Tennis, P. D. and Jennings, H. M. (2000). A model for two types of calcium silicate hydrate inthe microstructure of portland cement pastes. Cement and Concrete Research, 30(6):855–863.

Trtik, P., Dual, J., Muench, B., and Holzer, L. (2008). Limitation in obtainable surfaceroughness of hardened cement paste: ‘virtual’ topographic experiment based on focussedion beam nanotomography datasets. Journal of Microscopy, 232(2):200–206.

Trtik, P., Kaufmann, J., and Volz, U. (2012). On the use of peak-force tapping atomic forcemicroscopy for quantification of the local elastic modulus in hardened cement paste. Cementand Concrete Research, 42(1):215–221.

Trtik, P., Munch, B., and Lura, P. (2009). A critical examination of statistical nanoindentationon model materials and hardened cement pastes based on virtual experiments. Cement andConcrete Composites, 31(10):705–714.

Ulm, F., Constantinides, G., and Heukamp, F. (2004). Is concrete a poromechanics materials?- A multiscale investigation of poroelastic properties. Materials and Structures, 37(1):43–58.

Ulm, F.-J., Le Maou, F., and Boulay, C. (2000). Creep and shrinkage of concrete-kineticsapproach. Creep and shrinkage: Structural design Effects, pages 134–153.

Ulm, F.-J., Vandamme, M., Bobko, C., Alberto Ortega, J., Tai, K., and Ortiz, C. (2007).Statistical indentation techniques for hydrated nanocomposites: Concrete, bone, and shale.Journal of the American Ceramic Society, 90(9):2677–2692.

Ulm, F.-J., Vandamme, M., Jennings, H. M., Vanzo, J., Bentivegna, M., Krakowiak, K. J.,Constantinides, G., Bobko, C. P., and Van Vliet, K. J. (2010). Does microstructure matterfor statistical nanoindentation techniques? Cement and Concrete Composites, 32(1):92–99.

Vallée, D. (2013). Mechanical and chemical characterization of the heterogeneous microstruc-ture of green concrete with mineral additions. Master’s thesis, Universitée Laval.

Vallée, D., Sorelli, L., and Chen, J. (In preparation). Chemo-mechanical Characterization ofCement Paste Microstructures: a One-to-One Coupling of SNT and SEM-EDS. Cementand Concrete Research.

Vandamme, M. (2008). The nanogranular origin of concrete: A nanoindentation investigationof microstructure and fundamental properties of calcium-silicate-hydrates. PhD thesis, Mas-sachusetts Institute of Technology.

Vandamme, M., Tweedie, C., Constantinides, G., Ulm, F., and Van Vliet, K. (2012). Quanti-fying plasticity-independent creep compliance and relaxation of viscoelastoplastic materialsunder contact loading. Journal of Materials Research, 27(1):302–312.

141

Vandamme, M. and Ulm, F. J. (2009). Nanogranular origin of concrete creep. Proceedings ofthe National Academy of Sciences of the United States of America, 106(26):10552–10557.

Vandamme, M. and Ulm, F.-J. (2013). Nanoindentation investigation of creep properties ofcalcium silicate hydrates. Cement and Concrete Research, 52(0):38–52.

Vandamme, M., Ulm, F.-J., and Fonollosa, P. (2010). Nanogranular packing of C–S–H atsubstochiometric conditions. Cement and Concrete Research, 40(1):14–26.

Vanzo, J. (2009). A nanochemomechanical investigation of carbonated cement paste. Master’sthesis, Massachusetts Institute of Technology.

Velez, K., Maximilien, S., Damidot, D., Fantozzi, G., and Sorrentino, F. (2001). Determinationby nanoindentation of elastic modulus and hardness of pure constituents of portland cementclinker. Cement and Concrete Research, 31(4):555–561.

Wand, M. and Jones, M. (1995). Kernel smoothing. Chapman and Hall.

Wittmann, F. (1982). Creep and shrinkage mechanisms. Creep and shrinkage in concretestructures, pages 129–161.

Wu, C. F. J. (1983). On the convergence properties of the em algorithm. The Annals ofStatistics, 11(1):95–103.

Wu, M., Johannesson, B., and Geiker, M. (2012). A review: Self-healing in cementitiousmaterials and engineered cementitious composite as a self-healing material. Constructionand Building Materials, 28(1):571–583.

Yee, A. (1979). Record span box girder bridge connects Pacific Islands. Concrete International,1(6):22–25.

Zanjani Zadeh, V. and Bobko, C. P. (2013). Nanoscale mechanical properties of concretecontaining blast furnace slag and fly ash before and after thermal damage. Cement andConcrete Composites, 37(0):215–221.

Zaoui, A. (2002). Continuum micromechanics: Survey. J. Eng. Mech., 128(8):808–816.

Zhang, Q., Le Roy, R., Vandamme, M., and Zuber, B. (2013). Long-term creep propertiesof cementitious materials - comparing compression tests on concrete with microindenta-tion tests on cement. In Poromechanics V, pages 1596–1604–. American Society of CivilEngineers.

142