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This is to certify that the
dissertation entitled

DEVELOPMENT OF STRUCTURE-PROPERTY
RELATIONSHIPS FOR HYDRATED CEMENT PASTE,
MORTAR AND CONCRETE

presented by

TEWODROS TEKESTE GHEBRAB

has been accepted towards fulﬁllment
of the requirements for the

Ph.D. degree in Civil and Environmental
Engineering

 

 

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DEVELOPMENT OF STUCTURE-PROPERTY RELATIONSHIPS FOR
HYDRATED CEMENT PASTE, MORTAR AND CONCRETE

By

Tewodros Tekeste Ghebrab

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Civil & Environmental Engineering

2008

ABSTRACT

DEVELOPMENT OF STUCTURE-PROPERTY RELATIONSHIPS FOR
HYDRATED CEMENT PASTE, MORTAR AND CONCRETE

By

Tewodros Tekeste Ghebrab

A mechanistic approach was adopted to develop models for the mechanical properties of
hydrated cement paste, mortar and concrete. These models reﬂect the fundamental
structure of concrete, and represent a departure from the predominantly empirical models
available for cement-based materials. The intrinsic elastic modulus and fracture
toughness of hydrated cement paste were determined based on the interatomic
interactions between calcium silicate hydrate (C-S-H) particles, which are the primary
binder among the hydration products of cement. The elastic modulus model of hydrated
cement paste was developed by introducing the effect of elliptical capillary pores into the
corresponding intrinsic model. The fracture toughness model of hydrated cement paste
was also developed by introducing the energy dissipation associated with the C-S-H/C-S-
H debonding and the phononic frictional pullout of calcium hydroxide (CH) crystals ,
with the latter phenomenon found to be the major contributor. The strength of hydrated
cement paste was determined using the elastic fracture mechanics principles.
Experimental results and available empirical models were used to validate the models.

Mechanical models for cement mortar were developed by introducing the effects of ﬁne

aggregate on hydrated cement paste. The elastic modulus model of cement mortar
accounts for the positive effects of the high elastic modulus ﬁne aggregates, and the
negative effects of the interfacial transition zone. The ﬁacture toughness model of cement
mortar was developed by introducing the energy dissipation associated with the pull-out
of ﬁne aggregates into the fracture toughness model of hydrated cement paste. The
strength model of cement mortar was developed using the elastic modulus and fracture
toughness models based on fracture mechanics concepts, with due consideration given to
the effects of ﬁne aggregates in restrained shrinkage microcracking of cement mortar.
Experimental results and empirical models were used to verify the mechanistic models of
cement mortar. The mechanistic models of concrete were developed based on the
corresponding models of cement mortar by introducing various effects of coarse
aggregates. The fracture toughness model of concrete accounted for the energy
dissipation associated with pullout of coarse aggregates from cement mortar. The
approach to development of elastic modulus and strength models was similar to that used
for development of the corresponding models of cement mortar. The predictions of the
mechanistic models of concrete compared favorably with experimental results. The
mechanistic models of concrete were tailored for application to high-volume ﬂy ash
concrete, and were used to determine the effects of high-volume replacement of cement
with coal ﬂy ash on the mechanical properties of concrete. Some comparisons were also
made between the predictions of the mechanistic models and the test data available for

high-volume ﬂy ash concrete.

DEDICATION

This dissertation is fondly dedicated to my dear wife, children, father, mother, sisters,

brothers, and the Baha’i Community of East Lansing for their support and prayers.

Many Thanks be to The Almighty God!

iv

ACKNOWLEDGEMENT

The work of this research was supported partly by HE, Fulbright, and Washtenaw County
Road Commission. The author thanks the research guidance committee for their
comments and contributions. Special thanks goes to my advisor, Professor Parviz

Soroushian, for his understanding and support.

TABLE OF CONTENTS

LIST OF TABLES ........................................................................... ix
LIST OF FIGURES ......................................................................... xi
KEY TO SYMBOLS OR ABREVIATIONS ........................................... xvii
INTRODUCTION ........................................................................... 1
CHAPTER 1
LITERATURE REVIEW .................................................................. 6
1.1 Chemistry of Hydration of Cement ............................................... 6
1.1.1 Introduction ............................................................ 6
1.1.2 Hydration of Tricalcium Silicate (C3S) and Dicalciurn Silicate
(CZS) ...................................................................... 7
1.2 Capillary Pores ...................................................................... 14
1.3 Structure-Property Relationship of Cementitious Materials .................. 19
1.4 Fly Ash ............................................................................... 23
1.4.1 Type and Trends in Quality of Fly Ash ............................ 23
1.4.2 Effect of Fly Ash on Chemistry of Cement Hydration and
Cement Hydration Products .......................................... 24
1.4.3 Effect of Fly Ash on Capillary Pore System ....................... 27
1.4.4 Air-Entrainment of Concrete with Fly Ash ........................ 30
1.4.5 Effect of High Fly Ash Volumes on Engineering Properties of
Concrete ................................................................ 32
1.4.6 Effect of High Fly Ash Volumes on Hydration of Cement. . 32
1.4.7 Effect of High Fly Ash Volumes on Strength of Concrete ...... 35
CHAPTER 2
MECHANICAL MODELS OF HYDRATED CEMENT PASTE ................... 38
2.1 Introduction .......................................................................... 38
2.2 Hydrated Cement Paste Free From Capillary Pores and Microcracks
(Compact Hydrated Cement Paste) .............................................. 39
2.2.1 Molecular Interactions Between C-S-H Particles ................... 39

2.2.2 Modulus of Elasticity of Compact Hydrated Cement Paste (E0).. 42

2.2.3 Fracture Toughness of Compact Hydrated Cement Paste (Go). . .. 48
2.2.3.1 Fracture Toughness Due to Debonding Between C-S-H

Globules (Gd) ................................................... 48

2.2.3.2 Fracture Toughness Due to CH Crystal Pull-out (GP). . 51

2.2.4 Intrinsic Tensile Strength of Compact Hydrated Cement Paste
(Om) ........................................................................
2.3 Hydrated Cement Paste ............................................................
2.3.1 Introduction ...............................................................
2.3.2 Elastic Modulus of Hydrated Cement Paste .........................
2.3.3 Fracture Toughness of Hydrated Cement Paste .....................
2.3.4 Tensile Strength of Hydrated Cement Paste .........................
2.3.4.1 Shrinkage Microcracks .........................................
2.3.4.2 Tensile Strength of Hydrated Cement Paste ................
2.3.4.3 Parametric Study of the Effects of Various Parameters
on the Tensile Strength of Hydrated Cement Paste. . .. .
2.3.4.4 Comparing the Predictions of the Mechanistic Tensile
Strength Model of Hydrated Cement Past with those of
Empirical Models .............................................
2.3.5 Compressive Strength Model of Hydrated Cement Paste ..........

CHAPTER 3
MODELING THE MECHANICAL PROPERTIES OF MORTAR BASED
ON MECHANISTIC APPROACH .......................................................
3.1 Introduction ..........................................................................
3.2 Modeling .............................................................................
3.2.1 Modulus of Elasticity of Cement Mortar .............................
3.2.2 Parametric Study of the Effect of the Fine Aggregate Elastic
Modulus on the Modulus of Elasticity of Cement Mortar .........
3.2.3 Comparing the Predictions of Mechanistic Model of the Elastic
Modulus of Cement Mortar with those of Empirical Models. .
3.2.4 Mechanistic Model for the Fracture Toughness of Cement
Mortar .....................................................................
3.2.4.1 Energy Released During Debonding of Sand Particles
from Hydrated Cement Paste .................................
3.2.4.2 Energy Released During Pull-out of Sand Particles from
Hydrated Cement Paste ........................................
3.2.5 Tensile Strength Model of Cement Mortar ...........................

CHAPTER 4
MODELING THE MECHANICAL PROPERTIES OF CONCRETE BASED
ON MECHANISTIC APPROACH .......................................................
4.1 Introduction ..........................................................................
4.2 Modeling .............................................................................
4.2.1 Modulus of Elasticity of Concrete ....................................
4.2.1.1 Parametric Study on the Inﬂuence of the Modulus of
Elasticity of Coarse Aggregates on the Concrete Elastic
Modulus ..........................................................
4.2.1.2 Comparing the Predictions of the Mechanistic Model of
Concrete Elastic Modulus with those of Empirical
Models ............................................................

vii

6O
61
61
61
66
66
67
70

71

89
91

94
94
95
95
106
116
117
120

122
130

135
135
137
138

148

156

4.2.2 Mechanistic Model for the Fracture Toughness of Concrete ...... 158
4.2.2.1 Energy Released During Debonding of Coarse
Aggregates from Hydrated Cement Paste in the
Interfacial Transition Zone of Concrete ..................... 162
4.2.2.2 Energy Released During Pull-out of Coarse Aggregates
from Cement Mortar in the Interfacial Transition Zone

of Concrete ...................................................... 163
4.2.3 Mechanistic Model for the Tensile Strength of Concrete .......... 168
4.2.4 Compressive Strength Model of Concrete ........................... 174
4.2.5 Comparing the Predictions of the Compressive Strength Model
of Concrete with those of Empirical Models ........................ 179
CHAPTER 5
MECHANICAL MODELS OF HIGH-VOLUME FLY ASH CONCRETE ........ 181
5.1 Introduction .......................................................................... 1 81
5.2 Development of a Strength Model for Hi gh-Volume Fly Ash Concrete. . .. 182
5.3 Concrete Containing 50% Replacement of Cement with Fly Ash ........... 183
5.3.1 Fracture Toughness of Concrete Containing 50% Fly Ash as
Replacement for Cement ............................................... 185
5.3.2 Modulus of Elasticity of Concrete Containing 50% Fly Ash As
Replacement for Cement .............................................. 193
5.3.3. Tensile Strength of Concrete Containing 50% Fly Ash ........... 198

5.3.4 Compressive Strength of Concrete Containing 50% Fly Ash. . 202
5.3.5 Comparing the Predictions of the Mechanistic Model of the
Strength of Concrete Containing 50% Fly Ash Cement with

Experimental Results ................................................... 206
SUMMARY .................................................................................. 209
REFERENCES .............................................................................. 218

viii

LIST OF TABLES

Table 1.1 - Calculated porosities based on the Poweres-Brownyard model

Table 1.2 - Degree of reaction of ﬂy ash in ﬂy ash concrete pastes

Table 2.1 - Effects of the distance between C-S-H globules on bond strength and modulus
of elasticity of compact hydrated cement paste free from capillary pores and
microcracks

Table 2.2 - Calculation of the fracture toughness (due to debonding of C-S-H globules) of
compact hydrated cement paste free from capillary pores and microcracks

Table 2.3 - Summary of the variations of the fracture toughness due to C-S-H/C-S-H
debonding, (Gd) in J/mz, with radius of globules, Rg, and dew

Table 2.4 - Signiﬁcance of CH diameter and thickness on fracture toughness due to CH
pull-out (Gp)

Table 3.1 - Volume fractions of the interfacial transition zone and the hydrated cement
paste for different aggregate sizes (at constant aggregate volume) in cement
mortar

Table 3.3 - The Eh and E; values calculated for different values of capillary porosity in
hydrated cement paste

Table 3.4 - The bulk and shear moduli of ﬁne aggregate (sand) in mortar

Table 3.5 - The upper and lower bounds of bulk modulus and shear modulus of cement
mortar with ﬁne aggregate having modulus of elasticity of 70 GPa

Table 3.6 - The upper and lower bounds of bulk modulus and shear modulus of cement
mortar with ﬁne aggregate having modulus of elasticity of 80 GPa

Table 3.7 - The upper and lower bounds of bulk modulus and shear modulus of cement
mortar with ﬁne aggregate having modulus of elasticity of 90 GPa

Table 3.8 - The upper and lower bounds of the elastic moduli of three cement mortars
each containing a ﬁne aggregate with a speciﬁc modulus of elasticity

ix

Table 4.1 - Volume fractions of the interfacial transition zone and the cement mortar for
different sizes of coarse aggregates in concrete

Table 4.2 - Em and Eic values calculated for different values of hydrated cement paste
capillary porosity

Table 4.3 - Bulk and shear moduli of coarse aggregates in concrete.

Table 4.4 - Upper and lower bounds of the bulk and shear moduli of concrete for
different sizes of coarse aggregates having an elastic modulus of 70 GPa

Table 4.5 - Upper and lower bounds of the bulk shear moduli of concrete with coarse
aggregates of different size having elastic modulus of 80 GPa

Table 4.6 - Upper and lower bounds of the bulk and shear moduli of concrete with coarse
aggregates of different size having elastic modulus of 90 GPa

Table 4.7 - Upper and lower bounds of the modulus of elasticity of three concretes with
coarse aggregates of different elastic moduli

Table 4.8 - Tensile strength of concrete with different maximum aggregate sizes
Table 4.9 - Compressive strength of concrete with different maximum aggregate sizes
Table 5.1 - Reduction of CH and ﬂy ash from hcp during pozolarric activity

Table 5.2 -Effect of unreacted ﬂy ash and cement particles on fracture toughness of
hydrated cement paste

Table 5.3 - Modulus of elasticity of hcp, mortar and concrete and ﬁ'acture toughness and
tensile strength of concrete, containing 50% ﬂy ash, in 28-days

Table 5.4 - Modulus of elasticity of hcp, mortar and concrete and fracture toughness and
tensile strength of concrete, containing 50% ﬂy ash, in 90-days

Table 5.5 - Modulus of elasticity of hcp, mortar and concrete and fracture toughness and
tensile strength of concrete, containing 50% ﬂy ash, in l80-days

Table 5.6 - Modulus of elasticity of hcp, mortar and concrete and ﬁ'acture toughness and
compressive strength of concrete, containing 50% ﬂy ash, in 28-days

Table 5.7 - Modulus of elasticity of hcp, mortar and concrete and fracture toughness and
compressive strength of concrete, containing 50% ﬂy ash, in 90-days

Table 5.8 -Modu1us of elasticity of hep, mortar and concrete and fracture toughness and
compressive strength of concrete, containing 50% ﬂy ash, in 180-days

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

LIST OF FIGURES

1.1 - Microstructure of a hydrated cement paste. It shows that the structure is not

homogenous; some areas are dense and the others are porous. Massive crystals
of calcium hydroxide, long and slender needles of ettringite, and aggregation of
small ﬁbrous crystals of calcium silicate hydrate can be observed by magnifying
the porous arrears

1.2 - Model of a well-hydrated Portland cement paste ‘A’ represents aggregation of

poorly crystalline C-S-H particles. ‘H’ represents hexagonal crystalline products
such as CH, C4ASH13, C4AH19. ‘C’ represents capillary pores

1.3 - Schematic representation of the internal structure of a cement paste in various

phases of hydration (not to scale) (a) Dormant period (b) and (c) Setting ((1)
Hardening; Black mass: unhydrated cement; Fibers: hydration products. White
mass: pores

1.4 - Degree of hydration of the constituents in a type I cement as a function of time

1.5 - (A,B) Types I and II C-S-H, respectively (SEM of fracture surfaces, courtesy

K.L. Scrivener). (C,D) SEM/STEM pair of ion beam thinned section, showing
Type III C-S-H and Type IV C-S-H. (A) is of an ordinary Portland cement
paste, w/c = 0.5, aged 10 h. (B) is of a paste of an oil well cement, w/c = 0.44,
with 2.4% of CaClz on the weight of cement, aged 1 day. (C) and (D) are of a
C33 paste, w/c = 0.47, aged 330 days.

1.6 - Pore size distribution in fresh cement paste (C indicates capillary pores; the

black spots are gels; and the spaces between them are gel pores).

1.7 - Relations between compressive strength (log scale) and porosity; data from

various sources for Portland cement pastes cured at ordinary temperatures (open
circles), hot pressed Portland cement paste (ﬁlled circle), autoclaved pastes of
Portland cement, sometimes with added sulfur (ﬁlled square) and autoclaved
pastes of Portland cement with 50% (open squares) or 30% (cross) of added ﬂy
ash.

1.8 - Typical Class F ﬂy ash: (a) spherical and glassy particles; (b) a Plerospher

1.9 - Cement Particles

xi

Fig. 1.10 - Diagrammatic representation of well-hydrated cement pastes made with a
Portland pozzolana cement. As compared to Fig. 1.2, the capillary voids are
either eliminated or reduced due to the pozzolanic reaction.

Fig. 1.11 — Change in pore size distribution (determined by mercury porosimetry) in
cement paste containing 30% of Class F ﬂy ash by mass of total cernentitious
material

Fig. 1.12 - Ca(OH2) content relative to the cement content in PC and FC pastes (based on
ignited weight).(a) w/b = 0.3, and (b) w/b = 0.5

Fig. 2.1 - C-S-H morphology and geometric arrangement

Fig. 2.2 - (a) Bonding energy as a function of the distance of separation; (b) force-
displacement curve

Fig. 2.3 - Stress-strain curve approximated as a half sine wave
Fig. 2.4 - Hexagonal CH crystal

Fig. 2.5 - CH orientation with ﬂat surface parallel to the direction of loading (a) side view
(b) front view

Fig. 2.6 - Schematic model of CH pull-out

Fig. 2.7 - Schematics of the effective contact area between CH crystal and C-S-H
globules

Fig. 2.8 - Capillary pore shape and interactions, and the resulting stress concentration
Fig. 2.9 - Modulus of elasticity-porosity relationships

Fig. 2.10 - Schematic presentation of the formation of microcracks in otherwise compact
hydrated cement paste

Fig. 2.11 - Inﬂuence of CH thickness on the strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.12 - Inﬂuence of CH thickness on the strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.13 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 8 nm, distance between

xii

C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.14 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 8 nm, distance between
C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.15 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.16 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.17 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 8 nm, distance between
C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.18 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 8 nm, distance between
C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

Fig. 2.19 - Inﬂuence of the C-S-H globule diameter on the strength-porosity relationships
of hcp for spherical capillary pores, distance between C-S-H globules of 0. 1 8
nm, and CH crystal diameter of: (a) 1000 nm; (b) 10,000 nm

Fig. 2.20 - Inﬂuence of the C-S-H globule diameter on the strength-porosity relationships
of hcp for spherical capillary pores, distance between C-S-H globules of 0.32
nm, and CH crystal diameter of: (a) 1000 nm; (b) 10,000 nm

Fig. 2.21 - Inﬂuence of the C-S-H globule diameter on the strength-porosity relationships
of hcp for ellipsoidal capillary pores, distance between C-S-H globules of 0.18
nm, and CH crystal diameter of: (a) 1000 um; (b) 10,000 nm

Fig. 2.22 - Inﬂuence of the C-S-H globule diameter on the strength-porosity relationships
of hcp for ellipsoidal capillary pores, distance between C-S-H globules of 0.32
nm, and CH crystal diameter of: (a) 1000 nm; (b) 10,000 nm

xiii

Fig. 2.23 - Inﬂuence of the distance between C-S-H globules on the strength-porosity
relationships of hcp for spherical capillary pores with CH diameter of: (a)
1000 nm; (b) 10,000 nm

Fig. 2.24 - Inﬂuence of the distance between C-S-H globules on the strength-porosity
relationships of hcp for ellipsoid capillary pores with CH diameter of: (a)
1000 nm; (b) 10,000 nm

Fig. 2.25 - Comparing the predictions of the mechanistic and empirical tensile strength
models of hcp

Fig. 2.26 - Crack formation due to compressive in hydrated cement paste specimen

Fig. 2.27 - Predications of the compressive strength model of hcp versus those of
empirical models derived based on experimental results

Fig. 3.1 - Variation of the interfacial transition zone thickness with aggregate size
Fig 3.2 - Schematic presentation of the geometric model of a sand particle

Fig. 3. 3- The relationship between the moduli of elasticity of the hydrated cement paste
and the interfacial transition zone in cement mortar

Fig. 3.4 - The upper and lower bounds of the elastic moduli of three cement mortars each
containing ﬁne aggregates with different elastic moduli

Fig. 3.5 - Comparing the predictions of theoretical model for the elastic modulus of
cement mortar with those of empirical model based on substantial test data

Fig. 3.6 - Surface roughness of ﬁne aggregate, and interaction of the hydrated cement
paste (1GC) at the interfacial transition zone

Fig 3.7 - The cracking pattern of the hydrated cement paste dendrite due to force P
generated by sand pull-out

Fig. 3.8 - Surface of sand particle subjected to ﬁictional pull-out

Fig. 3.9 - Fracture toughness of mortar as a ﬁrnction of the capillary porosity of hydrated
cement paste

Fig. 3.10 - Fracture toughness of mortar as a function of the volume ﬁaction of sand for a
capillary porosity of 0.3

Fig. 3.11 - Fracture toughness of mortar as a function of the volume fraction of sand for a
capillary porosity of 0.05

xiv

Fig. 3.12 - Fracture toughness of mortar as a function of the volume fraction of sand for a
capillary porosity of 0

Fig. 3.13 - The critical shrinkage crack in cement mortar bridging between two adjacent
sand particles

Fig. 3.14 - Tensile strength of cement mortar as a function of the capillary porosity of
hydrated cement paste

Fig. 3.15 - Tensile strength of mortar as a function of the volume fraction of sand for a
capillary porosity of 0.3

Fig 4.1 - The coarse aggregate shape model

Fig. 4.2 - Relationship between the elastic moduli of cement mortar and interfacial
transition zone in concrete

Fig. 4.3 - Upper and lower bounds of the modulus of elasticity of three concretes with
coarse aggregates of different elastic moduli

Fig. 4.4 - Comparing the predictions of theoretical model for the elastic modulus of
concrete with those of empirical model based on substantial test data

Fig. 4.5 - Surface roughness of the coarse aggregate, and bonding of cement mortar at the
interfacial transition zone

Fig 4.6 - The cracking pattern of the cement mortar dendrite due to the force P associated
with the coarse aggregate pull-out stress

Fig. 4.7 - Average surface area of the aggregate particle subjected to frictional pull-out
Fig. 4.8 - Fracture toughness of concrete as a function capillary porosity
Fig. 4.9 - Inherent shrinkage crack in cement mortar bridging between two sand particles

Fig. 4.10 - Tensile strength-capillary porosity relationship for concrete with different
maximum aggregates sizes

Fig. 4.11 - Schematic of conical wedges forming above and below coarse aggregates as
concrete fails

Fig. 4.12 - Compressive strength-capillary porosity relationship for concrete with
different maximum aggregates sizes

Fig. 4.13 - Validation of the prediction of compressive strength model of concrete by
experiment based empirical models and results

XV

Fig. 5.1 - Schematic presentation of approximate surface area of ﬂy ash particle subjected
to pull-out as hcp cracks

Fig. 5.2 - Effect of age on fracture toughness of high-volume ﬂy ash hcp
Fig. 5.3 - Tensile strength of high-volume ﬂy ash concrete at different ages
Fig. 5.4 - Compressive strength of hi gh-volume ﬂy ash concrete at different ages

Fig. 5.5 - Validation of the 28-day compressive strength model of concrete containing
50% ﬂy ash

Fig. 5.6 - Validation of the 90-day compressive strength model of concrete containing
50% ﬂy ash

xvi

KEY TO SYNIBOLS OR ABBREVIATIONS

A = empirical coefﬁcient

Ac = effective projected contact area of C-S-H globule

Acad = surface area of a coarse aggregate subjected to debonding and pull-out stresses
A ’60,; = effective surface area of a coarse aggregate subjected to debonding stresses
A 'cap = effective surface area of a coarse aggregate subjected to pull-out friction

A 'capm = total surface area of the cement mortar subjected to shear stress during pull-out
of coarse aggregate

A EC” = effective are of CH subjected to pull-out ﬁiction

A fa = projected plane area of the ﬂy ash surface are subjected to pull-out

Ag = total projected area of C-S-H globule

A s = surface area of CH subjected to pull-out friction

A 3d: interfacial surface are of sand subjected to debonding stress

A ’3‘; = effective interfacial surface area of sand subjected to debonding stress
A ’3’, = effective interfacial surface area of sand subjected to pull-out friction
A ’sph = total surface area of hcp subjected to ﬁictional stress during sand pull-out
A 5k = surface area of CH subjected to skin ﬁiction

a = half of major axis length of elliptical capillary pore

01,: 23 = degree of hydration at the age of 28-days

[70 = equilibrium atomic spacing (minimum energy)

b = half of minor axis length of elliptical capillary pore

,6 = empirical coefﬁcient

C = empirical coefﬁcient

CH = calcium hydroxide

CHp = CH produced during hydration

CH, = CH reacted during pozolanic activity

C-S-H = calcium silicate hydrate

DCH = diameter of CH crystal

d = clear distance between two globules

dc = clear distance between capillary pores

dg = diameter of C-S-H globule

E = modulus of elasticity

Ei = modulus of elasticity of interfacial transition zone

E; = energy dissipated during pull-out of a single CH crystal

Eim = modulus of elasticity of interfacial transition zone in mortar

xvii

E}. = modulus of elasticity of hcp

Em- = lower bound modulus of elasticity of cement mortar

Em+ = upper bound modulus of elasticity of cement mortar

E0 = modulus of elasticity at zero porosity

E3 = modulus of elasticity of sand (ﬁne aggregate)

EC = modulus of elasticity of concrete

Em = modulus of elasticity of coarse aggregate

Eccf= modulus of elasticity of high-volume ﬂy ash concrete under compression

Eaf = modulus of elasticity of high-volume ﬂy ash concrete under tension

Ehf = modulus of elasticity of high-volume ﬂy ash cement paste

Etc = modulus of elasticity of interfacial transition zone in concrete

Emf= modulus of elasticity of high-volume ﬂy ash mortar

8 = strain

63;, = shrinkage strain

F f= phononic ﬁiction force

F g = force required to break the bond strength between two C-S-H globules

F W = van der waals force per unit area

Gadd = additional ﬁacture toughness due to the wedging action of mortar cones forming
on top of coarse aggregates in concrete under compression

G; = fracture toughness of concrete under tension

Gcad = fracture toughness of concrete due to debonding of coarse aggregates from mortar

Gcc = fracture toughness of concrete under compression

Gchsp = energy released per unit fractured area of interfacial transition zone in mortar

G, = fracture toughness of hep due to debonding of C-S-H globules

Gfa = the contribution to fracture toughness of a single ﬂy ash particle

0,6 = total energy release rate per unit fractured area of the interfacial transition zone in
concrete

G, = fracture toughness of hcp

th= fracture toughness of high-volume ﬂy ash concrete

Gm = total energy released per unit fractured area of interfacial transition zone in mortar
Gm = fracture toughness of cement mortar

G ’= shear modulus

G; = shear modulus of interfacial transition zone

G; = shear modulus of hcp

G;_ = lower bound shear modulus of cement mortar
GI,” = upper bound shear modulus of cement mortar
G; = shear modulus of concrete

G2,, = shear modulus of coarse aggregate

xviii

G; = shear modulus of interfacial transition zone in concrete
G; = lower bound shear modulus of concrete

0;, = upper bound shear modulus of concrete
Go = intrinsic fracture toughness of hcp
Gof = intrinsic fracture toughness of high-volume ﬂy ash concrete

Gp = ﬁacture toughness of hcp due to CH pull-out

Gsd = ﬁ'acture toughness during debonding of sand from interfacial transition zone
)1, = surface energy

hcp = hydrated cement paste

hw = dielectric constant

K ’= bulk modulus

K1 = stress concentration ampliﬁcation factor for a single capillary pore

K 2 = stress concentration ampliﬁcation factor for two capillary pores

K; = bulk modulus ofhcp

K: = bulk modulus of interfacial transition zone

K ;,_ = lower bound bulk modulus of cement mortar

K :n + = upper bound bulk modulus of cement mortar

K i = bulk modulus of sand (ﬁne aggregate)

K z. = bulk modulus of concrete

K Ea = bulk modulus of coarse aggregate

K; = bulk modulus of interfacial transition zone in concrete
K; = lower bound bulk modulus of concrete

K 2 + = upper bound bulk modulus of concrete

1 = crack length

[6 = critical crack length

NCHP = number of CH crystals per plane section

NCHV = number of CH crystals per unit cubic cell volume

Na, = number of coarse aggregates per unit volume of concrete
N, = number of sand particles per unit volume of cement mortar
P = porosity

P}, = capillary porosity

P,- = porosity of interfacial transition zone

Pm = porosity of mortar

PC = porosity of concrete

PCR = zero-strength porosity

xix

PCH = perimeter of CH crystal

R = radius of CH crystal

RFA = percent reduction of ﬂy ash after t day of hydration

Rg = radius of C-S-H globules

rc = radius of the effective projected contact area of C-S-H globule
rfa = is the radius of the ﬂy ash particle

Sc = surface area of one of the mortar conical wedges covering the coarse aggregate
Sfa = pulled-out surface area of ﬂy ash

SGp = speciﬁc gravity of ﬂy ash particles

S Gpc = speciﬁc gravity of unhydrated cement grains

t, = radius of a sphere equivalent to ﬁne aggregate

a = strength or stress

0'“ = compressive strength of concrete

ac, = tensile strength of concrete

a'ccf = compressive strength of high-volume ﬂy ash concrete

06¢ = tensile strength of high-volume ﬂy ash concrete

0'0 = strength at zero porosity

am = maximum stress to break the bond between C-S-H globules
mm = compressive strength of hcp

ch, = tensile strength of hcp

tCH = thickness of CH crystal

t,- = thickness of interfacial transition zone

, = ratio of the thickness of the interfacial transition zone to a radius of a sphere
equivalent to ﬁne aggregate

tn. = ratio of interfacial thickness to a radius of a sphere equivalent to coarse aggregate

tic = thickness of interfacial transition zone in concrete

rm = radius of radius of a sphere equivalent to coarse aggregate

tf= phononic ﬁiction stress

Ucad = energy released due to debonding of coarse aggregate from mortar

UCG = unhydrated cement grains

UCH = unreacted calcium hydroxide crystals

Uchcapl = energy released due to CH pull-out when A ’capm fractures

Uchcap = energy released due to CH pull-out at the bottom surface of the aggregate
particle

Uchcap = total energy released due to CH pull-out during the pull-out of the aggregate
particle ﬁom the cement mortar

Uchspl = energy released due to CH pull-out upon fracture over A 'sph

Uchsp = energy released due to CH pull-out at the bottom surface of sand particle

Uchsp = total energy released due to CH pull-out during the pull-out of sand from
hydrated cement paste in mortar

XX

U, = strain energy released during cracking

U,- = energy released during cracking

U ,4 = energy released during debonding of sand from interfacial transition zone
VCH = volume of CH crystal

Vf'a = volume fraction of ﬂy ash in hcp

V; = volume fraction of hcp

V}. = volume of hcp

V,- = volume of interfacial transition zone

V", = volume of cement mortar

me = pore volume of mortar

Vph = pore volume of hcp

Vpi = pore volume of interfacial transition zone

V, = volume of sand

V],- = volume of interfacial transition zone surrounding a single aggregate
me = pore volume of mortar

Vpic = pore volume of interfacial transition zone in concrete

VC = volume of concrete

Va, = volume of coarse aggregate

Vic = volume of interfacial transition zone in concrete

Vh-c = volume of interfacial transition zone surrounding a single coarse aggregate
V100 = volume of a single coarse aggregate

V; = volume fraction of sand
V": = volume ﬁaction of mortar
V; = volume fraction of coarse aggregate

V; = volume ﬁaction of interfacial transition zone in concrete

v = poisons ratio

w/b = water-to-binder ratio

y = crack width at the periphery of CH crystal

Z = distance from the geometric center of CH crystal to the line marking the exposed part
of the crystal from the adhering C-S-H globules during pull-out

INTRODUCTION

Portland cement concrete, henceforth referred to as concrete, is a composite material
widely used in the construction industry. It is composed of aggregates and cement binder.
Concrete technology was put into practice by ancient Romans in about 2000 BC,1 using
natural pozzolans as binder. Intensive use of concrete as construction material started
during the 19th century.2 Nowadays, concrete is widely used all over the world.
Worldwide, about six billion tons of concrete are produced annually. Production of
concrete is a $100 billion industry, which provides job opportunities for over two million

individuals.3

Concrete has some drawbacks which limit its effective use. Large weight-to-strength
ratio, sensitivity to environmental attack and brittleness are some of the disadvantages of
concrete. The two main advantages of concrete are its strength and durability. Its low
tensile-to-compressive strength ratio requires generally necessitates the use of reinforcing
materials, which add to the cost of concrete structures. Further, concrete (in spite of
desirable durability characteristics) eventually deteriorates under environmental effects.
The growing emphasis on energy and environmental issues” also demand close
consideration of the energy consumption and polluting effects of cement production. The

addition of pozzolans (e. g., coal ﬂy ash) can help improve the durability of concrete and

reduce the overall energy consumption and polluting effects associated with production

of concrete.

Most models developed for the mechanical properties of cementitious materials are
empirical, and derived based on experimental results. Empirical models are helpful, but

may not provide comprehensive solutions that are generally applicable.

Given the porous nature of concrete, strength-porosity relationships have been
emphasized in modeling of concrete performance. The Balshin model (Eqn. 1.2) is

among the strength-porosity relationships commonly applied to concrete.‘

O'=0'°(1-P)A (1.2)

where, 0', is compressive strength at zero porosity; P is porosity, and A is an empirical

coefﬁcient.

Ryshkewitch conducted experimental investigations to study the inﬂuence of porosity on
ceramic materials, and developed another empirical strength-porosity relationship (Eqn.
1.3).5 The Ryshkewitch model ﬁts well to strength at high porosity levels, but tends to

overestimate strength at relatively low porosities.6

0' = 0' e (1.3)

 

where, a“, is compressive strength at zero porosity; and B is an empirical coefﬁcient.

Another empirical strength-porosity relationship developed by Schiller (Eqn. 1.4)
produces results comparable to those obtained with the Ryshkewitch model except for the

extremes of porosity.’

 

P
0' = Cln I? (1.4)

where, P is porosity; PCR is zero-strength porosity; and C is an empirical coefﬁcient.

 

The purpose of this research was to develop models for the mechanical properties of
concrete which are based on structure-property relationships and account for the
molecular interactions of the phases present in the concrete structure. Hydrated cement
paste (the binder in concrete) is a complex assembly of nano-scale semi-crystalline and
micro-scale crystalline solids bound together by physical interactions, and a host of nano-
to micro-scale defects (pores, cracks). The introduction of aggregate further complicate
the structure of concrete. In spite of these complexities, concrete cannot defy the laws of

physics and mechanics; it should thus be possible to model concrete starting from

 

intermolecular interactions, with mechanics laws employed to incorporate various
complexities into the basic model. A sequential approach is adopted in this research to
develop ﬁrst intrinsic models (neglecting defects) for hydrated cement paste (hep),

followed by modeling of hep with defects (capillary pores and microcracks);

 

——

 

subsequently, the effects of ﬁne and then coarse aggregates on the structure, defects and
failure mechanisms of the hep binder are introduced in order to develop the cement
mortar and concrete models. The hydrated cement paste is modeled as an assembly of
nanostructured calcium silicate hydrate globules and micro-scale crystalline products
(predominantly calcium hydroxide). The hep models provides the basis for development
of cement mortar and concrete models. Cement mortar and concrete are modeled as
three-phase materials comprising hydrated cement paste, aggregates, and the interfacial
transition zone. The models developed in this research are also tailored to evaluate the

mechanical properties of high-volume ﬂy ash concrete.

The theoretical predictions of the fundamental models developed in this research
compare favorably with experimental results, conﬁrming that the complexity of the
concrete structure is not an unsurrnountable obstacle against development of structure-

property relationships based on ﬁrndamental laws of physics and mechanics.

This research. represents an effort to develop models for concrete which start with
interatomic interactions and the nano- and micro-structure of cement hydration products,
and then incorporate such micro-structural features as the capillary pore system and
microcracks, and ﬁnally consider the micro- and macro-structural implications of
introducing ﬁne and coarse aggregates. The key inspiration for development of these
models was to depart from the empiricism which has governed past efforts to model the
behavior of concrete under load. The models developed here represent structure-property

relationships which highlight the need to further explore some key structural features of

concrete, particularly those at nano- and micro-scales. The models can be used to guide
new developments towards improvement of concrete properties through reﬁnement of its

structure.

The mechanistic models developed in this project can be used to provide preliminary
guides in development of new concrete materials (e. g., additives) which reﬁne the
structure and thus properties of concrete. Knowledge of the mechanisms of action of
these materials (additives) in concrete can be used to predict any changes in the concrete
structure (at nano-, micro— and/or macro-scales) resulting ﬁ'om their introduction. The
mechanistic models can then be used to product the effects of such changes in concrete
structure on its material properties. This would guide the selection of additive types and

dosages, and reduce the extent of trial-and-adjustment experimental work.

CHAPTER 1

LITERATURE REVIEW

1.1 Chemistry of Hydration of Cement

1.1.] Introduction

Cement is a complex material composed of different phases which are Tricalcium Silicate
(C38), Dicalcium Silicat B-(CZS), Tricalcium Alurninate (C3A) and Tetracalcium
Aluminoferrite (C4AF). Hydration of cement, as the name implies, is a mere combination
of anhydrate cement phases with water to form hydrate phases.9 The cement particles
disperse in water and form a plastic type material called paste.lo Therefore, hydration of
cement implies the hydration of its constituent phases, which is a very complex chemical
reaction that takes place between water and cement grains to produce hydrates.9 The
amount of water needed to hydrate a certain amount of cement has to be proportioned for
it affects the property of concrete both as fresh as well as hardened paste.lo As the volume
of the hydration products increases in a paste, it creates an interlock mechanism and as a

result the porosity of the paste reduces and leads to setting and hardening.lo

 

Fig. 1.1 - Microstructure of a hydrated cement paste. It shows that the structure is not
homogenous; some areas are dense and the others are porous. Massive crystals
of calcium hydroxide, long and slender needles of ettringite, and aggregation
of small ﬁbrous crystals of calcium silicate hydrate can be observed by
magnifying the porous arrears. ”

1.1.2 Hydration of T ricalcium Silicate (C36) and Dicalcium Silicate (CpS)

Tricalcium silicate or Alite (C38) and Dicalcium silicate or Belite B-(CZS) produce
calcium hydroxide (CH) and calcium silicate hydrates of amorphous or poorly crystalline

and porous ge1(C-S-H) (Fig. 1.1). 9

During the early stage of hydration, the solution becomes saturated with the CH produced
from C38 and C3A. At the later age, C38 continues to produce C-S—H gel. The solution at

this period is saturated with CH and C-S-H (Fig. 1.2).”

 

l m
t——-"—-—l

Fig. 1.2 - Model of a well-hydrated Portland cement paste. ‘A’ represents aggregation of
poorly crystalline C-S-H particles. ‘H’ represents hexagonal crystalline products
such as CH, C4ASH13, C4AH19. ‘C’ represents capillary pores. “

C38 has a more predominant effect on the setting process of cement paste than C3A, for a

large amount of C-S-H gels were observed to form and interlock at the initial set, which

grows into an interlocking mass at the ﬁnal set (Fig. 1.3).12

\\

~

 

Fig. 1.3 - Schematic representation of the internal structure of a cement paste in various
phases of hydration (not to scale). (a) Dormant period. (b) and (c) Setting. ((1)
Hardening. Black mass: unhydrated cement. Fibers: hydration products. White

mass: pores. ”

There are. two fundamental stages in the rate of hydration of cement. The ﬁrst stage is
that at early stage, hydration is controlled by chemical reaction and thus each particle
hydrates at its own rate. The second stage occurs when the outer surfaces of gains are
coated with hydrate C-S-H gels, and as a result the mechanism of hydration is governed
by the diffusion of new hydrate products ﬁom the anhydrate gain surface. These newly
formed hydrate gels diffuse into the water-ﬁlled outer space through the pores of the
coating C-S-H gel. In this later stage, the difference in rate of hydration becomes small

for a given cement type, as shown in Fig. 1.4.‘3

 

O Alite - _
. Belite ‘
e C3A _ " °

0.8 " 9 Ferrite s.s.

1.0”

Degee of hydration

0.2"

 

 

 

 

0.1

Time in days

Fig. 1.4 - Degee of hydration of the constituents in a type I cement as a function of time12

10

Various studies have been made on C38 and B-C28 to determine their composition in
terms of Ca/Si ratio. Ca/Si ratio of C-S-H can be determined from the amount of CH and
C02 present in a fully hydrated paste, using TG (Therrnogavimetry) and DTA

(Differential Thermal Analysis). “

C-S-H gel or cement paste has a layered structure, which acquires rigidity upon the
occurrence of pore solutions that make interlayer spaces. In a saturated condition, where
the pores are ﬁlled with water, C-S-H can be hydrated to a higher degee. C-S-H
formation in cement paste is highly dependent on the availability of space to enable the
anhydrate gains produce hydrate gels form their internal and outer layers. Thus to
achieve almost a complete hydration and get ample pore space, an optimum amount of
water (42-44% by weight of cement) should be added.9 Different researchers came up
with almost similar composition of C-S-H: 1 .7CaO.SiOz.2. 1H20,”
1.64CaO.8iOz.1.96H20.l6 Based on D-drying procedure, the water content for a fully

saturated C38 is about 20.4-22% corresponding to C-H-S of 1.7CaO.Si02.1.3-1.5H20. ‘7

During hydration, some microscopic investigations have revealed that CH was observed
to accumulate as an isolated mass of small size at more concentration with C38 than with
C28. Further, a gel forms around the anhydrate gains and moves towards the water space
as a hydrate. The hydrates of calcium silicates were found to be of two types; those that
are hydrates of the inner part of the gains, which are large and amorphous in structures,

and those products of the outer part of the gain which forms in the water-ﬁlled space,

11

which have a column or ﬁber like structure.9 C—S-H gel have four morphological types
that develop with age of the cement or calcium silicate paste (Fig. 1.5). ‘3 Type I is ﬁbrous
and Type II is honeycomb like, and both occur at early age. Type III has a tightly packed

gains and Type IV is a featureless which occurs at older age as Type III.

 

 

Fig. 1.5 - (A,B) Types I and H C-S-H, respectively (SEM of ﬁ‘acture surfaces, courtesy
K.L. Scrivener). (C,D) SEM/STEM pair of ion beam thinned section, showing
Type III C-S-H and Type IV C-S-H. (A) is of an ordinary Portland cement
paste, w/c = 0.5, aged 10 h. (B) is of a paste of an oil well cement, w/c = 0.44,
with 2.4% of CaClz on the weight of cement, aged 1 day. (C) and (D) are of a
C38 paste, w/c = 0.47, aged 330 days. '3

1.2 Capillary Pores

Fresh cement paste is mainly composed of hydration products of different phases,
unhydrated cement gains, free water, capillary pores, gel pores and some other minor
components. Capillary pores, the water-ﬁlled spaces in ﬁ’esh cement paste, are the main
suppliers of water for hydration. During hydration, most of the water added to dry cement
is temporarily stored in the capillary pores until it reacts with cement gains. Capillary
pores are part of the goss volume of concrete that has not been replaced by hydration
products. The size of capillary pores in cement paste varies with water/cement ratio and
degee of hydration.” The size and distribution of these pores, in addition to interfacial
cracks, constitute the microstructure of concrete or cement mortar, which is determines
concrete permeability as well as strength.” Capillary pores of highly saturated cement
past form an interconnected piping system distributed throughout the paste. This system
makes the cement paste more permeable and thus more susceptible to deterioration under
freezing and thawing cycles, and chemical attacks.21 Under prolonged curing, the degee
of hydration increases with time and the hydration products continue to precipitate into
the capillary space. As the newly formed hydration products gow in mass, they start to
block the capillary piping system and ﬁnally reduce it to the capillary pore system.”
Further, the newly formed gels suck up the water from the capillary system. This suction
of water reduces the capillary pores size.9 This indicates that strength and durability of

cementitious materials can be reduced by further progess in cement hydration.

14

The unhydrated cement gains react with the water in the capillary pores to produce
porous hydrate gels, having interstitial spaces called gel pores. Gel pores are ﬁlled with
water sucked from the capillary pores. Gel pores are relatively smaller in size than
capillary pores (Fig. 1.6). The water in gel pores is referred as gel water.19 The formation
of hydrate gels depends on the availability of sufﬁcient water for chemical reactions and
for ﬁlling the gel pores that form after reactions. If the water is insufficient, gels form

until the capillary pores dry up.l9

 

Fig. 1.6 - Pore size distribution in fresh cement paste. (C indicates capillary pores;
the black spots are gels; and the spaces between them are gel pores). 22

Under sufﬁcient water/cement ratio, a unit volume of cement produces about double of

its size hydration products. At critical water/cement ratio, which is 1.2 by volume (0.4 by

weight), the hydration products ﬁll up all the available space and no further reaction takes
place. On the other hand, if sufﬁcient water is added, a capillary pore space is created to

keep the extra water.23

Hydration gels have a limiting minimum size beyond which new particles are, due to
insufﬁcient space, unable to hydrate. If the water/cement ratio is low and no water
movement is allowed during hydration, the gel pores do not get sufﬁcient water and the
capillary pores become empty. This is a phenomenon called self desiccation, where the
hydration becomes very slow under the resulting low relative humidity, even though
there is enough space available. Emptying the capillary pores can result in permanent
change of the pore structure, which can affect the strength of the hardened paste. This
demonstrates the necessity of providing sufficient water during curing.9 Compressive
strength of concrete is adversely affected by its porosity (Fig. 1.7).24 The weakening
effect of pores can be simulated to that of tension cracks. These pores minimize the bond
area between hydration products formed by interlocking. Further, for a given porosity,
saturated cement pastes are weaker than dry ones. This is attributed to the fact that water

molecules in pores weaken the forces of attraction between particles.9

16

 

 

 

 

 

VCOMPRESSIVE STRENGTH

2 I l l ‘ I
0 10 20 30 1+0 50
POROSITY (°/ol

Fig. 1.7 - Relations between compressive strength (log scale) and porosity; data from
various sources for Portland cement pastes cured at ordinary temperatures
(open circles), hot pressed Portland cement paste (ﬁlled circle), autoclaved
pastes of Portland cement, sometimes with added sulfur (ﬁlled square) and
autoclaved pastes of Portland cement with 50% (open squares) or 30% (cross)
of added ﬂy ash. 2‘

Taking the total water porosity as the sum of capillary porosity and gel porosity, Powers
and Brownde have demonstrated (Table 1.1) that capillary and total water porosities

decrease while gel porosity increases as hydration proceeds.”

 

Table 1.1 - Calculated porosities based on the Poweres-Brownyard model

 

 

 

 

 

 

w/c Fraction of cement Capillary Gel porosity Total water
Ratio hydrated (a) porosity porosity
0.3 0.00 0.49 0.00 0.49
0.3 0.79 0.00 0.27 0.27
0.4 0.00 0.56 0.00 0.56
0.4 1.00 0.03 0.29 0.32
0.5 0.00 0.61 0.00 0.61
0.5 1.00 0.15 0.26 0.41
0.6 0.00 0.65 0.00 0.65
0.6 1.00 0.24 0.23 0.47

 

 

 

Capillary porosity affects the mechanical properties of cementitious products. The
strength and modulus of elasticity of cement paste, for example, is inversely proportional

to capillary porosity; they are related by the following formula: 2‘

E = BOO—Pk)3 (1.1)

where, E = Elastic modulus; E0 = E at zero porosity; and Ph = capillary porosity.

Unlike strength, modulus of elasticity is larger in saturated pastes than in dry ones. This

is because the water inﬁlling capillary pores cause an increase in stiffness.”

 

 

1.3 Structure-Property Relationships of Cementitious Materials

Though concrete has many properties that need to be addressed during quality assurance,
strength is the most common property used in the evaluation of concrete. The reason
could be that strength can be measured conveniently, and most parameters which affect
concrete strength also inﬂuence other major properties of concrete such as permeability
and durability. Thus, concrete is generally formulated and produced to meet targeted

strength requirements.

Concrete comprises different phases at various scales. These phases include hydrated
cement paste and aggegates at macro-scale, interfacial transition zone, microcracks and
capillary pores at micro-scale, and hydration products such as calcium silicate hydrate
(C-S-H) at nano-scale. This complex composition of concrete makes the theoretical
prediction of its mechanical and physical properties very complicated. Concrete is a
quasi-brittle material and most of the theories applied in predicting the mechanical
properties of brittle ceramics do not directly apply to concrete. The theories developed
based on the nano-scale structure of concrete will have to be reﬁned for application at

micro- or macro-scales.

Lack of proper understanding of the concrete behavior under stress has led to

predominantly empirical approached to modeling of concrete. Empirical models are

helpful, but may not provide comprehensive solutions that are generally applicable.

19

Given the porous nature of concrete, strength-porosity relationships have been
emphasized in modeling of concrete performance. The Balshin model (Eqn. 1.2) is

among the strength-porosity relationships commonly applied to concrete.‘
A
0 = 00(1—P) (1.2)

where, 0', is compressive strength at zero porosity; and A is an empirical coefﬁcient.

Ryshkewitch had conducted experimental investigations to study the inﬂuence of
porosity on ceramic materials, and developed another empirical strength-porosity
relationship (Eqn. 1.3).5 The Ryshkewitch model ﬁts well to strength at high porosity, but

tends to overestimate strength at relatively low porosities.‘5
0' = 0' e (1.3)

where, 0', is compressive strength at zero porosity; and B is an empirical coefficient.

Another empirical strength-porosity relationship developed by Schiller (Eqn. 1.4)
produces results comparable to those obtained with the Ryshkewitch model except for the

extremes of porosity.7

0' = C111 —I(;i- (1.4)

20

where, P is porosity; PCR is zero-strength porosity; and C is an empirical coefﬁcient.

All above models have empirical parameters which depend on material properties. This
makes the models material-speciﬁc, and experiments are required to determine those
parameters for every new material. This limits the range of applicability of such models,
and does not make them useful in exploratory development of new concrete materials. In
the strength models of concrete, porosity is considered as a sole parameter affecting the
property of concrete. New concrete materials, including those incorporating large
volumes of coal ﬂy ash, could reﬁne the size distribution of capillary pores, thereby
altering concrete strength while minimally changing its total porosity. A more
fundamental approach to modeling of concrete is needed, which accounts for the
interatomic interactions at nano-scale, capillary porosity and microcracks at micro-scale,

and the effects of the interfacial transition zone at macro-scale.

With a more comprehensive understanding of concrete structure emerging, several
theoretical models have been developed to predict the mechanical properties of concrete.
However, most of these models account for micro- or macro-scale structure of concrete,
but neglect the fundamental nano-scale features of the concrete structure. Some of the

methods and theories applied in developing these models are described below.

Ortiz and Popov modeled concrete as a two phase composite material, composed of

cement mortar and aggegates, by postulating the energy balance equation.8 First they

modeled the performance of the two phases separately, and then combined them using the

21

mixture theory to get one model for the mechanical performance of concrete. Ortize
accounted for the inelasticity of concrete in his model by applying the rate-independent
model of distributed damage for mortar, and then the mixture theory to account for the
composite nature of concrete.28 Rezende has also used damage mechanics concepts to
propose a rate-independent constitutive theory for the inelastic behavior of concrete,
considering shear and hydrostatic tension damages as the basic damage mechanisms.29
Nerma et. a1. has developed a model for the behavior of concrete under load based on
micromechanics. They modeled concrete at macro-scale, where it was considered as a

two-phase composite material composed of cement mortar and aggegates.30

Past efforts to model the properties of concrete and other cementitious materials have
been largely empirical. There are, however, some efforts reported to develop models
which predict concrete properties based on its basic structure. Mattei et. Al. developed a
multi-sclae model for the mechanical properties of concrete based on physical
interactions between various phases present in concrete structure. Concrete is modeled
here as a two-phase material comprising mortar and coarse aggegate, with mortar
comprising, in turn, of cement paste (with microcracks and pores) and ﬁne aggegates.
This model is oriented toward prediction of the constitutive (stress-strain) behavior of
concrete, and considers the loss of stiffness resulting ﬁ'om crack propagation in concrete.
The cracks are assumed to occur at aggegate interfaces prior to loading, and to propagate
ﬁrst at the interface and then into the mortar phase under load. Fine aspects of the

concrete structure, including the micro-structure of capillary pore system, and the nano-

22

and micro-structures of hydration products (calcium silicate hydrate, calcium hydroxide,

etc.) are not considered in this model.

1.4 Fly Ash

1.4.1 Types and Trends in Quality of Fly Ash

Pozzolana refers to naturally occurring or artiﬁcially produced materials which contains
highly reactive silica. These materials are not cementitious, but they have compounds
which produce cementitious products upon reaction with other compounds from different
sources. To get a stable cementitious compound, the pozzolana has to be divided ﬁnely
and reacted in the presence of water. The reactivity of pozzolanas depends partly on the
shape and ﬁneness of its particles. Pozzolanas of amorphous (glassy) silica react more

readily than those of crystalline ones.”

Fly Ash, the pulverized-fuel ash, is the most common artiﬁcial pozzolana obtained from
the coal used in power stations at combustion temperatures of about 1000 °C.” Fly ash is
different from the ash that collects at the bottom of furnace (bottom ash). Due to their
spherical shape and ﬁneness, ﬂy ash particles have large surface areas which allow them
to react readily.” The chemical properties of ﬂy ash depend on the mineral content of
coal and the burning condition in power station.9 According to ASTM C 618-94a, ﬂy ash
is classiﬁed based on the type and source of coal. Class F ﬂy ash, the most commonly

available which is mainly siliceous, originates form bituminous coal and anthracite. The

23

other type, Class C ﬂy ash, has high lime content and originates form sub-bituminous

coal and ligrite. The latter has more CaO content than the former, and is hydraulic.”

The cementitious property of Class C ﬂy ash increases with the rise in CaO content. With
10% to 20% of CaO content, Class C ﬂy ash acts as cementitious and pozzolanic
materials, respectively; more than 20% of CaO content makes the ash more
cementitious.32 The reason for Class C ﬂy ash, at high lime content, to become
cementitious is that lime can react with the silica and alumina constituents of ash in the
presence of water. Class C ﬂy ash has lower carbon content than Class F, and its ﬁneness
is higher. Class C ﬂy ash is not as popular as Class F in the construction industry. The
reason could be that the lime and sulfur constituents of ash might have a detrimental

effect in concrete.33

1.4.2 Eﬂ'ect of Fly Ash on Chemistry of Cement Hydration and Cement Hydration
Products

Fly ash, as mentioned above, is not cementitious as such; it can be classiﬁed under latent

hydraulic materials, and is always used in conjunction with Portland cement. During

cement hydration, the silica portion of ﬂy ash readily reacts with the calcium hydroxide

(CH), precipitating from the hydration process, to produce a cementitious material

(calcium silicate hydrate). Fly ash can be blended with Portland cement or added to it

even during mixing of concrete.”

24

Fly ash, particularly Class F, is relatively slow to react during early stages of reaction. It
starts to react more intensely almost a week after the initial mixing. This has been
explained by the observation that ﬂy ash becomes stimulated to react when the alkalinity
of pore solution increases.34 The rise in alkalinity is achieved by the concentration of
alkaline products of Portland cement hydration. Therefore, ﬂy ash has to wait until this
favorable condition is created.” When the alkalinity of pore solution reaches its required
level, ﬂy ash particles start to react with the calcium hydroxide produced from hydration
of Portland cement. This reaction takes place on the surface of ﬂy ash particles which

retain their original spherical shape for some time.

Pozzolanic cements, mostly Class F ﬂy ash, have generally low early strength. They are
low heat cements, and resist sulfate and weak acid attacks on concrete. Fly ash reacts
with the calcium hydroxide produced from hydration of Portland cement. This process
reduces the level of calcium hydroxide, and there will remain limited calcium hydroxide

to react with sulfate to produce ettringite, which is the cause for sulfate attack.”

Fly ash concrete absorbs more water, proportional to the ﬂy ash content, than ordinary
Portland cement concrete. This is due to the slow reaction resulting ﬁ'om its pozzolanic
property. Therefore, to achieve the desired effect of ﬂy ash, sufﬁcient curing is

necessary.”

Fly ash concrete demonstrates good workability as compared to Portland cement

concrete. This is attributed to the spherical shape of ﬂy ash particles which reduce

25

viscosity (Fig. 1.8). In Portland cement, under low water/cement ratio and with the
precipitation of new hydration products, the cement past becomes stiff and cement
particles get packed. This may increase the viscosity of the paste because of increased
surface friction, and particle interlocking resulting from the irregular shape of the cement
particles (Fig. 1.9). Unlike Portland cement particles, ﬂy ash particles have spherical
shape which renders less problem of particle interlocking and surface ﬁiction. This
property of ﬂy ash helps in increasing the workability of concrete at low water/cement
ratio. The amount of ﬂy ash required to deal with such problems is only about 20% by

mass.”

 

(a) (b)

Fig. 1.8 - Typical Class F ﬂy ash: (a) spherical and glassy particles; (b) a Plerospher ’5

26

 

Fig. 1.9 - Cement Particles ’6

Fly ash, of a deﬁnite amount, contributes to the later age strength of concrete. Some of
the main factors which contribute to the strength of concrete are the stiffness of the paste
and the paste-aggegate bond. For a given water/cement ratio, the stiffness of the past can
increases when more hydration products, from pozzolanic reaction, diffuse into and

minimize capillary pores.”

1.4.3 Effect of Fly Ash on Capillary Pore System

The pozzolanic activity of ﬂy ash starts when the pore water alkalinity is increased. This

level of alkalinity is reached through the accumulation of calcium hydroxide, produced

from hydration of cement, in the pore water. This level triggers the reaction of calcium

hydroxide with the silica part of the ﬂy ash to give off new hydration products. Over

27

time, the accumulated reaction products diffuse into the capillary pores and result in

narrowing of the capillary system (Figs. 1.10 and 1.11).“

 

Fig. 1.10 - Diagammatic representation of well-hydrated cement pastes made with a
Portland pozzolana cement. As compared to Fig. 1.2, the capillary voids are
either eliminated or reduced due to the pozzolanic reaction”

28

 

§

'é’

§

  
    

Volume of intruded Mercury- mmalg '
B
l l

 

 

 

 

6° " 1 year
0 i4 rrrrrrl r r rrrrrrl
0.001 0.01 0.1 1.0
Pore Diameter- um

Fig. 1.11 - Change in pore size distribution (determined by mercury porosimetry) in
cement paste containing 30% of Class F ﬂy ash by mass of total
cementitious material 3‘

Portland cement concrete with high water/cement ratio has a network of capillary pore

water. These interconnected pipes act as open channel for bleeding of concrete and upon

setting they retain a tube like structure which increases the permeability of concrete. This

phenomenon is considerably reduces in the presence of ﬂy ash. The newly formed C-S-

H, ﬁ'om the pozzolanic activity, start to ﬁll up the capillary pores and even block the

channel; the capillary piping system becomes discrete and small pores.38 The concrete

then gains impermeability and strength as a result of this pozzolanic activity. 39

29

1.4.4 Air-Entrainment of Concrete with Fly Ash

Freezing and thawing are the weather-induced problems which adversely affect the
performance of concrete, especially when it is saturated, unless concrete is properly
desigred to account for such deteriorating effects. The performance of concrete against
frost action can be improved when the following conditions are ﬁrlﬁlled: (1) the presence
of uniformly distributed entrained air; (2) use of coarse aggegates which are sound; and

(3) protecting concrete against frost until it acquires certain strength.”

Air-entrainment of concrete is done during mixing by adding a certain amount of an air-
entraining admixture required to give the desired volume, size and distribution of air
bubbles, with the dispersion of air bubbles remaining unaffected by the process of

transportation, placement and upon hardening of concrete.”

The range of adsorption of air-entraining admixtures depends on the carbon content,
organic matter content and loss-on-igrition of ﬂy ash. Concrete containing Class C ﬂy
ash requires less air-entraining admixture, and looses less entrained air when compared
with concrete containing Class F ﬂy ash. This could be attributed to Class C ﬂy ash
having low carbon and high lime contents. The higher the alkaline content, the lower
would be air-entraining admixture requirement. Concrete containing Class F ﬂy ash
looses about 59% of the total entrained air in about 90 minutes after mix preparation.

Therefore, the loss of entrained air is proportional to the demand for air-entraining

30

admixtures.‘l The excessive demand for air-entraining admixtures can be minimized by
deactivating the carbon content by exposing ﬂy ash to certain chemicals such as chlorine

gas, calcium hypochlorite and surfactants.42

The amount of air-entraining admixtures required in ﬂy ash concrete is higher when
compared with normal cement concrete. This is attributed to the carbon present in ﬂy ash,
which adsorbs the air-entraining agent. The degee of adsorption by carbon depends on
its surface area, type, and polarity. High-carbon-content ﬂy ash disrupts the air
entrainment process by adsorbing some air-entraining agents within porous carbon
particles.” The performance of air-entrained concrete against freezing and thawing has a
direct relation with the amount of ﬂy ash present in concrete. Concrete comprising up to
20% Class C ﬂy ash as replacement for cement demonstrates acceptable performance
against freezing and thawing, as far as the desired size and distribution of air bubbles are
entrained after satisfying the demand of the ﬂy ash carbon. As the ﬂy ash content
increases, the durability of concrete, with respect to frost action, reduces. Concrete with
50% Class C ﬂy ash replacement demonstrated scaling problems after 400 cycles of
freezing and thawing.“3 Further, Virtanen has concluded that the addition of ﬂy ash, to a
certain extent, has no considerable effect on the performance of concrete against freezing
and thawing, as far as the strength and the air content are maintained.“ However, Brown
et al. suggested that, owing to the later age strength development, the performance of
blended cement concrete has to be evaluated after a prolonged curing period. Blended

cements have demonstrated higher resistance against freezing and thawing than ordinary

31

cement concrete. This is attributed to the higher strength gained at later age due to

pozzolanic action."

1.4.5 Effects of High Fly Ash Volumes on Engineering Properties of Concrete

Fly ash, as discussed in the preceding chapters, has many advantages when used as a
cement replacement. It improves the workability and strength of concrete. Due to its
spherical shape, ﬂy ash reduces the water demand of fresh concrete for improved
workability, which results in changed structure and properties of concrete. This occurs
because the products of the pozzolanic reaction diffuse into the capillary pores.” The
effect of ﬂy ash on concrete properties depend on the amount of ﬂy ash utilized. The

optimum ﬂy ash content depends on the particular performance gains desired.

1.4.6 Effect of High Fly Ash Volumes on Hydration of Cement

Generally, ﬂy ash does not react with water as such. It reacts, after 3 to 7 days of
hydration, with calcium hydroxide, which precipitates from the hydration of Portland
cement. Theoretically, high-volume ﬂy ash concrete needs prolonged curing periods to
fully complete the hydration process. However, after about 91 days, still unreacted
calcium hydroxide and ﬂy ash exist in the hardened cement paste.“5 One may thus
consider high-volume ﬂy ash cement paste as a composite material with unreacted ash
particles behaving as reactive micro-aggegates.‘7 Table 1.2 shows the degee of reaction
of ﬂy ash at various volumes with different water/binder ratios of different curing

periods. At the age of 7 days, about 5% of ﬂy ash has started to react.48 This is attributed

32

to either the initial reaction of ﬂy ash, or to the reason that at early age ﬂy ash particles

are involved in the formation of ettringite. ‘9

Table 1.2 - Degee of reaction of ﬂy ash in ﬂy ash concrete pastes ‘8

 

The reduction in degee of reaction with high-volume ﬂy ash replacement of cement, as
in Table 1.2, is attributed to the low calcium ion concentration in pore water, resulting
from the high replacement of cement with ﬂy ash. Another factor could also be that the
high water/binder ratio dilutes the calcium ion concentration in pore water and reduces
the contact between particles. Therefore, high-volume ﬂy ash concrete needs prolonged

curing to overcome all these problems at a later age and provide the required strength.”

33

The degee of hydration of Portland cement can be known from the amount of calcium
hydroxide precipitated. The same approach can not be applied in the case of cement
partially replaced by with ﬂy ash. The reason is that calcium hydroxide also reacts with
ﬂy ash. However, ﬁ'om the level of calcium hydroxide in cement paste, the rate of
hydration of ﬂy ash can be estimated. During the period from 7 to 28 days of age, about
14 % of ﬂy ash particles react with calcium hydroxide.” Fig. 1.12 demonstrates that the
rate of reaction of ﬂy ash cement can be obtained from the reduction in calcium

hydroxide concentration of paste.’0

 

 

 

 

 

 

 

 

 

 

 

 

 

14
“5 a
no ' 12 -
Q
S.
r. 10. r
fag
G
‘3 i6
3 ° —+—-Pc
§ 4' +25%FA
9 2'M+55%FA
5 o . _ . .

o 25 so 75 100
Age(days)

(a)

34

 

 

 

 

 

 

 

 

 

cement
H
o

 

i
l
s

 

 

axon); content per 100g of

 

 

 

Ole-Race
U:
U.
39
"11
:3»

O 25 50 75 100
Age (days)

0))

Fig. 1.12 - Ca(OH2) content relative to the cement content in PC and FC pastes
(based on igrited weight).(a) w/b = 0.3, and (b) w/b = 0.5 ‘9

1. 4. 7 Effect of High Fly Ash Volumes on Strength of Concrete

High-volume ﬂy ash concrete is an ideal construction material from the economic point
of view. It replaces large amounts of cement which consumes substantial amounts of
energy for its manufacturing. When used in high strength concrete, ﬂy ash reduces the
demand for admixtures. However, there may be limits on the volume of ﬂy ash replacing
cement if certain combinations of properties are to be provided. As mentioned above,

reaction of ﬂy ash depends on the presence of calcium hydroxide. If the cement

35

replacement is too much, there will not be enough cement left to produce calcium
hydroxide for ﬂy ash to react. The strength of ﬂy ash concrete, at a given age, depends on
the degee of hydration. The degee of hydration also depends on other factors including
gel/space ratio.51 The low early strength of high-volume ﬂy ash cement is partly due to the
rise in net water/cement ratio resulting from the high-volume replacement of cement with
ﬂy ash.‘2 At the later stages of hydration, when hydration products precipitate on the
surface of ﬂy ash particles, the pozzolanic activity of ﬂy ash increases the bond strength
between ﬂy ash and cement particles. This makes the paste more homogenous, and
increases the stiffness of paste, which contributes to the increment in concrete strength.”
For strength gain, more hydration should take place. If there is limited pore space, the
hydration process slows down because when the pore spaces are ﬁlled with gel.23 The
other advantage of high-volume ﬂy ash is that it increases the strength of the bond
between particles and reduces the negative side effects of calcium hydroxide by

consuming it."

High-volume replacement of with ﬂy ash (e.g. 50% replacement) has yielded desirable
fresh mix workability providing relatively good strength increment at 28 days.”
Water/binder ratio affects the impact of hi gh-volume ﬂy ash replacement on concrete
strength. The early strength of high-volume ﬂy ash concrete is less adversely affected by

lower water/binder ratio than by higher water/binder ratio.48

The effect of high-volume ﬂy ash replacement of cement on the compressive strength of

cement mortar is different than for concrete. Generally, higher strengths are achieved in

36

concrete than in mortar. This is attributed to the effect of ﬂy ash on improving the

interfacial bond between the cement matrix and aggegates.48

Class C ﬂy ash, despite its high lime and sulfate contents which can adversely inﬂuence
concrete properties, increases the early strength of concrete. It yields desirable levels of
strength when used as cement replacement (30%) and also as aggegate (50% by weight

of cement). However, the ettringite formation problem has to be studied.“5

37

CHAPTER 2

NIECHANICAL MODELS OF HYDRATED CEMENT PASTE

2.1 Introduction

Hydrated cement paste is a complex composite material with multiple phases at micro-
and nano-scales. These features complicate development of structure-property
relationships for hydrated cement paste. Several empirical structure-property
relationships have been developed for hydrated cement paste based on laboratory
experiments." 5 Computational tools are also used toward modeling of hydrated cement

paste.

The multi-phase nature of hydrated cement paste at different scales distinguishes it from
technical ceramics. Unlike most ceramics, hydrated cement paste does not fracture in a
brittle mode. It is heterogeneous at micro-scale, where capillary pores, large CH crystals
and shrinkage microcracks are distributed randomly. The presence of these capillary
pores and microcracks produces stress concentrations which weaken the strength and
stiffness of material, but also contributes to the fracture toughness of hydrated cement
paste through arrest and deﬂection of microcracks, formation of microcrack processing

zone, and ﬁictional energy dissipation during pull-out of calcium hydroxide crystals.

38

These phenomena render hydrated cement paste semi-brittle with some level of ductility

and energy absorption capacity.

This paper presents analytical models for the mechanical performance of hydrated
cement paste. The models incorporate intermolecular interactions between nano-scale C-
S-H particles, effects of capillary pores and microcracks, and ﬁictional pull-out of

calcium hydroxide micro-scale platelets.

The models for hydrated cement paste are developed in two steps. The ﬁrst step focuses
on compact hydrated cement paste free from capillary pores and nricrocracks (henceforth
referred as “compact hydrated cement paste”), and the second step incorporates the
effects of capillary pores and microcracks into the models developed for compact

hydrated cement paste.

2.2 Hydrated Cement Paste Free from Capillary Pores and Microcracks (Compact
Hydrated Cement Paste)

2. 2.1 Molecular Interactions between C-S-H Particles

Mechanical models for compact hydrated cement paste were ﬁrst developed based on

phase interactions at molecular level. These models were later modiﬁed by introducing

the effects of capillary pores and microcracks for application to hydrated cement paste

with these micro-defects. This section focuses on compact hydrated cement paste.

39

The interactions between nano-scale calcium silicate hydrate (C-S-H) particles play key
roles in determining the mechanical properties of hydrated cement paste.‘4 C-S-H
particles are non-crystalline hydration products with large surface areas which enhance
their bonding to each other and to other particles within their reach through van der waals
interactions. Therefore, C-S-H plays a predominant role in rendering hydrated cement

paste its cementitious properties.”

Given the signiﬁcant binding role of C-S-H particles, the ﬁrst step in theoretical
modeling focused on the morphology and structural arrangement of C-S-H particles in
order to determine the bond strength between them. Research toward determination of
the detailed morphology of C-S-H is still in progess. C-S-H has been likened by several
researchers to Tobermorite mineral because of their similar compositions.“ Although C-
S-H particles gow to have a size of approximately 60 x 30 x 5 nm3, they have been
modeled as an agglomeration of unit particles making a spherical globule.‘7 This
geometrical modeling was adopted in the research repeated herein. An agglomeration of
basic C-S-H units is assumed to make a spherical C-S-H particles (globule) with diameter
ranging from 6 to 8 nm (Fig. 2.1).'"8 Due to the availability of water and space, the C-S-H
particles which form at the outer part of hydrating cement gain have lower density than
the ones which form at the inner part. A low density C-S-H is assumed to be an
agglomeration of globules bonding to each other and having a gel porosity of about 28%

(of the globule volume)?7

40

(a) Single C-S-H globule

 

.‘I‘.

(b) Agglomeration of globules

Fig. 2.1 - C-S-H morphology and geometric arrangement

The bonding between C-S-H globules involves van der waals interactions caused by
weak physical electrostatic attractions between molecules in neighboring globules. The
van der waals force per unit area (F vdw) between two ﬂat surfaces separated by a distance

d can be expressed as follows?9

3hw

vdw = 2_
16752d3 (1)

where, hw is the dielectric constant, approximately 10eV or 1.6022 x 10"8 J for C-S-H;59 d

is the clear distance between two C-S-H globules.

41

This equation can be applied to C-S-H globules as far as an effective (circular) area is
used with d taken as the minimum distance between two C-S-H globules (0.18 to 0.32

nm).”

Substituting the above values into Eqn. 2.1 gives a van der waals bond strength between
two adjacent C-S-H globules of 0.93 to 5.22 GPa, depending upon the size and spacing of
globules. The effective area of adjacent globules over which this stress operates will be

presented later in discussions on ﬁ'acture toughness.

2.2.2 Modulus of Elasticity of Compact Hydrated Cement Paste (Ea)

The modulus of elasticity of compact hydrated cement paste was determined by
considering the effect of relative movement of neighboring C-S-H globules against the

resistance of the bond strength between them. The relationship between the theoretical
cohesive stress (0'max ) required to debond two C-S-H globules at a distance d from each

other, the corresponding modulus of elasticity (E) and the minimum surface energy (73),

developed by Mindess et al.20 was applied to approximate the modulus of elasticity of

compact hydrated cement paste (Eqn. 2.2).

A typical curve representing the energy interaction between two C-S-H globules as a
function of their separation is shown in Fig. 2.2(a).‘50 The minimum energy state is

obtained at the equilibrium distance between two particles (b0). The minimum energy

required to separate two physically bonded surfaces is twice the surface energy, y,. The

42

force-displacement curve shown in Fig. 2.2(b) is obtained by differentiating the curve in
Fig. 2.2(a). The modulus of elasticity of the two bonded C-S-H globules can be estimated
from the initial curve in Fig. 2.2(b). The force-displacement curve can be transformed

into the stress-strain curve as shown in Fig. 2.3.60 Approximating the stress-strain curve
by a half sine curve, the total energy required to separate two surfaces, 2%. can be

calculated as the area under the curve, and the initial slope is modulus of elasticity, E0.
According to Mindess et al.,"0 the relationship between maximum tensile strength, 0m;
and modulus of elasticity, E0, can be approximated based on the energy interactions

stated above.

Stress, varying as a sine ﬁmction of x, can be expressed as:

. 272' x

 

The energy required to cause fracture can be calculated as:

2/2 27m 20'
2 = 0' sin—dxz—"ﬂ-x—
y. g .. , ,. (2.3)

Applying Hook’s law for small displacements,

43

Taking sin x z x for small angles, and substituting Eqns. 2.2 into 2.4 yields:

 

_ ,1 E
max 2 ﬂ" b O
Equating Eqns. 2.3 and 2.5 yields:
1/2
_ E 7.
O-max _ b0

(2.4)

(2.5)

(2.6)

 

Distance of separation. b

Energy, U
I

 

I
I
I
I
I
I
l
I
I
I
I
l
I
I
I
I
I
l
I
I
I

 

dU

db

Force =

 

 

A Displacement, b
I

bn

(b)

Fig. 2.2 - (a) Bonding energy as a function of the distance of separation; (b) force-
displacement curve 5°

45

Young’s modulus

Stress

 

 

 

Fig. 2.3 - Stress-strain curve approximated as a half sine wave 6°

The elastic modulus can thus be determined from the stress-strain relationship between
C-S-H globules, as shown above. This relationship can be reasonably approximated to be
about Boll 0 for cohesive solids.‘51 Therefore, the relationship between bond strength and

the intrinsic modulus of elasticity was approximated as follows:

Eo z 10 Umax (2.7)

where, am is the bond strength between particles under consideration, that is Fm, (the

van der Waals force per unit area); the relationship thus becomes:

46

E. z 10 dew (2.8)

Substituting the values of dew derived earlier into Eqn. 2.8 yields the values of E0, which
range ﬁ'om 9.3 to 52.2 GPa, depending upon the size and spacing of C-S-H globules
(Table 2.1). The bond strength and modulus of elasticity of compact hydrated cement
paste are inﬂuenced strongly by the distance between the C-S-H globules, irrespective of

their sizes, as shown in Table 2.1.

Table 2.1 - Effects of the distance between C-S-H globules on bond strength and modulus
of elasticity of compact hydrated cement paste ﬂee from capillary pores and

 

 

 

 

 

 

 

microcracks.
Diameter of Distance between Bond stress Modulus of
C-S-H globules C-S-H globules between globules Elasticity
dg (m) d (m) dew (Pa) Eo (Pa)
6.00E-09 1.80E-10 5.22E+09 5.22E+10
2.00E-10 3.80E+09 3.80E+10
3.20E-10 9.29E+08 9.29E+09
8.00E-09 1.80E-10 5.22E+09 5.22E+10
2.00E-10 3.80E+09 3.80E+10
3.20E-10 9.29E+08 9.29E+09

 

 

 

 

 

47

 

Fr

till

the

2. 2.3 Fracture Toughness of Compact Hydrated Cement Paste (Ga)

Fracture toughness of compact hydrated cement paste is deﬁned here as the energy per
unit crack area dissipated in the process of breaking C-S-H/C-S-H bonds and pull-out of
CH crystals. The latter was found to be the major contributor to the fracture toughness
(see the following sections). The energy dissipation mechanism in CH pull-out is related
to friction at nano-level. The ﬁictional stress was thus treated as phononic ﬁiction at
nano-scale.” The total intrinsic fracture toughness (G0) was taken as the sum of the
energy dissipated due to C-S-H/C-S-H debonding and frictional CH pull-out; however,

the contribution from the former is relatively insigniﬁcant.

2. 2. 3. 1 Fracture Toughness Due to Debonding Between C-S-H Globules (Gd)

The fracture toughness contributed by the rapture of interatomic bonds between two C-S-

H globules was determined using an equation which relates fracture toughness to van der

Waals strength as shown below.59

cult-o

3G,,
FVdW 272, 2k ZRg (2.9)

 

48

where, Rg is the radius of C-S-H globule (ranges between 3 and 4 nm); dew was found

earlier to range from 0.93 to 5.22 GPa (Table 2.1); and

_ 150- v2)
— E (2.10)

0

k

 

where, E0 is intrinsic modulus of elasticity, and v is Poisson’s ratio (~ 0.2) of compact

hydrated cement paste.

1n the above model, R8 and dew are independent variables which can vary over a range; a
parametric study was conducted in order to assess the sigriﬁcance of their effects on the
fracture toughness model. The parameters calculated for arriving at the Gd value
(fracture toughness due to C-S-H/C-S-H debonding) are presented in Table 2.2. The
variations of G3 with the radius of C-S-H globule (Rg) and the bond strength (dew) for

compact hydrated cement paste is summarized in Table 2.3.

49

Table 2.2 - Calculation of the fracture toughness (due to debonding of C-S—H globules) of
compact hydrated cement paste free from capillary pores and microcracks.

 

 

 

 

 

 

 

E dgtm) % d(m‘) FVMPa) Eo(Pa) ‘ k 4; G.(J{m2)
6.00E-09 1.80E-10 5.225+09 5.225+10 2.76E-11 2.14
2505-10 1.955+09 1.955+10 7895-11 0.80
3205-10 9.295+08 9.29E+09 1.55510 0.38
8.00E-09 1.80E-10 5.225+09 5.225+10 2.76E-11 2.85
2005-10 3.80E+09 3.80E+10 3.78E-11 2.08
3205-10 9.295+08 9.29E+09 1.55540 0.51

 

 

 

 

 

 

 

 

Table 2.3 - Summary of the variations of the fracture toughness due to C-S-H/C-S-H
debonding, (03) in J/mz, with radius of globules, kg, and 5......

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dew (GPa)

0.93 1.0 2.0 3.0 4.0 5.0 5.22

3.0 0.38 0.41 0.82 1.23 1.64 2.05 2.14

E 3.2 0.41 0.44 0.87 1.31 1.75 2.18 2.28
g 3.4 0.43 0.46 0.93 1.39 1.86 2.32 2.42
go 3.6 0.46 0.49 0.98 1.47 1.96 2.46 2.56
g 3.8 0.48 0.52 1.04 1.56 2.07 2.59 2.71
g 4.0 0.51 0.55 1.09 1.64 2.18 2.73 2.85

 

 

 

It can be observed from Table 2.3 that the maximum contribution of C-S-H/C-S-H

debonding to the fracture toughness of compact hydrated cement paste is 2.85 J/mz,

50

which is quite small when compared with the contribution of CH pull-out (see next
section). Hence, the variations in the C-S-H/C-S-H debonding contributions (resulting

from changes in different parameters) to ﬁ'acture toughness are not practically sigriﬁcant.

2. 2. 3. 2. Fracture Toughness Due to CH Crystal Pull-Out (6,)

CH crystals make about 20-25 % by volume of the hydration product in hydrated cement
paste;"2 they are generally hexagonal in shape (Fig. 2.4).63 The width of CH crystals in
hydrated cement paste ranges from 1000 to 10,000 nm, and their thickness from 220 to
260 nm.“ ‘5 Frictional pull-out of CH plates from hydrated cement paste is another factor
contributing to the fracture toughness of hydrated cement paste. For ease of computation,
CH plates were assumed to be circular (instead of hexagonal). The contribution of CH
pull-out to ﬁ'acture toughness represents the energy dissipated in the form of heat due to
phononic ﬁiction (associated with atomic vibration) between the surfaces of CH crystals
and the C-S-H nanoparticles.“ This ﬁictional energy dissipation during pull-out of CH
crystals depends on the orientation of the CH crystals relative to the direction of loading.
The maximum and minimum energy dissipations occur when the ﬂat surface of CH
crystal is at an angle with and perpendicular to the direction of loading, respectively.
Therefore, a vertical CH orientation, with its ﬂat surface parallel to the direction of
loading (Fig. 2.5), is assumed to account for the random orientation of CH crystals in

hydrated cement paste.

51

 

 

Fig. 2.4 - Hexagonal CH crystal ‘3

P

      

/ a

‘. l ' 'r
1 .{}ﬂ%’ ‘,
v ‘. A

a»

(0) (b)

Fig. 2.5 - CH orientation with ﬂat surface parallel to the direction of loading
(a) side view (b) front view

52

The portion of the width (diameter) of a single CH crystal which could be subjected to
phononic ﬁiction as the CH pulls out from a cluster of C—S-H globules ranges between 0
and 0.5 (of the width). Therefore, a quarter of the diameter of CH crystals was assumed
to be subject to phononic friction for the purpose of estimating an average CH pull-out
length. The part of CH crystal cross-section subjected to ﬁictional pull-out is a segnent
bound by a chord 0.5R away ﬁ'om the center of the circle as shown in Fig. 2.6. The area
of this segnent (As) can be obtained as the difference between the OABC and the OAC

areas.

A. = 2R2 cos’l[%]—Z\IR2 — 22 (2.11)

where, R is the radius of CH crystal; and Z is the half of the radius plus the distance

moved by CH as it pulls-out. The value of Z increases during the pull-out process of CH.

P

  

C

Fig. 2.6 - Schematic model of CH pull-out

53

The actual contact area between spherical C-S-H globules and the ﬂat surface of circular
CH crystal in compact hydrated cement paste was determined based on the assumed

morphology and structural arrangement of C-S-H globules (Fig. 2.7).

  
 

Area OFCH
SUPFQCQ

 

Fig. 2.7 - Schematics of the effective contact area between CH crystal and C-S-H
globules

The effective contact area (Ac) subjected to bond energy (Gd) per unit area can be

determined as followsz”

Ac = 7: Fe2 (2.12)
/3
rc = (1.57rkRg2 Gd)‘ (2.13)

where, Rg is the radius of the C-S-H globule.

54

The values of rc and A; range from 1.4 to 1.8 nm and 5.8 to 10 nmz, respectively. The
total effective contact area of all C-S-H globules contacting both faces of a single CH

crystal can thus be calculated as follows:

A
A = 0.8 c
ECH A

 

As (2.14)

g

where, A3 is the projected area of a C-S-H globule.

The value of Aecn calculated above ranges from 5.15 to 0.00515 nmz, as shown in Table

2.4. Eqn. 2.14 is multiplied by 0.8 to account for gel porosity.

The next step was to determine the frictional force acting on the CH surface as it pulls
out. For the atomically sharp surfaces of CH crystals, nono-scale phononic friction

dominates, where collision of vibrating molecules as two ﬂat surfaces slide against each
other dissipate energy in the form of heat. The phononic frictional stress (If) associated

with CH pull-out can be estimated at about 1 GPa.“
The ﬁictional force (Ff, newtons) due to phononic ﬁiction can be computed as the

effective area of CH subjected to frictional pull-out (AECH, m2) times the phononic

friction stress acting on it:

Ff = TfAECH =1x109x AECH (2.15)

55

The energy dissipated by pull-out of a single CH crystal, En, is computed as the work

done by F f during pull-out of CH crystal over a distance equal to half its radius:

R R
15,, = (F, dz = (109921,“ dz (2.16)

R/2 R/2

The next task was to approximate the possible number of CH crystals per unit area of
hydrated cement paste in order to calculate the amount of energy dissipated per unit
cracked area. The number of CH crystals in a unit cubic cell can be estimated by
assuming a uniform distribution of vertically oriented CH crystals. The amount of CH in

mature Portland cement paste ranges ﬁ'om 20 to 25 percent of the solid volume

(estimated at 22% on average).64 The volume of a single CH crystal (V ) can be
CH

calculated as:

 

_ 2
VCH " 75R t CH (2.17)
The number of CH crystals per unit cubic cell volume (Ncrrv) can be obtained as:
N _ 0 .22 * 1
CHV _ (2.18)
VCH

56

The number of CH crystals in a plane parallel to any side of the unit cubic cell, Ncnp, is:

Ncyp = (Nan/)2” (2.19)

The total energy per unit area (Gp) due to pull-out of CH crystals can thus be calculated

as follows:

Gp = E le CHP (2-20)

The values of GI) calculated using the above approach range between 5.42 and 281 J/m2

(See Table 2.4).
Therefore, the total intrinsic fracture toughness of compact hydrated cement paste (Go)

can be determined as the sum total of energy dissipation by CH pull-out (Gp) and

debonding of C-S-H nanoparticles (Gd):
G. = Gd + Gp (2.21)

The values of Go calculated using the above equation range between 5.82 and 283 J/m2
(Table 2.4), compared to measured values which range from about 16.9 to 41.0 J/m2 for

fracture toughness (Glc) of compact cement pastes.“ Table 2.4 summarizes the sensitivity

57

of the calculated value of the ﬁ'acture touglmess due to CH pull-out (Gp) and the total
fracture toughness (Go) to variations in the parameters of model. It can be observed ﬁ'om
the table that variations in the size of C-S-H globules (dg), the distance between C—S-H
globules (d), and the thickness of CH crystals (tCH) do not sigriﬁcantly affect the
calculated values of Gd and Go. Major variations in the calculated values of fracture
toughness were observed with changes in the width of CH crystals (Den). An average
size of CH crystals, obtained ﬁ'om a reasonable CH size distribution in hydrated cement

paste, was used in this research.

58

 

Rev 59:8 mo 9 as mans—was 8883 8 ”883“ e8 sense no .8 85836 - E 833

59

2.2.4 Intrinsic Tensile Strength of Compact Hydrated Cement Paste (am)

The tensile strength of compact hydrated cement paste was determined from the force
required to break the bond between two basic C-S-H globules. The total force required to
separate two surfaces of physically bonded C-S-H globules was determined as the sum of
the forces required to break each bonded C-S-H globule divided by the total debonded
surface area.

The tensile force required to break the bond between two basic C-S-H globules (F g) can

be determined as:

g c vdw (2.22)

The intrinsic tensile strength (0,) can be determined, assuming 20% gel porosity, as

follows:

o ' A (2.23)

The values of 0'0 calculated using the above equations range between 152 and 854 MPa,

compared with the measured value of about 930 MPa for the tensile strength of compact

hydrated cement paste.”

60

2.3. Hydrated Cement Paste

2. 3. 1. Introduction

The term “hydrated cement paste” is used here to refer to a conventional hydrated cement
paste which incorporates capillary pores and microcracks in its microstructure. The
mechanical and physical properties of hydrated cement paste are affected by the presence
of these micro defects, which should be accounted for in calculation of their strength,
modulus of elasticity and fracture toughness. Capillary pores and microcracks affect the
modulus of elasticity of hydrated cement paste by reducing the effective contact area and
generating stress concentration. The shape and size distribution of the capillary pores
sigriﬁcantly affect the modulus of elasticity of hydrated cement paste. Capillary pores
are assumed hence to have an elliptical shape with the ratio of the maj or-to-minor axis

about 2.69

2. 3. 2. Elastic Modulus of Hydrated Cement Paste

In modeling the elastic modulus of hydrated cement paste, the ﬁrst step involves

determining the relationship between capillary porosity and pore size. Uniform size

capillary pores distributed unifome in space were assumed for ease of computations

(Fig. 2.8).

61

 

rrrirrrrr

 

 

 

 

 

.L 1 til i iii

(a) Adjacent capillary pores

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

”P90 ! (oil) I PO—J

(b) Effect of adjacent capillary pores on stress concenuation

Fig. 2.8 - Capillary pore shape and interactions, and the resulting stress concentration

62

If a unit volume of hydrated cement paste, enclosing a single pore, is considered to have
2a+dc width, 2b thickness and a unit length (see Fig. 2.8b for notation), then porosity can

be approximated as:

 

 

_ 7m
1? — 40 + 2‘18 (224)
d. _ (7r -4p)
7— 217 (2.25)

The effect of interactions between two adjacent capillary pores, separated by a distance
(1,, on stress concentration in hydrated cement paste can be determined suing the
relationship between clc and the ampliﬁcation factor (K2) with the stress concentration

due to a single elliptical capillary pore (K1).”

 

(2.26)

where, K; is the stress concentration factor due to a single ellipse; and K2 is the stress
concentration ampliﬁcation factor due to the interaction of adjacent capillary pores. The

following expression can be used to calculate K1:

63

K, = 1+ 2 3
b (2.27)

From Fig. 2.8, it can be observed that (3’o = K26 , and since for constant strain in the linear

elastic range, E/Eo is equal to G/O'o, one concludes that E/Eo = 1/K2.

E — i(€€]37
E. a. K. K. a (”8)

g 1

Substituting Eqn. 2.25 and K1 into Eqn. 2.28, and simplifying the resulting equation,

gives the model for the modulus of elasticity of hydrated cement paste,

 

_ _ 0.37
Eh = O77E0 (1 ph)(7z- 4ph)

1 + 2% p237 (2.29)

The analytical model is plotted as a function of porosity and compared with empirical
models in Fig. 2.9. The analytical model is observed to compare reasonably well with

empirical models which reﬂect the trends observed in experiments.

64

 

0.8
0.7 -

 

 

\ —Analytical
— -x— — Empirical 1

0'5 d . \ —9— Empirical 2

 

 

 

EIEo
O
.A

0.3 4
0.2 -

0.1 - i \
0 I I I ~ —__
0 0.2 0.4 0.6 0.8 1

Capillary Porosity

 

 

 

 

 

 

Fig. 2.9 - Modulus of elasticity—porosity relationships

Cement-based materials incorporate entrapped air voids which reﬂect imperfect
compaction of these materials. Entrapped air voids in concrete are comparable in size to
the largest microcracks considered in our models. When compared with microcracks,
however, the entrapped air voids have relatively large radii of curvature, which makes
them less critical than microcracks with sharp tips in terms of stress concentration.
Hence, tensile strength would still be governed by the largest microcracks in lieu of

entrapped air voids. The infrequent occurrence of entrapped air voids in concrete as well

65

as the relatively small stress concentrations associated with them indicates that their

effects on mechanical properties of concrete are relatively small.
2. 3.3. Fracture Toughness of Hydrated Cement Paste (G)

The fracture surface of hydrated cement paste is not highly tortuous, and thus the fracture
toughness of hydrated cement paste (G1,) can be assumed to vary linearly with porosity as

shown in Eqn. 2.30.

Gh = Go (1 "' Ph) (2.30)

where, Go is the fracture toughness of compact hydrated cement paste; and pk is

capillary porosity.
2. 3. 4. Tensile Strength of Hydrated Cement Paste

The Grifﬁth’s equation for linear elastic fracture mechanics (LEFM) is used to predict the
tensile strength of hydrated cement paste in order to use this model, one needs to
determine the critical crack length for hydrated cement paste. Microcracks develop in
cementitious materials even before the application of loads. These microcracks result
from the restraint of thermal and drying shrinkage of hydrated cement paste at early age.

Calculation of the critical crack length in hydrated cement paste is presented below.

66

2. 3. 4. 1. Shrinkage Microcracks

The formation of microcracks in hydrated cement paste is due to the differential
shrinkage that exists between the relatively stiff CH crystal and the softer C-S-H gel.
Restraint of the shrinkage strains in C-S-H gel by CH crystals lead to microcracking of
the young hydrated cement paste. Restrained shear stresses cause debonding of C-S-H

from the CH surface and subsequent cracking of hydrated cement paste (Fig. 2.10).

 

 

 

 

 

(a) Shrinkage cracking around CH crystal

67

 

 

 

 

 

 

 

 

 

(b) Simpliﬁed crack model

Fig. 2.10 - Schematic presentation of the formation of microcracks in otherwise compact
hydrated cement paste

The strain energy released during cracking under the effect of shrinkage strain can be

calculated as follows:

Ue = gshEoAsky (2.31)

where, Ue = strain energy released during cracking; 85,, = shrinkage strain; E = modulus
of elasticity of C-S-H; Ask (surface area of CH subjected to skin friction) = (PCH)(tCH); y
(crack width at the periphery of CH crystal) = 85,, PC“; and Pen and to" are the perimeter

and thickness of CH crystal, respectively.

68

Substituting the above expressions into Eqn. 2.31 yields:

2
Ue = (5.9121) CH) EotCH (2.32)

The energy dissipated upon cracking (U i) can be calculated as follows:

Ui = GdltCH (2.33)

where, Gd is the ﬁ'acture toughness due to C-S-H debonding; and l is crack length.

Equating U.3 (Eqn. 2.32) and U; (Eqn. 2.33) yields the following expression for crack

length, I :

 

= (ashPCH )2E0
Gd (2.34)

1

Due to the unavailability of sufﬁcient information on the relationship between capillary
porosity (p) and shrinkage strain (85,, ), an empirical relationship was determined using

experimental results71 as follows:

fish 2 0.0148ph —8.8x10'4 (2.35)

69

After substituting Eqn. 2.35 into Eqn. 2.34 and simplifying it, the following expression

can be derived for crack length:

 

1 39.8%er (0.0148 p, —8.8x10"4)2
- Gd (2.36)

where, rm is the radius of the largest possible CH crystal; a 100 pm diameter CH was

considered.“ The critical crack length l' is half of I plus the diameter of CH:

[C : 0.5(1 + 2rCH) (2.37)

2. 3. 4.2 Tensile Strength of Hydrated Cement Paste

The Grifﬁth equation deﬁnes the relationship between tensile strength (at) and critical

crack length (l'):

 

EhGh

0' .2
ht
7: [C

(2.38)

Substituting Eqns. 2.29 for E, 2.30 for G and 2.37 for 1’ into Eqn. 2.38 yields:

70

 

0'77EoGo(1_ ph)2(77 " 41700.37

 

 

V 7’1.- 1+2; p237 (2.39)

2.3.4.3 Parametric Study of the Eﬂ'ects of Various Parameters on the Tensile Strength
of Hydrated Cement Paste
An extensive parametric study was conducted to identify parameters deﬁning the
characteristics of C-S-H, CH and capillary pores which signiﬁcantly inﬂuence the
strength of hydrated cement past. The effects of CH thickness and diameter are shown in
Figs. 2.11 through 2.18. It can be observed from these ﬁgures that (for constant capillary
pore size, distance between C-S-H globules and diameter of C-S-H globule and CH
crystal) the variation of CH thickness (within a reasonable range) only minimally affect
the strength. The capillary pore aspect ratio (a/b) and the CH crystal diameter, on the
other hand, have important effects on strength. The effect of capillary pore aspect ratio
can be observed by comparing the graphs in Figs. 2.ll(a) and 2.15(a). The strength
values of the former are almost twice those of the latter. This could be attributed to the
higher stress concentrations that occur when capillary pores are modeled as ellipsoids
rather than spheres. The effect of CH crystal diameter on strength-porosity relationship
can be assessed by comparing graphs (a) and (b) in any one of these ﬁgures. For example,
in Fig. 2.11, the strength of hydrated cement paste with 10,000 nm CH diameter is about

three times that of hydrated cement paste with 1000 nm CH diameter. This signiﬁcant

71

effect could be attributed to the effect of CH size on the fracture toughness of hydrated
cement paste. The larger the diameter of CH crystal, the higher is the fracture toughness
due to the high contact surface for phononic friction as the CH crystal pulls out.
However, in real sense, not all the CH crystals have such a large size. CH crystals grow
in size only when there is adequate free space. From this perspective, it is unlikely that

the majority of CH crystals will reach the larger size.

72

 

alb=1; dg=6 nm; d = 0.18 nm; DCH=1OOO nm

 

 

NM
01

O
l...

 

\ tCH (nm)

-— + - 220
240
- - -A- - - 260

 

 

 

_l
O

 

 

 

 

 

Tensile strength (MPa)
51
/

 

OU‘I

 

O 0.5 1
Capillary porosity

 

 

 

(a)

 

alb=1; d9=6 nm; d = 0.18 nm; DCH=10000 nm
80

 

K to“ (nm)

 

 

—o—220

240
- - a— - -260

0 M

0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

 

N
O

 

Tensile strength (MPa)
4:- a:
o o
-/

 

 

 

 

 

 

(b)

Fig. 2.11 - Inﬂuence of CH thickness on the strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

73

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=1; dg=6 nm; (1 = 0.32 nm; DCH=1OOO nm
12
T6
% ‘° ‘r
E 8 tCH (nm)
g 6 \ -- '0 - 220
5 4 \ 240
g .. .- . .260
2
o 2
l—
O
O 0.5 1
Capillary porosity
(a)
alb=1; dg=6 nm; (1 = 0.32 nm; DCH=1OOOO nm
A 40
i r
E 30 ten (nm)
5 \
O) — —
* :23
173
_a_> \ - - t- - -260
7;, 10
C
'93 w
0
0 0.5 1
Capillary porosity
(b)

Fig. 2.12 - Inﬂuence of CH thickness on the strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 6 nm, distance between
C-S-H globules of 0.32 run, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

74

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=1; dg=8 nm; d = 0.18 nm; DCH=1OOO

30
”a?
a. 25
:2: 1
@320 tCH (nm)
E) 15 \ — '0 — 220
'53 \ 240
3 1° - - a- - .260
9 5
g2

0
_ 0.5 _ 1
Capillary porosrty

 

 

 

(a)

 

alb=1; dg=8 nm; d = 0.18 nm; DCH=1OOOO nm
a“: 75 x tCH
2
v60 \ — o — 220

240
“’30 - - a- - .260

 

 

 

 

 

 

.C

 

 

 

 

 

 

 

0 0.5 1
Capillary porosity

(b)
Fig. 2.13 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for

spherical capillary pores, C-S-H globule diameter of 8 nm, distance between
C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nrn

 

 

 

75

 

alb=1; dg=8 nm; d = 0.32 nm; DCH=1000 nm

 

 

 

 

 

 

 

 

 

15
’5
0.
E10 ‘ tCH(nm)
3% \ —o—220
c
g 5 240
E, "‘""260
'7)
5
l— O

 

 

O

0.5 1
Capillary porosity

(a)

 

 

 

 

alb=1; dg=8 nm; (1 = 0.32 nm; DCH=1OOOO nm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g 50 tel-101m)
v 40 "
g 30 1 — a — 220
§ \ 240
'2): 20 - - a— - 260
a 10
E 0
O 0.5 1
Capillary porosity
(b)

Fig. 2.14 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
spherical capillary pores, C-S-H globule diameter of 8 nm, distance between
C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

76

 

alb=3; (1 9:6 nm; (1 = 0.18 nm; DCH=1000 nm

 

 

 

 

 

 

 

 

 

 

 

12
It“? t
2 g _ CH
5 - o — 220
g: 6
9 \ 240
4., - - ‘- - |
3 3 260
'7)
E, 0

0 0.5 1

 

Capillary porosity
(a)

 

 

 

alb=3; dg=6 nm; (1 = 0.18 nm; DCH=1OOOO nm

 

.h.
C

tCH (nm)

\ --o—220

240

03
O

 

 

 

 

 

 

 

 

Tensile strength (MPa)
N
o

 

 

 

 

 

1O
0 M
0 0.5 1
Capillary porosity
(b)

Fig. 2.15 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 6 nm, distance
between C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000
nm; (b) 10,000 nm

77

 

alb=3; dg=6 nm; d = 0.32 nm; DCH=1OOO nm

 

 

 

 

 

 

 

 

 

 

 

It? 5.0

% 4.0 ‘\ tCH (nm)
g 3 0 \ — T _ 32
5 24
g 2 0 0
.3

g

I'-

 

o...x
OO
O

0.5 1
Capillary porosity

(a)

 

 

 

 

alb=3; dg=6 nm; d = 0.32 nm; DCH=1OOOO nm
16

 

?\ tCH (nm)

\
240

\ nan-260

.3
N

 

 

 

 

 

 

 

 

 

 

 

Tensile strength (MPa)
00

 

 

 

4
0 \M—H
O 0.5 1
Capillary porosity
(b)

Fig. 2.16 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 6 nm, distance
between C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000
nm; (b) 10,000 nm

78

 

 

 

 

 

 

 

 

 

 

 

 

alb=3; dg=8 nm; (1 = 0.18 nm; DCH=1OOO nm
A 12
a“: ‘ tCH (nm)
E 9 X — o — 220
.C
:3: 6 240
g - - A- - -260
w 3
g
g 0
'— 0 0.5 1

Capillary porosity

 

 

(a)

 

alb=3; dg=8 nm; d = 0.18 nm; DCH=1OOOO nm

 

 

 

 

 

 

 

.1.
O

 

 

 

A 40

d0 ‘ tCH (nm)
a 30 \ — o — 220
.C

F" 20 240
g \ - - a— - -260
(D

32

a

C

O)

'—

 

M

0.5 1
Capillary porosity

O
O

 

 

 

(b)

Fig. 2.17 - Inﬂuence of CH thickness on strength-porosity relationship of hcp for
ellipsoidal capillary pores, C-S-H globule diameter of 8 nm, distance
between C-S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000
nm; (b) 10,000 nm

79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=3; dg=8 nm; d = 0.32 nm; DCH=1000 nm
136'0
35.0 X ten (nm)
333:3 \ — O — 220
@2'0 \ 240
iii ' — at - 260
g 1.0
g 0.0 ‘ﬂa-o—H—n
'— 0 0.5 1

Capillary porosity

 

(a)

 

alb=3; dg=8 nm; d = 0.32 nm; DCH=1OOOO

 

 

 

 

 

 

 

 

 

 

 

$20

2 ‘ tCH("m)
:15

“Q3 \ -o—220
$10 \ 240
_a_> 5 "rt-"260
'2 8*, I

|_a_> 0

O

0.5 1
Capillary porosity

 

 

 

(b)

Fig. 2.18 - Inﬂuence of CH thickness on strength-porosity relationship of hep for
ellipsoidal capillary pores, C-S-H globule diameter of 8 nm, distance
between C-S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000
nm; (b) 10,000 nm

80

A study on the effect of C-S-H globules diameter (dg) on the strength-porosity
relationship in hydrated cement paste was conducted using an average CH thickness of
240 nm. The results are summarized in Fig. 2.19 through Fig. 2.22. It is obvious that the
inﬂuence of the range of dg is not signiﬁcant. Therefore, an average value of 7 nm was

safely considered in developing the mechanical model of hydrated cement paste.

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=1; d=0.18 nm; DCH=1000 nm
30

76
E
:5: 20 ‘ d9
8’
2 --o-6
*" +
a"; 10 8
'6
C
o
1..

0 I r

0 0.2 0.4 0.6 0.8 1
Capillary porosity
(a)
alb=1; d=0.18 nm; DCH=10000 nm
80

 

.1: c»
o o
/

9 .9
a:

 

 

 

 

 

 

 

 

Tensile strength (MPa)
N
O
/
i

 

O 0.5 1
Capillary porosity

 

 

 

(b)

Fig. 2.19 - Inﬂuence of the C-S-H globule diameter on the strength-porosity
relationships of hcp for spherical capillary pores, distance between C-S-
H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

82

 

alb=1; d=0.32 nm; DCH=1000 nm

1 “9
9

--o.-.6
+8

 

.3
N

 

 

 

 

 

 

00

 

Tensile strength (MPa)
0)

 

0 I 1

0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
alb=1; d=0.32 nm; DCH=10000 nm
A 40
a -\ d9
2 30
5 X ' ' 0' 6
D)
g, 20 +3
‘1 \
d)
a 10
C
a M
0 1 r
O 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

(b)

Fig. 2.20 - Inﬂuence of the C-S-H globule diameter on the strength-porosity
relationships of hcp for spherical capillary pores, distance between C-S-
H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

83

 

 

 

 

 

 

 

 

 

 

 

alb=3; d=0.18 nm; DCH=1000 nm

12
i"
E 9 1 d9
5 \
O)
a 6 "°' 6
‘63 \ +8
0:
77', 3
C
o
l— M

0 l l

0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=3; d=0.18 nm; DCH=10000 nm

40
EB dg
g, 30
5 --0-6
G)
E, 20 +8
‘65
2 \
E5 10
C
Q)
I‘— w

0 1 r
0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

(b)

Fig. 2.21 - Inﬂuence of the C-S-H globule diameter on the strength-porosity
relationships of hcp for ellipsoidal capillary pores, distance between C-
S-H globules of 0.18 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 run

84

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=3; d=0.32 nm; DCH=1000 nm
6.0
S 5.0 dg
2 ‘1
5 4.0 - . o. . 6
E) 3.0 \\ +8
E, 2 0
E
a, 1.0
l—
0.0 I .
0 0.2 0.4 0.6 0.8 1
Capillary porosity
(a)
alb=3; d=0.32 nm; DCH=10000 nm
18 d
EL" 15 4‘ g
E
5 12 - - o- - -6
8’ \ +3
2 9 \
E 5
’17)
E, 3
'—
0
0 0.5 1 1.5
Capillary porosity

 

 

 

(b)

Fig. 2.22 - Inﬂuence of the C-S-H globule diameter on the strength-porosity
relationships of hcp for ellipsoidal capillary pores, distance between C-
S-H globules of 0.32 nm, and CH crystal diameter of: (a) 1000 nm; (b)
10,000 nm

85

The next step in the parametric study was to investigate the inﬂuence of the distance
between C-S—H globules (d) on the strength of hydrated cement paste. In this
investigation, the number of independent parameters is reduced by taking C-S-H
diameter and CH thickness to be 7 nm and 240 nm, respectively. The resulting strength-
porosity relationships are summarized in Figs. 2.23 and 2.24. The results suggest that the
inﬂuence of the variation of d within the speciﬁed range on the strength-porosity
relationship is highly pronounced at lower porosities. An average value of 0.2 nm was

adopted for d in the model based on the recommendations in the literature.”

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

alb=1; DCH=1000 nm
14
a 12 a
n.
2 \ 11
‘6'» 8 —-0.18
C
g 6 1 "' ' '0.32
"’ \
g 4
c \
0 l T l
0 0.2 0.4 0.6 0.8 1
Capillary porosity
(a)
alb=1; DCH=10000 nm
80
E \
é 6° (1
5 \ 018
O) — .
C
g 40 \ - - -0.32
w
0
7-,; 20
C
d)
'- ¥
0 1 1 1
0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

(b)

Fig. 2.23 - Inﬂuence of the distance between C-S-H globules on the strength-porosity
relationships of hcp for spherical capillary pores with CH diameter of: (a)
1000 nm; (b) 10,000 nm

87

 

a/b=3; DCH=1000 nm

 

 

 

 

 

 

 

 

 

 

 

6
7'?
D. 5 ‘
é \ d
5 4
23:3 \ —0.18
b - - -0.32
E 2 k
51
'— ¥
0 l I l

 

 

0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

(a)

 

alb=3; DCH=10000 nm

 

 

 

 

 

 

 

 

 

 

 

 

A35

(U

5:: 1 .
€320 \ ——0.18
g 15 \\ - - -0.32
95> 10 K

g 5

F1\—~ 1

 

 

0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

 

(b)

Fig. 2.24 - Inﬂuence of the distance between C-S-H globules on the strength-porosity
relationships of hcp for ellipsoid capillary pores with CH diameter of: (a)
1000 nm; (b) 10,000 nm

88

2. 3. 4.4 Comparing the Predictions of the Mechanistic Tensile Strength Model of
Hydrated Cement Past with those of Empirical Models

The values of different parameters governing the tensile strength of hcp were determined
using the ranges suggested in the literature (presented earlier), with parametric studies
conducted to assess the signiﬁcance of various parameters in order to assist with ﬁnal
selection of values within the suggested ranges. The ﬁnal selections were as follows: rCH
(radius of largest possible CH crystal) = 50,000 nm; E0 = 38 Gpa; Gd = 1.56 J/mz; and the
ratio of the major-to-minor axis of capillary pores was 2.69 E0 and Gd were calculated by
considering the average values of d (the distance between two C-S-H globules) and Rb
(the radius of C-S-H globule)57 to be 0.2 nm and 3.5 nm, respectively. Average thickness
and diameter of CH crystal were assumed to be 240 nm and 2500 nm, respectively, where
the later is close to mean value of CH size distribution. Substituting the above values into

Eqn. 2.37 yields:

 

)0.37

 

(77—41%
0' 26.7 l— ‘ ‘
"t 3( 1") p2‘37(106(0.0148ph—8.8x10‘4)2+0.041) (“0)

where, em = is the tensile strength of hop in MPa.

The above equation is observed in Fig. 2.25 to compare favorably with an empirical

model which reﬂects experimental result. The empirical models developed by Balshin’l

89

(Eqn. 2.41) and Ryshkevitc 52 (Eqn. 2.42) were used for evaluating the prediction of the

theoretical model for the tensile strength of hydrated cement paste.
9 .97
0'11: = 46(1— P) (2.41)

_ —14.47 p
0' hr — 52 e (2.42)

 

 

—Theoretical - - A- - 'Balshin - e — Ryshkevitch

 

 

 

 

20.00
E 16.00

 

 

 

 

 

Tensile strength (M
4. 00 KS
0 O O
O O O

 

 

0.00 1
0 0.2 0.4 0.6 0.8 1

Capillary porosity

 

 

 

Fig. 2.25 - Comparing the predictions of the mechanistic and empirical tensile strength
models of hcp

90

2.3.5 Compressive Strength Model of Hydrated Cement Paste

Several experimental studies have demonstrated that the mechanism of failure of
hydrated cement paste subjected to compressive stress is by forming bundles of cracks
parallel to the direction of loading. The crack lines are nearly straight and do not have
pronounced tortuosity. Based on this observation, one can conclude that the compressive

failure is due to the lateral tensile stresses caused by the poison’s effect (Fig. 2.26).

 

 

 

O’hc
llllllll D»:
v
Y)? -
"in" 9x :2" i l‘ l ">

 

 

 

 

 

 

o’hc

Fig. 2.26 - Crack formation due to compressive in hydrated cement paste specimen

The Poisson’s effect under compression can be expressed as follows,

(The

 

x

8 = —V8 = —V
y h (2.43)

and

91

x Eh (2.44)

Substituting Eqn. 2.42 into Eqn. 2.41 yields,

 

c _ (2.45)

Substituting Eqn. 2.40 into Eqn. 2.45 and taking the value of v to be 0.2 gives the

following compressive strength model of hydrated cement paste:

 

(n—4p1>°-”

0' =33.6 1— - ‘
1.. 3( ‘0") p2’37(106(0.0148p—8.8x10‘4)2+0.04l) (146)

where, 0,“, is the compressive strength of hcp in MP3.

The predictions of the above model are evaluated by comparing them with those of
empirical models developed based on experimental results (Fig. 2.27). The empirical
models developed by Balshin‘l (Eqn. 2.47) and Ryshkevitch52 (Eqn. 2.48) were used for
evaluating the predictions of the theoretical model for the compressive strength of

hydrated cement paste.

92

O'hc 2' 540(1— P)14'47 (2.47)

O'hc = 6366—17'04p (2,433)

It can be observed in Fig. 2.27 that there is no signiﬁcant difference between the
theoretical and empirical models except at smaller porosities, where even the empirical

models are also considered to be less accurate.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— i - Balshin Theoretical - - 0- - Ryshkevitch
g 100 ‘\
.C
3) 80 \\
9
173 60 ‘-
q; \
.2
:3 40
9
g- 20
O
U 0 1
0 0.2 0.4 0.6 0.8 1
Capillary porosity

 

 

Fig. 2.27 - Predications of the compressive strength model of hcp versus those of
empirical models derived based on experimental results.

93

CHAPTER 3

MODELING THE MECHANICAL PROPERTIES OF MORTAR
BASED ON MECHANISTIC APPROACH

3.1 Introduction

Mortar is a composite material with general properties that are affected by the
arrangement, characteristics and proportions of the constituent materials. These factors
also inﬂuence the characteristics and arrangement of the three phases at micro-level.
These phases are the hydrated cement paste, ﬁne aggregate (sand), and interfacial
transition zone. The interfacial transition zone is usually expressed as a region having a
thickness of 30—50 pm, in which the cement paste composition differs signiﬁcantly from
that of the bulk cement paste.72 The interfacial transition zone has much higher porosity
and CH volume ﬁ'action, and lower contents of C-S-H gel and unhydrated cement paste,
than the bulk cement paste. This is attributed, to some extent, to the low packing of C-S-
H gels near the surface of the aggregate particle.”73 This phenomenon renders interfacial
transition zone the weakest zone in mortar or concrete.74 Therefore, the presence of the
interfacial zone adds to the complexity of mortar or concrete. From the component
materials point of view, the major factors affecting the bond strength of mortar include
the water/cement ratio of the hydrated cement paste, the mineralogical nature and origin
of ﬁne aggregates, the surface roughness of aggregates, the age of the cement paste, and

the aggregate size.”

94

In this chapter, an attempt has been made to develop mechanical models of cement

mortar based on mechanistic (theoretical) approach.

3.2 Modeling

The mechanical models of cement mortar were developed by expanding upon the models
of hydrated cement paste. The addition of ﬁne aggregates to hydrated cement paste
changes the paste into a composite material. In this process, mortar is considered as a
composite material composed of the hydrated cement paste matrix and ﬁne-aggregate
ﬁllers. At micro-scale, another third phase is considered - the interfacial transition zone.
The interfacial transition zone in mortar is a thin porous zone at the interface between
hydrated cement paste and the surface of ﬁne aggregate. The inclusion of aggregates into
cement, unlike any other ceramics, does not improve the strength of the hydrated cement
paste; rather, it reduces the strength considerably. This strength loss could be attributed to
the inﬂuence of the interfacial transition zones, which are called the zones of weakness in

cementitious materials where the porosity and the CH concentration are relatively high.

3.2.1 Modulus of Elasticity of Cement Mortar

Cracks propagate in cement mortar mostly through the hydrated cement paste and along

the interfaces between hydrated cement paste and ﬁne aggregate surfaces. This indicates

95

that the interfacial transition zone inﬂuences the mechanical properties of cement mortar.
In modeling the modulus of elasticity of cement mortar, the effect of interfacial transition
zone was considered. Cement mortar was modeled as a composite material composed of
three phases: the hydrated cement paste, ﬁne aggregates, and the interfacial transition

zone.

In order to develop the modulus of elasticity model of cement mortar, it was necessary to
determine the modulus of elasticity and shear modulus of every constituent phase of the
composite material. For hydrated cement paste, the modulus of elasticity model
developed in the preceding chapter (Eqn. 2.29) was used. The modulus of elasticity of the
interfacial transition zone of cement mortar (Eim) was derived using the modulus of
elasticity of hydrated cement paste with some modiﬁcation. The porosity of the
interfacial transition zone was considered to be about twice that of the hydrated cement
paste.76 This variation in porosity was accounted for in developing the modulus of
elasticity of the interfacial transition zone (Em) based on that of the hydrated cement
paste (E). The modulus of elasticity of ﬁne aggregates (E,) ranges from 70 GPa to 90
GPa, depending on the aggregate type.77 A parametric study was conducted to evaluate

the signiﬁcance of this range in determining the modulus of elasticity of cement mortar.

The upper and lower bounds of the modulus of elasticity of cement mortar were
calculated from an equation relating the modulus of elasticity of a composite material to
its shear modulus (0)") and bulk modulus (K 'm) (Eqn. 3.1).78 The upper and lower bounds

of the shear modulus and bulk modulus were calculated using the Hashin’s modulus of

96

elasticity model for three-phase composite materials (Eqns. 3.2 -— 3.5).78 The steps
followed in the computation of the upper and lower bounds for the modulus of elasticity

of cement mortar are presented below.

The upper and lower bounds of the modulus of elasticity of mortar, Em, were

approximated as follows: 7“

9Km-Gm- S E S. 9Km+Gm+
3K}n_ + G;_ "’ 3K},+ + Gfm, (3'1)

 

 

Where K j", and K 111— are the upper and lower bounds of the bulk modulus of cement
mortar, respectively; and GI", and an_ are the upper and lower bounds of the shear

modulus of cement mortar, respectively. 73 These parameters were calculated as follows:

Lower bound of the bulk modulus of cement mortar, K 1'11— , 7’

 

 

 

1 V,’ V; 3V3,
r : r +—T+ I r (32)
Km_ K, K, K,+1.3BG,. '

Upper bound of the bulk modulus of cement mortar, K :11 + , 73

97

V1'K§
+ 3Ks't, (3.3)
K; + 1.33 G,’

 

K1,.” =V,:K,',+

 

Lower bound of the shear modulus of cement mortar, G,',,_ , 7‘

 

 

1 =—L+L—+0.4Vs'tr 2 +3.
G111— Gi ; K,'+1.33G,' a; (3.4)

Upper bound of the shear modulus of cement mortar, G; , 78

Vs'Gi
+ 2.5th, (35)
K,’ + 3.33 G,’

 

G'... = Vh'G; +

 

where, V; is volume fraction of sand; and V1: is volume fraction of hcp.

The next step was to determine all the parameters involved in the above equations.

The bulk modulus and shear modulus of any material are related to its modulus of

elasticity through Eqns. 3.6 and 3.7, respectively: 78

98

, E
K = (3.6)
3(1— 21/)

 

, E
G = _" (3.7)
2(1+ v)

where, v is the poison’s ratio of cement mortar.
From the above relationships, the bulk modulus and shear modulus of hydrated cement

paste and ﬁne aggregates were calculated as follows (considering v = 0.2 for both the

hydrated cement paste ’9 and ﬁne aggregates 80):
The bulk modulus of hydrated cement paste, K): ,

;:_ﬂ_: Eh =O'56Eh (38)
3(1— 2V) 3(1— 2(0.2)) °

 

The bulk modulus of sand (ﬁne aggregate), K; ,

S

K; = i— : Es = 0.565, (3 9)
3(1— 2v) 3(1— 2(0.2)) -

 

99

The shear modulus of hydrated cement paste, G, ,

 

 

2(1+ v) 2(1+ 0.2) '
The shear modulus of sand (ﬁne aggregate), G; ,
;= ————E‘ = E‘ = 0.42 E, (311)
2(1+v) 2(1+O.2) '

Based on the above relationships, the bulk modulus and the shear modulus of the

interfacial transition zone of cement mortar could also be determined as follows:

Ki, : 0.56 Ei (3.12)

G; = 0.42 Ei (3.13)

For the value of the modulus of elasticity of sand (Es) ranging from 70 to 90 GPa, the

corresponding values of K g and G; range from 39.2 to 50.4 and 29.4 to 37.8,

respectively. A parametric study conducted to assess the signiﬁcance of these ranges in

determining the mechanical properties of mortar is presented below. The modulus of

100

elasticity of the interfacial transition zone (E) is a function of the modulus of elasticity
(E) and capillary porosity (Pk) of the hydrated cement paste in cement mortar. The ratio
of the thickness of the interfacial transition zone to a radius of a sphere equivalent to ﬁne

aggregate, tr, 7‘ can be expressed as follows:

’ r (3.14)

where, ti is thickness of the interfacial transition zone, and is a ﬁmction of aggregate
size;“ and rs is the radius of an equivalent spherical aggregate. Thickness of interfacial

transition zone (ti) is proportional to the size of the aggregate it envelops. 8‘

No information could be found in the literatures on the approximate thickness of the
interfacial transition zone in mortar. Therefore, it was approximated, based on the
maximum interfacial transition zone thickness in concrete. A linear relationship between
the aggregate size and the thickness of the interfacial transition zone was assumed as
shown in Fig. 3.1. The aggregates size was considered to range from 0.1 to 4 mm and
from 4 to 25 mm, for ﬁne and coarse aggregates, respectively. The maximum thickness
of the interfacial transition zone in concrete is about 50 um.” 72 This size was considered
to occur for the 25 mm aggregate size. No interfacial transition zone was assumed to
occur for an aggregate size lower than 0.1 mm. This assumption is based on the fact that
no interfacial transition zone has been detected so far around CH crystals of the same

size. Based on this relationship, the value of ti ranged from O to 3.9 pm for aggregate size

101

ranging from 0.1 to 4 mm. A parametric study was conducted to investigate the
signiﬁcance of this range on the mechanical properties of mortar, and the results are

presented in the next section.

 

 

 

 

 

30 /
20 /
10 /

 

 

 

 

 

0 T 1
0 10 20 30
Aggregate size (mm)

 

 

 

Fig. 3.1 - Variation of the interfacial transition zone thickness with aggregate size.

The signiﬁcance of the three phases (hydrated cement paste, ﬁne aggregates and
interfacial transition zone) in determining the mechanical properties of cement mortar

also depends upon their respective volume proportions in the mortar. The calculation of

the volume fractions of hydrated cement paste (V; ), ﬁne aggregate (V5') and interfacial

transition zone (V; ) in cement mortar is discussed below.

102

Porosity of interfacial transition zone (Pi) is about two to three times that of hydrated
cement paste, and 25% of the volume of interfacial transition zone is ﬁlled with CH

crystals. 83 P; was assumed to be about twice the porosity of hydrated cement paste.

The pore volume of mortar (me) was expressed as the sum of the pore volumes of

hydrated cement paste and the interfacial transition zones, as follows:

V = VP, + VP,- (3.15)

pm

where, VP). and VP,- are the pore volumes of the hydrated cement paste and the interfacial

transition zone, respectively.

Porosity of mortar (Pm) can also be expressed as the ratio of the pore volume of mortar to

its total volume, as follows:

 

 

m V = V (3.16)

where, V", is the total volume of mortar, which is the sum of the volumes of ﬁne
aggregates and the bulk hydrated cement paste (the term bulk is used to refer to the
hydrated cement paste and the interfacial transition zone as one entity). Vm can be

expressed as follows:

103

V». = V. + V}. (3.17)

When very large volume fractions of sand are used in cement mortar, the workability of
the ﬂesh mix tends to deteriorate. This damage to workability results in the formation of
large voids and micro-defects, which would undermine the performance of the mortar. A
reasonable range for the volume fraction of sand considered in this research is from 0.5 to
0.6. A mortar mix proportion, where sand and bulk hydrated cement paste constitute 55%
and 45%, respectively, of the mortar volume was considered to represent conventional
mortar mixtures. The volume of the interfacial transition zone surrounding a single sand
particle in cement mortar (V1,) was calculated using Eqn. 3.18 based on the relationship
between aggregate size and the interfacial transition zone thickness, as shown in Fig. 3.1.
A circular cylinder with aspect (height-to-diameter) ratio equal to 1 was considered as a

geometric model of sand (Fig. 3.2).

HEM

.-
o

  

 

 

Fig 3.2 - Schematic presentation of the geometric model of a sand particle.

104

This geometric model implies that:

Vli = 2”((rs + ti)3 _ 7:93) (3.18)

Assuming a uniform aggregate size, the number of sand particles per unit volume of

cement mortar (N,) was calculated as follows:

N = Vs = Vs 3
27rrs (3.19)

 

 

where, V1, is the volume of a single cylindrical sand particle of radius r, and height 2r3, as

shown in Fig. 3.2.

The total volume of the interfacial transition zone (V,) was then calculated as the product

of N, and V1,, as follows:
Vi : Vli X Ns (3.20)

Since the thickness of the interfacial transition zone is proportional to the size of ﬁne

aggregate, for a constant aggregate volume, the volume fraction of the interfacial

105

transition zone and the hydrated cement paste remain constant with a change in aggregate

size (at constant aggregate volume), as shown in Table 3.1.

Table 3.1 - Volume ﬁactions of the interfacial transition zone and the hydrated cement
paste for different aggregate sizes (at constant aggregate volume) in cement

 

 

 

 

 

 

 

 

 

 

mortar.
rs t, tr V" N, V, V'h
(m) (m) m“
3.00E-04 1 2015-06 4.00E-03 2.04E-12 3.24E+09 0.007 0.443
1 .20E-03 4.80E-06 4.00E-03 1 .31 E10 5.07E+07 0.007 0.443
2.00E-03 8.00E-06 4.00E-03 6.06E-10 1.09E+07 0.007 0.443

 

 

 

3.2.2 Parametric Study of the Effect of the Fine Aggregate Elastic Modulus on the
Modulus of Elasticity of Cement Mortar
Since the modulus of elasticity of ﬁne aggregates used in cement mortar ranges from 70
to 90 MPa, an extensive parametric study was conducted to assess the signiﬁcance of this
range of aggregate elastic modulus in determining the modulus of elasticity of cement
mortar. It could be observed from the results of the study presented below that the cement
mortar elastic modulus does not change signiﬁcantly as the aggregate elastic modulus
varies within the applicable range. In this parametric study, all parameters related to
hydrated cement paste were obtained from the results presented in preceding chapters (for

example, E0 = 38 GPa, and capillary pore aspect ratio, a/b, equalto 2). Based on the

results presented in Table 3.1, V}: =O.443, Vi, =0.007 and V; =O.55 were considered for

determining the modulus of elasticity of cement mortar. The elastic modulus of hydrated

106

cement paste was calculated by substituting the above values into Eqn. 2.29, and

simplifying it to obtain:

 

0.37
E, = 5.852(1— ph)(” ‘41“)
p

0.37 .21
h (3 )

The modulus of elasticity of the interfacial transition zone was calculated by modifying
Eqn. 3.21 assuming that the porosity of the interfacial transition zone is twice that of the
hydrated cement paste."5 The value of the modulus of elasticity of the interfacial
transition zone was multiplied by a factor of 0.75 to account for the 25% volume ﬁaction

of the directionally oriented CH crystals in the transition zone.

 

— — \(7: _ SP )037
Er — 3.40 2P}, } p1.0.3117 (3.22)

E). and E; were calculated using Eqns. 3.21 and 3.22, for a reasonable range of capillary

porosity in cement mortar. The results are presented in Table 3.3.

107

Table 3.3 - The By, and Bi values calculated for different values of capillary porosity in
hydrated cement paste

Pr.

 

The next step was to ﬁnd a relationship between E). and E). This relationship was
obtained by plotting the corresponding values of E. and E.- ﬁom Table 3.3, and ﬁtting a

curve as shown in Fig. 3.3. The resulting equation relating E1. and E.- is shown below.

E, = 0.642 E, — 3.15 (3.23)

108

 

30

 

25

E, = 0.6423Eh - 3.15?

 

20

/

 

15

 

Ei (GPa)

10

 

/

 

 

/

 

 

 

l

10

20

1

30
E, (GPa)

40 50 60

 

 

Fig. 3.3 - The relationship between the moduli of elasticity of the hydrated cement paste
and the interfacial transition zone in cement mortar.

The values of K; and G; for sand were calculated using Eqns. 3.9 and 3.11, for

various values of the modulus of elasticity of ﬁne aggregates. The resulting values are

shown in Table 3.4.

Table 3.4 - The bulk and shear moduli of ﬁne aggregate (sand) in mortar.

 

 

 

 

 

E, (GPa) K’s (GPa) G’s (GPa)
70 39.2 29.4
80 44.8 33.6
90 50.4 37.8

 

 

 

109

 

Tables 3.5, 3.6 and 3.7 present the results of the parametric studies concerned with
determining the effects of the variations in E,. From these tables, one can also conclude
that the variation in aggregate size does not have a considerable effect on the modulus of
elasticity of cement mortar. Three aggregate sizes (0.6 mm, 2.3 mm and 4 mm) were
compared, and the results suggest that the upper and lower bounds of the shear modulus
and the bulk modulus did not change considerably with aggregate sizes. The reason for
this is that the bulk modulus and the shear modulus of a material, as shown in
corresponding equations, depend on volume fractions and elastic moduli of the individual
phases in the composite material and not on aggregate size. In the case of the third phase
in cement mortar, the interfacial transition zone, though its volume depends on aggregate
size, its volume fraction remains constant because its thickness changes linearly with

aggregate size as discussed previously.

110

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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113

The upper and lower bounds of the modulus of elasticity of cement mortar (Em & E...)
for different elastic moduli of ﬁne aggregates were calculated using Eqn. 3.1. The results
(as shown in Table 3.8) were plotted (Fig. 3.4) in order to assess the inﬂuence of the
variation in ﬁne aggregate elastic modulus (E,) on the modulus of elasticity of mortar

(Em)-

Table 3.8 -The upper and lower bounds of the elastic moduli of three cement mortars
each containing a ﬁne aggregate with a speciﬁc modulus of elasticity.

 

114

 

 

+ Em - (70) + Em - (80) + Em - (90)
-0-Em + (70) +Em + (80) -I-Em + (90)

 

 

 

 

80.00
70.00 -
60.00 -
.3 50.00 —
a.
9 40.00
E
l“ 30.00
20.00
10.00

0.00 I T T Y I I I
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Capillary porosity

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3.4 - The upper and lower bounds of the elastic moduli of three cement mortars each
containing ﬁne aggregates with different elastic moduli.
It can be observed in Fig. 3.4 that the variations in the upper bounds for Em tend to be
more pronounced relative to those in the lower bounds. However, it could generally be
concluded that the effect of the variation in the ﬁne aggregate elastic modulus, within the
given range, on the modulus of elasticity of cement mortar is not of great signiﬁcance.
Based on this observation, an average value for the modulus of elasticity of ﬁne
aggregate, 80 GPa, was considered as a representative value in development of the

model.

115

3. 2.3 Comparing the Predictions of Mechanistic Model of the Elastic Modulus of
Cement Mortar with those of Empirical Models

Once the upper and lower bounds of the elastic modulus of cement mortar were
established, the next step was to narrow down these bounds and develop a reasonable
model for the modulus of elasticity of cement mortar that can also be applied toward
development of the corresponding strength model. For this purpose, the predictions of an
empirical“ model for elastic modulus (based on substantial experimental data) were
introduced into Fig. 3.4, with the results shown in Fig. 3.5. This ﬁgure suggests that the
predictions of the empirical model fall within the upper and lower bounds of the

theoretical model, leaning more toward the lower bounds.

 

-O—Em - (70) -t— Em - (80) -I—Em - (90) —O—Em + (70)
-II—Em + (80) —O—Em + (90) -K- Empirical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80.00
70.00 4
60.00 1 }
50.00 - \\
Q —
85 40 00 v : A
30.00
‘ ~90— _
20.00 - ""‘ - .x
10.00
0.00 I l T I I I l
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Capillary porosity

 

 

 

Fig. 3.5 - Comparing the predictions of theoretical model for the elastic modulus of
cement mortar with those of empirical model based on substantial test data

116

Based on the trends observed in Fig. 3.5, it was concluded that the lower-bound model of
elastic modulus reasonably represents the trends observed in experimentally obtained
elastic modulus of cement mortar. The modulus of elasticity of cement mortar was then

represented as follows:

~ 9K;_G;,,_
’" ~ 3K,’,,_ + G,’,_ (324)

 

With aggregate elastic modulus (E,) of 80 GPa, Eqn. 4.35 yields the following expression
for the modulus of elasticity of cement mortar in terms of the elastic modulus of hydrated

cement paste.

~ 5.7813,? —28.35Eh
’" 0.04Ef +2.43E, —12.46 (3.25)

 

where, E). is the modulus of elasticity of hydrated cement paste, given by Eqn. 3.21.

3. 2.4 Mechanistic Model for the Fracture Toughness of Cement Mortar

It was noted earlier that the key source of fracture toughness in hydrated cement paste is
the pull-out of CH crystals. In the case of cement mortar, one should add the

contributions of sand (ﬁne aggregate) pull-out to fracture toughness. This phenomenon is

117

expected to make major contributions to fracture toughness by increasing the surface area
involved in ﬁictional pull-out. Sand particles are assumed to have rough surfaces, with

the hydrated cement paste interacting with sand particles as shown in Fig. 3.6.

 

Fig. 3.6 - Surface roughness of ﬁne aggregate, and interaction of the hydrated cement
paste (hep) at the interfacial transition zone

The computation of the fracture toughness of cement mortar was based upon the energy

dissipated during the pull-out of sand particles, which involves phononic ﬁiction as the

dendrites of hydrated cement paste bonding to the sand surface shear off (Fig. 3.6).

Further, the crack-shielding effect associated with sand particles contributes to fracture

toughness by enhancing formation of process zones in front of the crack tip. Actually, it

is assumed that each dendrite of hydrated cement paste is subjected to shearing force

118

ﬁom the sand surfaced dendrites, acting at the median contact surface. This causes a
diagonal tension stress on the hydrated cement paste dendrites, generating a 45 degree
diagonal crack as shown in Fig. 3.7. When sand particle pulls out, its rough surface
exerts a stress on the dendrites of the hydrated cement paste causing them to shear off.
The surface area of the hydrated cement paste subjected to this type of stress was

approximated to be about half of the pulled-out surface area of the sand.

d'loos45

Fig 3.7 - The cracking pattern of the hydrated cement paste dendrite due to force P
generated by sand pull-out.

The surface area of sand subjected to pull-out and debonding was computed by

considering a circular cylinder sand particle model as shown in Fig. 3.8. The shaded

region in Fig. 3.8 is the mean surface area of sand which would be subjected to ﬁiction

during pull-out. This region is assumed to occur, on the average, at the lower quarter of

the cylinder height.

119

 

051g:

Fig. 3.8 - Surface of sand particle subjected to frictional pull-out

 

3. 2. 4.1 Energy Released During Debonding of Sand Particles from Hydrated Cement
Paste

The interfacial surface area of sand that is subjected to debonding (followed by pull-out),

AM, can be expressed as follows:

S

_ 2
A d _ 2727s (3.26)

In order to account for the porosity of the interfacial transition zone (estimated at twice

that of hydrated cement paste, Ph), equation 3.26 was further modiﬁed as follows:

120

A'sd = 277420-213.) (3.27)

Debonding of sand occurs essentially through cracking of the interface zone, and
involves debonding of C-S-H globules. The energy lost due to debonding of C-S-H
globules, calculated earlier, is about 1.56 J/mz. Therefore, the energy (in Joule) released
(Usd) due to the debonding of sand ﬁ'om hydrated cement paste (i.e., due to debonding of

C-S-H globules in the interfacial transition zone) can thus be calculated as follows:

Usd : 1'56A'sd (3.28)

:> Usd =3.12727'32(1—2Ph) (3.29)

The ﬁ'acture energy released per unit area of mortar (6,4, in J/mz), due to C-S-H/C-S-H
debonding, can be calculated as the total debonding energy released due to one sand
particle divided by the projected area of the sand on a plane perpendicular to the pull-out

direction (see equations 3.30 and 3.31).

 

3.12 2
= m; (1—2P,)

717' 2 (3.30)
s

Gsd

121

:3 Gd =3.12(1—2P,) (33,,

S

3. 2. 4.2 Energy Released During Pull-out of Sand Particles from Hydrated Cement
Paste

The surface area of sand subjected to ﬁictional pull-out (A .310) is the shaded area of the

cube in Fig. 3.8 excluding the bottom surface.
A' _— 7172(1— 2P)
sp 3 h (3.32)

When the dendrites of hydrated cement paste shear off due to sand pull-out, the ﬁ'actured
diameter of the dendrites is about 1.414 (1' (due to diagonal shear effect), as shown in Fig.
3.7. Due to the assumption that the dendrites of both hydrated cement paste and sand are
interlocked (Fig 3 .6), the total surface area of hydrated cement paste subjected to shear

stress (A Q.) is about 1.414 times A ,p. Therefore,

A'SP, =1.414 mi (1 — 210,)

(3.33)

The energy (Uchsp 1) released due to CH pull-out upon fracture over A’sph can be calculated

as the product of Gp and A'sph (see equation 3.34). Gp is the energy released per unit

122

fractured surface area (due to CH pull-out in hydrated cement past), and is considered to

be equal to 27.88 J/mz, as discussed in Chapter 2.

U = 27.88/1's”, (3,4)

chsp l

:3 U =39.42mf(1—2P,)

chsp 1 (3.35)

Let Uchspz represent the energy released due to CH pull-out at the bottom surface of sand

particle. It can be calculated as the product of GP and the base area of sand particle:

U =27.88mf(1—2P,)

chsp 2 (3.36)

The total energy released due to CH pull-out during the pull-out of sand from hydrated

cement paste in mortar (Uchsp) is, therefore, the sum of Uchspl and Uchspz, and is given as:

U =U +U

chsp chsp] chsp 2 (3.37)

U = 673717120 " 2P1.) (3.38)

chsp

123

The energy released per unit ﬁactured area of interfacial transition zone (Gem, in J/mz),
due to CH pull-out, can be calculated as the total energy released as one sand particle
pulls out divided by the projected area of the sand on a plane perpendicular to the pull-out

direction (see equations 3.39 and 3.40).

67.3w,2 (1 — 210,)

Ch” 7172 (3.39)

S

 

:30

67.3(1— 210,)

chsp (3.40)

The total energy released rate per unit fractured area of the interfacial transition zone in

mortar (Gm) can now be calculated as the sum of Gchsp and Gsd:

Gim : Gchsp + Gsd (3.41)

:> 6,, = 7040- 210,) (,4,

The other factor which contributes to the ﬁacture toughness of cement mortar is the crack
growth shielding effect of sand particles. This crack shielding effect results from bridging
of the two crack surfaces by sand particles near the crack tip. This phenomenon promotes

multiple microcracking ahead of the crack tip in hydrated cement paste. The region where

124

these microcracks form is called crack process zone. For typical volume fractions of sand
used in mortar, formation of the process zone further increases the fracture toughness of
hydrated cement paste by about 65%. The total ﬁacture toughness of hydrated cement
paste (Go) was calculated in Chapter 2 at about 29.44 J/mz. Therefore, the modiﬁed
fracture toughness of hydrated cement paste in mortar will be about 1.65 x 29.44 = 48.58

J/mz.

Fracture toughness of mortar (0,) was calculated as the sum of the ﬁacture toughness of
the interfacial transition zone and that of hydrated cement paste, proportional to their

respective volume fractions in cement mortar:
_ I r
Gm _ Gim Vs + Gh Vh (3.43)

where, V; is the sand (and the small interfacial transition zone) volume fractions in

mortar; and Vh' is the hydrated cement paste volume fraction in mortar. As noted in

previous sections, V; = 0.557 and Vh'= 0.443 are used as reasonable examples. It was

also shown in Chapter 2 that G), = Go(I-Ph) and G, = 48.58 J/mz. Substituting these

values into equation 3.43 yields:

G... = 70.4(1—210,)+ 21.38(1-P,,) (3...,

125

:3 Gm = 91.78(1—1.75P,,) (3.45)

The fracture toughness from this model for a reasonable capillary porosity ranges from
44 to 76 J/mz, which is in conformance with the experimental values of the fracture

toughness of cement mortar which ranges from 55 to 75 J/ 2.35

The resulting values of G,,, are plotted as shown in Fig. 3.9 versus the capillary porosity

of hydrated cement paste.

 

 

«b 01 O) N on
O O O O O
l L l J

OJ
C
1

Fracture toughness of mortar, G m (J/mz)

 

 

N
O

 

1 I

0.15 0.2 0.25 0.3
Capillary Porosity (Ph)

9
_L

 

 

 

Fig. 3.9 - Fracture toughness of mortar as a ﬁrnction of the capillary porosity of hydrated
cement paste

126

A parametric study was conducted to investigate the signiﬁcance of sand volume fraction

in determining the fracture toughness of cement mortar. The relationships between the

fracture toughness of cement mortar (Gm) and the volume fraction of sand (V,') for

capillary porosities (P1,) of 0.3, 0.05 and O are plotted as shown in Figs. 3.10, 3.11 and
3.12, respectively (for the typical conditions considered earlier). The addition of sand is
observed to yield marked improvements in fracture toughness; however, the increase in

sand volume fraction have relatively small effects on the ﬁacture toughness of cement

 

 

 

 

 

mortar.
32
’E
9
._ _ 0
g 30 . O
O
2
O 28 a
C ‘3
.C
9 26 -
O
l.-
a
.3 24 . . .
“9: 0.45 0.5 0.55 0.6 0.65
Voume Fraction of Sand (V‘s)

 

 

 

Fig. 3.10 - Fracture toughness of mortar as a function of the volume fraction of sand for a
capillary porosity of 0.3

127

 

 

 

 

 

’E

Q, 62

E 58 4

o

2 o

"5 54 . o

a) N

m E 9

2 >3

.5, 50 —

3

,2

e 46 T

a

O

E 42 . T .
0.45 0.5 0.55 0.6 0.65

Volume Fraction of Sand (V's)

 

 

 

Fig. 3.11 Fracture toughness of mortar as a function of the volume ﬁaction of sand for a

 

 

 

 

 

capillary porosity of 0.05.

’E

(p 64 —

T: o
E 6

g 60 — O

“5

8 "E .

2 § 56

.C

3’

13 52 -

Q)

.5

g 48 . . .

Li: 0.45 0.5 0.55 0.6 0.65

Volume Fraction of Sand (V‘s)

 

 

 

Fig. 3.12 Fracture toughness of mortar as a function of the volume fraction of sand for a
capillary porosity of 0.

128

3. 2.5 Tensile Strength Model of Cement Mortar

The tensile strength model of cement mortar was developed in the same manner as that
for hydrated cement paste. The Grifﬁth equation is applied to develop the strength-

porosity relationship of cement mortar as follows:

7,] (3.46)

where, Em is the modulus of elasticity of cement mortar; Gm is the fracture toughness of
cement mortar; and I," is half the critical crack length at which unstable crack

propagation occurs in mortar. Many microcracks form in cement mortar prior to any
loading due to restrained shrinkage effects. The restraint of drying shrinkage partly
results ﬁ'om the presence of the dimensionally stable and high-modulus aggregates within
the cement paste. Microcracks tend to initiate at the interfacial transition zone, and then
propagate towards the hydrated cement paste. These cracks may extend over and bridge
between two adjacent sand particlesn'“ In this research, the critical crack length in

cement mortar is considered to extend between two sand particles, as shown in Fig. 3.13.

129

 

 

 

 

 

Fig. 3.13 - The critical shrinkage crack in cement mortar bridging between two adjacent
sand particles

The interfacial transition zone is the weakest region where restrained shrinkage cracks

initiate. Therefore, from Fig. 3.13, initial crack length in mortar was determined as the

sum of the center-to-center distance between two sand particles (1”) and the size of sand

(a) (Eqn. 3.47).
lm = O.5(l"+2rs) (3.47)

Shrinkage cracks are more likely to occur around larger aggregates due to the fact that
their interfaces are weaker (due to attraction of greater quantities of bleeding water) and
also because restrained shrinkage stresses tend to be greater in their vicinity. For this
reason, the aggregate size in mortar where crack is likely to initiate was considered to be

4 mm (i.e., close to the upper bound of ﬁne aggregate particle size). If we assume a

130

uniform sand particle size in cement mortar, the center-to-center spacing of sand particles

(1") can be calculated for the case with sand volume fraction of 0.55 as follows:

2) 2717‘s3 = 0.551"3 (3.48)

For a 4 mm diameter sand,

%
._ 24(2)’ _
l — —O——55— —4.50 mm (3.49)

Substituting the result of Eqn. 3.51 into Eqn. 3.49 yields 1,, = 4.24 mm.

The tensile strength model of cement mortar was then obtained by substituting Eqns.
3.25, 3.47 and 3.49, for Em, Gm and 1",, respectively, into Eqn. 3.48. The resulting model

is shown below.

m-289 (1— 17510,)[1 Zﬂ J6]
,6 +10. 4ﬂ- 9.23 (350)

 

where, P). is porosity of hydrated cement paste; and ,6 is deﬁned as:

,3 = a (1 — P1) (3.51)

131

and a is deﬁned as:

 

a = P1. (3.52)

The relationship between the tensile strength of cement mortar (am) and the porosity of

hydrated cement paste (Ph) (Eqn. 3.50) is plotted in Fig. 3.14.

 

 

.a
C)

A
.h
1

A
N
1

A
O
1

oo
1

O5
1

h
1

 

Tensile Strength of Mortar (MPa)

 

 

N

l l f

0 0.1 0.2 0.3 0.4
Capillary Porosity (Ph)

 

 

 

Fig. 3.14 - Tensile strength of cement mortar as a function of the capillary porosity of
hydrated cement paste.

132

A parametric study was conducted to investigate the effect of sand volume fraction on the

tensile strength of cement mortar. Figs. 3.15 shows the relationships between the tensile

strength of cement mortar (am) and the volume fraction of sand (7..) for a capillary
porosities (Pk) of 0.3 (covering a typical range of sand volume fraction). Theoretical
predications indicate a slight increase in tensile strength with increasing sand volume
fraction within the range considered here. It should be noted that excess quantities of
sand can compromise workability (at constant water/cement ratios), thereby complicating
production of cement mortar and potentially introducing large defects which are

damaging to tensile strength.

 

 

Tensile Strength of Mortar (MPa)
h 01 O) \l on
O
o
o
o
o
o

(A)
1

 

 

 

2 I l l l I T
0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62

Volume Fraction of Sand (Vs)

 

 

 

Fig. 3.15 - Tensile strength of mortar as a function of the volume fraction of sand for a
capillary porosity of 0.3

133

CHAPTER4

MODELING THE MECHANICAL PROPERTIES OF CONCRETE
BASED ON MECHANISTIC APPROACH

4.1 Introduction

Well-compacted concrete prepared with dense aggregates may be considered as a
multiphase ceramic material consisting of coarse aggregates embedded in mortar matrix.
The mortar matrix consists of ﬁne aggregates, the hydrated cement paste, and the pore
system. “7 Hydration of cement produces a product consisting of solid and a pore phases. 8“
Pores are thus inherent to cement-based materials; inadequate compaction further
increases the pore content (and pore size) of cement-based materials. The pore system
inﬂuences most important properties of concrete, including strength and stiffness. 39' 9° The
pore system present in the mortar of concrete, however, is different from the pores of
cement mortar prepared independently using identical proportions of various ingredients.
This difference is due to the presence of the interfacial transition zone at the mortar-
aggregate interfaces. 9" 92' 93 The capillary porosity of hardened cement paste depends upon
water-to-cement ratio, curing time and methods, and other factors. Similar parameters
partly determine the porosity of the interfacial transition zone in concrete where the
accumulation of bleeding water and the geometric constraints against full development of

hydration products lead to increased porosity. ‘7

134

There are several strength versus water—cement ratio relationships, which indirectly relate
the strength of concrete to its pore system characteristics. 87' 9" 95 These relationships serve
their purpose very well in the design of concrete mixes. However, such indirect
relationships cannot consider the effects of various factors (curing conditions, aggregate
size, etc.) on pore system characteristics and thus strength and other mechanical

properties of concrete.

The most important characteristics of pore system are porosity and pore size distribution,
which can be determined through mercury intrusion porosimetry (MIP). However, MIP
results are affected by a number of factors. 96' 97' 98' 99' '°° For example, the smallest size of
pores, in which mercury can intrude, depends upon the maximum intrusion pressure
applied. Consequently, the extent of porosity that can be determined by the porosimetry
test depends upon the nature of the pores, the size of the smallest pores likely to be
encountered in the material, and the maximum intrusion pressure applied. While
developing an empirical strength-porosity relationship for concrete, the contributions of
gel pores to the overall porosity and pore size distribution of concrete can be neglected

' Hence, to determine the pore system

without introducing signiﬁcant errors.10
characteristics inﬂuencing strength, the maximum pressure in the porosimetry test must
be sufﬁcient to cause intrusion of mercury into the smallest capillary pores. In mercury
porosimetry, a major part of gel pores remains non-intruded. Further, the closed pores
also remain non-intruded. One other limitation of mercury porosimetry is that it measures
the entry sizes rather than the true pore size. Thus, the porosity determined using the

101

mercury porosimetry test represents an apparent (but not actual) porosity.

135

A number of relationships relating the strength of cement-based materials to their pore
system characteristics are available in the literature. In this research, a theoretical
strength-capillary porosity relationship model is developed for concrete. This model was
evaluated by comparing its predictions with those of empirical models. Some minor
discrepancies are found between the theoretical and empirical models, which may be

attributed to the differences in estimation of porosity.

4.2 Modeling

The mechanical models of cement were developed by expanding upon the models of
cement mortar. While cement mortar contains only ﬁne aggregates, concrete incorporates
both ﬁne and coarse aggregates. The presence of coarse aggregates beneﬁts the stiffness
of concrete but negatively affects its strength. Concrete is a three-phase composite
material at macro-level, comprising hydrated cement paste, ﬁne aggregates and coarse
aggregates. At micro-level, concrete can also be treated as a four-phased composite

material, with the interfacial transition zone constituting the fourth phase.

Our approach involved modeling of hydrated cement paste, and then mortar as a three-
phase material comprising the hydrated cement paste, ﬁne aggregates and interfacial
transition zone. Concrete is considered here to comprise three phases: mortar, coarse
aggregate and the interfacial zone, noting that mortar itself is a three-phase material. The

interfacial transition zone in concrete is the region at the interface between cement mortar

136

and surfaces of the coarse aggregates. The introduction of coarse aggregates improves the
modulus of elasticity of concrete because coarse aggregates have relatively large elastic
moduli and occupy a relatively large volume of concrete. However, the strength of
concrete is lower than either hydrated cement paste or mortar. This could be attributed to
the presence of larger intrinsic cracks at the interface and also the introduction of the
interfacial transition zone which is highly porous with oriented CH crystals and low C-S-

H gel packing.

4. 2.1 Modulus of Elasticity of Concrete

To develop the modulus of elasticity model of concrete, it was necessary to determine the
modulus of elasticity, bulk modulus and shear modulus of every constituent phase of the
composite material. For cement mortar, the modulus of elasticity model developed in the
preceding chapter (Eqn. 3.25) was applied. A model for the elastic modulus of the
interfacial transition zone of concrete (E ic) was developed in a way similar to that of
cement mortar with some modiﬁcations. The porosity of the interfacial transition zone
was considered to be about twice that of hydrated cement paste. 7‘ This rise in porosity
was accounted for in modeling the elastic modulus of the interfacial transition zone (E ic)

using that of hydrated cement paste (Eh), as was the case for cement mortar.

As for the ﬁne aggregates, the modulus of elasticity of coarse aggregates (Em) ranges

from 70 GPa to 90 GPa, depending upon the type of aggregate. 77 A parametric study was

137

conducted to evaluate the signiﬁcance of this range in determining the modulus of

elasticity of concrete.

The upper and lower bounds of the modulus of elasticity of concrete were calculated
from an equation relating the modulus of elasticity of a composite material to its shear
modulus (G 'c) and bulk modulus (K 'c) (Eqn. 4.1). 7‘ The upper and lower bounds of the
shear modulus and bulk modulus were calculated using Hashin’s modulus of elasticity
model for three-phase composite materials (Eqns. 4.2 — 4.5). 78 The steps followed in the
computation of the upper and lower bounds of the modulus of elasticity of concrete are

presented below.

The upper and lower bounds of the elastic modulus of concrete (EC) was approximated as

follows: 7‘

9K'_G;_ 9K;+Gc'+
.6 . S E.- -<- , , (4.1)
3Kc_ + Gc_ 3K6+ + Gc+

 

 

where K 2, and K £_ are the upper and lower bounds of the bulk modulus of concrete,

respectively; and 06+ and G; are the upper and lower bounds of the shear modulus of

concrete, respectively. 73 These parameters were calculated as follows:

Lower bound of the bulk modulus of concrete, K ;_ , 78

138

 

 

 

1 _ VJ. + Vc'a + 3Vc'atrc
K1 K1. Kg, K,;+1.33G,;

C

Upper bound of the bulk modulus of concrete, K 2+ , 73

VCLKL.
3Kéatm

1+
K;c+l.33G,.'c

K2. = V,,ZK,',, +

 

 

Lower bound of the shear modulus of concrete, G ;_ , 7‘

 

 

—1—= V’" + V“ +0.4Vc'atrc 2
G' G; ,

9" ca

Upper bound of the shear modulus of concrete, G; , 7‘

V4.02.
2.50' t

CH rc

+
K121“ 3.33 Gi'c

 

0;, = V":an +

 

139

(4.2)

(4.3)

6
+—
K; +1336; 0,; j (44)

(4.5)

The next step was to determine all the parameters included in the above equations. The
bulk modulus and shear modulus of any material are related to its modulus of elasticity

through an Eqns. 4.6 and 4.7, respectively. 7‘

 

 

, _ E
3(1— 2v) “-9
, E
G = (4.7)
2(1 + v)

where, v is the poison’s ratio of cement mortar for the model of concrete.
From the above relationships, the bulk modulus and shear modulus of cement mortar and

coarse aggregates were calculated as follows (v = 0.2 was considered for both cement

mortar and coarse aggregates”):

The bulk modulus of cement mortar, K m ,

m m

"’ 3(1—2v) 3(1—2(0.2)) "’ (4-8)

 

 

K, E E

The bulk modulus of coarse aggregate, K ;a ,

140

, E E

CG CG

ca 2 = = 0'56Eca
3(1—2v) 3(1—2(0.2))

 

 

The shear modulus of cement mortar, GI" ,

 

 

G}, = E’" = E’" =O.42Em
2(1+v) 2(1+O.2)

The shear modulus of coarse aggregate, G5,, ,

, E E

CG CG

a, =———= = 0.42Ew
2(1+ v) 2(1+ 0.2)

 

(4.9)

(4.10)

(4.11)

Based on the above relationships, the bulk modulus and the shear modulus of the

interfacial transition zone of concrete could be determined as follows:

K; = 0.56E,.c

q; = 0.42E,,

141

(4.12)

(4.13)

For the elastic modulus of coarse aggregate (Em) ranging from 70 to 90 GPa, the
corresponding values of K 1.0 and G La range ﬁom 39.2 to 50.4 GPa and 29.4 to 37.8 GPa,
respectively. A parametric study conducted to evaluate the signiﬁcance of these ranges
on the mechanical properties of concrete. The modulus of elasticity of the interfacial
transition zone in concrete (EC) is a function of the modulus of elasticity of cement
mortar (Em) and capillary porosity. tn. is the ratio of the thickness of the interfacial
transition zone in concrete to the radius of an equivalent sphere representing coarse

aggregates,78 and was calculated as follows:

,- (4.14)

where, tic is thickness of the interfacial transition zone in concrete, which is a function of
the coarse aggregate size;81 and rm is the radius of an equivalent spherical coarse
aggregate. Thickness of the interfacial transition zone in concrete (tic) is proportional to

the size of the coarse aggregate it envelops.81

The next step was to determine the relationship between the interfacial transition zone
thickness and aggregate size in concrete. This relationship was approximated based on a
linear relationship developed in the preceding chapter. The linear relationship between
aggregate size and thickness of the interfacial transition zone, shown in Fig. 3.1, was
adapted for concrete. The size of coarse aggregates considered in this analysis ranges

from 4 to 25 mm. The maximum thickness of the interfacial transition zone is about 50

142

um.”2 This thickness was considered to occur around 25-mm aggregates which represent
a common maximum coarse aggregate size in concrete. No interfacial transition zone was
assumed to occur for aggregates that are smaller than 0.1 mm in size. This assumption is
based on the fact that no interfacial transition zone has been detected so far around CH
crystals of the same size. From this relationship, it was calculated that the value of tr
ranges from 3.9 to 50 um for aggregate size ranging from 4 to 25 mm. A parametric study
conducted to investigate the signiﬁcance of this range in deciding the mechanical

properties of concrete is presented in the next sections.

The signiﬁcance of the three phases (cement mortar, coarse aggregates and interfacial
transition zone) in determining the mechanical properties of concrete also depends on

their respective volume proportions in concrete. Calculations of the volume ﬁactions of
cement mortar (VJ. ), coarse aggregate (VOL) and interfacial transition zone (Viz) in

concrete are discussed below.

Porosity of the interfacial transition zone in concrete (Pic) is about two times that of
hydrated cement paste, and 25% of the volume of interfacial transition zone is ﬁlled with

CH crystals. 83

The pore volume of concrete (Vpc) could be expressed as the sum of the pore volumes of

cement mortar and the interfacial transition zones, as follows:

V =V +V- (4.15)

pc pm pro

143

where, VP," and Vpic are the pore volumes of cement mortar and the interfacial transition

zone in concrete, respectively.

The porosity of concrete (PC) can also be expressed as the ratio of the pore volume of

concrete to its total volume, as follows:

 

c V V (4.16)

where, Vc is the total volume of concrete, which is the sum of the volumes of coarse
aggregate and cement mortar. In this statement, the volume of the interfacial transition

zone is included in the volume of cement mortar. Vc can be expressed as follows:

V = V + V (4.17)

In a normal concrete mix, the volume of aggregates comprises about 75% of the total
volume of concrete. Since the ﬁne-to-coarse aggregate volumetric ratio in concrete is
about 2:3, ﬁne and coarse aggregates typically comprise about 30% and 45%,
respectively, of the total volume of concrete. Therefore, a typical volume of fraction of
cement mortar in concrete (including the interfacial transition zone), calculated as the

sum of the volume fractions of the hydrated cement paste (0.25) and the ﬁne aggregate

144

(0.3), can be estimated at about 0.55. The volume of interfacial transition zone
surrounding a single coarse aggregate in concrete (Vb-C) can be calculated using Eqn. 4.18
based on the relationship between the aggregate size and the thickness of interfacial
transition zone, as discussed in the preceding chapter (Fig. 3.1). The shape of coarse
aggregate has been modeled as spherical, cylindrical or cubic in the literature. Evaluation
of the geometry of a random sample of concrete coarse aggregates has indicated that a
cylindrical shape with aspect ratio (height-to-diameter ratio) of about 1.25 would
reasonably represent the concrete coarse aggregate. The assumed shape of the coarse

aggregate is shown in Fig. 4.1.

 

2.5rca

 

 

Fig 4.1 - The coarse aggregate shape model

The volume of the interfacial transition zone of thickness tic surrounding a single coarse

aggregate with a shape shown in Fig. 4.1 can be calculated as follows:

145

I[lie : mic (7r; + 6'5ratic + 2tii) (4-18)

Assuming a uniform coarse aggregate size, the number of coarse aggregates per unit

volume of concrete (Nca) can be calculated as follows:

V V

Cd _ Cd

Nca _ V _ 2.5”].3 (4.19)

lca

 

where, VIM is the volume of a single cylindrical coarse aggregate of radius rm and height

2.5rca.

The total volume of the interfacial transition zone (Vic) can then be calculated as the

product of Na, and V1 rel

Vic = V1 ic X Nca (4-20)

Since the thickness of the interfacial transition zone is proportional to the size of the
coarse aggregates in concrete, the volume fractions of the interfacial transition zone and

the cement mortar remain constant as the aggregate size changes, as shown in Table 4.1.

146

Table 4.1 - Volume fractions of the interfacial transition zone and the cement mortar for
different sizes of coarse aggregates in concrete

 

 

rca tic trc Vm Nca I7; 17;!I
(m) (m) m
2.00E-03 8.00E-06 4.00E-03 7.06E-10 7.16E+06 0.005 0.545
7.00E-03 2.80E-05 4.00E-03 3. 03E-08 1.67E-l-05 0.005 0.545
1.20E-02 4.80E-05 4.00E-03 1.53E-07 3. 32E +04 0.005 0.545

 

 

 

 

 

 

 

 

 

 

 

 

4.2.1.1 Parametric Study on the Inﬂuence of the Modulus of Elasticity of Coarse
Aggregates on the Concrete Elastic Modulus

Since the elastic modulus of the coarse aggregate used in concrete ranges from 70 to 90

MPa, a parametric study was conducted to assess the sensitivity of concrete elastic

modulus to that of coarse aggregate over this range. In this parametric study, all the

parameters related to hydrated cement paste and cement mortar were taken constant at

typical levels established in the preceding chapters (for example, E0 equal to 38 GPa, and

the aspect ratio of capillary pores, a/b, equal to 2). From Table 4.1, we considered

V": =0.545, K; =0.005 and V6; =0.45 for determining the modulus of elasticity of

concrete. The elastic modulus model of cement mortar developed in the preceding
chapter (Eqn. 3.25) was used for evaluation of the elastic modulus of concrete. The
equation for the elastic modulus of the interfacial transition zone in concrete (Eic) was

based on the similar equation developed for the interfacial transition zone in mortar (E ,-):

147

_ __ 8 0.37
Er. = 3-4(1— 2P1)(ﬂ—T€3’i—)_ (4.21)

h
E". and EC were calculated, using Eqns. 3.25 (from the preceding chapter) and 4.21, for a
reasonable range of capillary porosity in concrete. The result is shown in Table 4.2.

Table 4.2 - Em and Eu. values calculated for different values of hydrated cement paste
capillary porosity

Ph

 

The next step was to ﬁnd a relationship between E". and Etc. This relationship was
developed by plotting the corresponding values of E". versus Etc from Table 4.2, and
ﬁtting a curve as shown in Fig. 4.2. The resulting equation, relating Em and Etc, is shown

below.

Eic = 0.447Em — 5.01 (4.22)

148

 

 

 

 

 

 

 

 

16
y - 0 4469x 5 0075 °
12 /
15
O.
9, 8
111’
O
4 c
0 ‘ r r 1 I r 1
10 15 20 25 30 35 40 45
E... (GPa)

 

 

 

Fig. 4.2 - Relationship between the elastic moduli of cement mortar and interfacial
transition zone in concrete

The bulk and shear modulus (K '60 and G .ca) values for coarse aggregates were calculated

using Eqns. 4.9 and 4.11, respectively, for various values of the ﬁne aggregate elastic

modulus. The resulting values are shown in Table 4.3.

149

Table 4.3 - Bulk and shear moduli of coarse aggregates in concrete.

 

 

 

 

E... (GPa) K}. (GPa) GmGPa)
70 39.2 29.4
80 44.8 33.6
90 50.4 37.8

 

 

 

 

Tables 4.4, 4.5 and 4.6 show the outcomes of a parametric study for determining the
signiﬁcance of the coarse aggregate elastic modulus, E“, and size (varying within typical
ranges) in determining the concrete elastic modulus. The data presented in these tables
indicate that variation of the coarse aggregate size within a typical range does not
signiﬁcantly alter the elastic modulus of concrete. Three aggregate sizes (8 mm, 16 mm
and 20 mm) were considered, and the results showed that the upper and lower bounds of
concrete shear and bulk moduli did not change considerably with aggregate sizes. The
reason for this is that the bulk and shear moduli of a material, as shown in their
corresponding equations, depend upon the volume ﬁactions and elastic modulus of the
individual phases in the composite material, and not on aggregate size. The volume of the
third phase in cement mortar, the interfacial transition zone, remains constant because of

the opposite effects of coarse aggregate size on its thickness and surface area.

150

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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153

The upper and lower bounds of the elastic modulus of concrete (E04. & E..-) for different
coarse aggregate elastic moduli were calculated using Eqn. 4.1. The results (shown in
Table 4.7) are plotted in Fig. 4.3 to determine the inﬂuence of the variation in coarse

aggregate elastic modulus (E...) on the concrete elastic modulus (EC).

It can be observed from Fig. 4.3 that the variations in the upper bounds for E; with
changes in coarse aggregate elastic modulus seem to be more pronounced with compared
with those in lower bounds. However, the overall variations in concrete elastic modulus
with changes in coarse aggregate elastic modulus are small enough to justify selection of

an average value of 80 GPa for the coarse aggregate elastic modulus.

Table 4.7 - Upper and lower bounds of the modulus of elasticity of three concretes with
coarse aggregates of different elastic moduli.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E.. i E...

P" GP a , 3 GP? ‘ ,, ,, GP?” 5.. GPa - 9?? ‘ .99? , “
0.05 49.54 51.58 53.28 53.40 57.77 62.12
0.10 41.14 42.53 43.69 48.08 52.37 56.62
0.15 35.45 36.48 37.33 44.75 48.93 53.07
0.20 30.88 31.66 32.30 42.09 46.13 50.13
0.25 26.89 27.48 27.96 39.62 43.48 47.25
0.30 23.19 23.63 23.98 36.93 40.47 43.89
0.35 19.40 19.70 19.94 33.07 35.98 38.73
0.36 18.54 18.82 19.04 31.89 34.59 37.12
0.37 17.58 17.83 18.03 30.37 32.78 35.03

 

154

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ Ec - (70) + Ec - (80) + Ec - (90)
-0- EC + (70) + Ec + (80) + Ec + (90)
70
60 K
50 - ‘\\\ —
,. N
g \
u? 30 -\
20
1O
0 . . .
0.00 0.10 0.20 0.30 0.40
Capillary porosity

 

 

 

Fig. 4.3 - Upper and lower bounds of the modulus of elasticity of three concretes with
coarse aggregates of different elastic moduli.

4.2.1.2 Comparing the Predictions of the Mechanistic Model of Concrete Elastic
Modulus with those of Empirical Models

Aﬁer establishing the upper and lower bounds of the elastic modulus of concrete, the next
step was to narrow down these bounds and develop a reasonable model of concrete
elastic modulus for use toward development the concrete strength model. This was
performed by introducing the values of concrete elastic modulus obtained from an

empirical model based on comprehensive experimental results84 into Fig. 4.3, as shown in

155

Fig. 4.4, where the experimental results (represented by the empirical model) are

observed to lean towards the lower-bound values of the theoretically predicted elastic

moduli.

 

 

+Ec-(70) +Ec-(80) +Ec-(90) —0-Ec+(70)

 

 

-t—Ec+(80) —-—Ec+(90) - - - Ece

 

 

70.00

 

60.00 '\

 

50.00 -

 

 

40.00

 

Ec (GPa)

30.00

 

 

20.00

 

10.00

 

 

 

0.00 I l .
0.00 0.10 0.20 0.30 0.40

Capillary porosity

 

 

 

Fig. 4.4 - Comparing the predictions of theoretical model for the elastic modulus of
concrete with those of empirical model based on substantial test data

Based on the data presented in Fig. 4.4, the lower-bound theoretical model was chosen to
represent the elastic modulus of concrete. The modulus of elasticity of concrete was thus

approximated as follows:

156

N 9K£_G£-
C N 3K'_ +G'_ (4.23)

 

With the aggregate elastic modulus (Em) estimated at 80 GPa, substituting the parameters
in Eqn. 4.23 with their relevant equations expressed earlier yields the following
expression for the concrete elastic modulus as a function of the cement mortar elastic

modulus.

N 4.02193, —45.09Em
6 0.02315; +1.96Em —24.38 (424)

 

where, Em is the elastic modulus of cement mortar, given by Eqn. 3.25 in the preceding

chapter.

4. 2.2 Mechanistic Model for the Fracture Toughness of Concrete

It was shown in the previous chapters that the major contributor to the fracture toughness
of hydrated cement paste and cement mortar is the frictional pull-out of CH crystals. In
the case of cement mortar, CH pull-out accompanies sand—pull out. Following the same
approach, in concrete also the CH pull-out is considered to be associated with coarse
aggregate pull-out. The latter phenomenon makes major contributions to ﬁ'acture

toughness by increasing the surface area involved in frictional pull-out. Coarse

157

aggregates have rough surfaces where cement mortar develops bonds through the

interfacial transition zone as shown in Fig. 4.5.

v‘filfv

(
( J
1 ) ( )
Coarse
mortar ) ( mortar
( Aggregate )
) (
( n
) (
c )
) (
K D
) (

'm v x,-
i H
Fig. 4.5 - Surface roughness of the coarse aggregate, and bonding of cement mortar at the
interfacial transition zone
The computation of the ﬁ'acture toughness of concrete was based upon the energy
dissipated during the pull-out of coarse aggregates. This process involves phononic
friction as the dendrites of the hydrated cement paste bonding to the coarse aggregate
surface shear off (Fig. 4.5). Further, the crack shielding effect associated with coarse

aggregates contributes to the fracture toughness by enhancing the formation of the

158

process zone in front of a crack tip. It was assumed that each dendrite of hydrated cement
paste is subjected to a shearing force from the dendrites deﬁning the coarse aggregate
rough surface, acting at the centers of their contact surfaces. This produces a diagonal
tensile stress within the dendrites of cement mortar, generating 45-degree diagonal cracks
as shown in Fig. 4.6. When the coarse aggregate pulls out from the cement mortar, its
rough surface exerts a stress on the dendrites of the mortar and shears them off. The
surface area of the cement mortar subjected to this type of stress was approximated to be

about half the pulled-out surface area of the coarse aggregate.

d'n/cos45

 

 

crack

Fig 4.6 - The cracking pattern of the cement mortar dendrite due to the force P associated
with the coarse aggregate pull-out stress

159

The surface area of the coarse aggregate subjected to pull-out and debonding was
computed by considering a cylindrical coarse aggregate model shown in Fig. 4.7. The
shaded region in Fig. 4.7 is the mean surface area of the coarse aggregate which could be
subjected to ﬁiction as the aggregate pulls out of the mortar. This region is reasonably
estimated to be at the lower quarter of the cylinder height, representing take an average

pull-out depth.

 

2.5rca

 

 

 

Fig. 4.7 - Average surface area of the aggregate particle subjected to frictional pull-out

160

 

 

4. 2. 2.1 Energy Released During Debonding of Coarse Aggregate from Hydrated
Cement Paste in the Interfacial Transition Zone of Concrete

The surface area of a coarse aggregate subjected to debonding and pull-out, Acad, is: '

_ 2
Acad — 2 '25 ”ca (4.25)

The calcium hydroxide (CH) crystals are oriented to make planes of weakness around the
aggregate surface. To account for this effect and also the porosity of the interfacial
transition zone (which is about twice that of the bulk hydrated cement paste), equation

4.25 is modiﬁed as follows:

:> A'md = 2.25 m} (1 — 2P,,) (4.2.)

a

The total energy dissipated during concrete cracking is the sum total of the energy
dissipated by the cracking of mortar and t the debonding/pull-out phenomena in the
interfacial transition zone of concrete. The contribution of mortar cracking to ﬁacture
toughness was calculated in the previous chapter. The energy lost due to the debonding of
C-S-H globules, which is equal to 1.56 J/mz, and the fracture toughness of cement mortar
(determined in chapter 3), which is 91.78 J/mz, are used here towards calculation of the

fracture toughness of concrete. The additional energy released (U004, Joules) due to the

161

debonding of coarse aggregate from the hydrated cement paste (in the interfacial

transition zone) is calculated using the above relationship as follows:

U = 1 '56 A 'cad (4.27)

cad

:1) Ucad :3°51ﬂTci(1_—2Ph) (4.28)

The energy released per unit fractured area of the interfacial transition zone (Gm, in
J/mz) due to C-S-H/C-S-H debonding can be calculated as the total debonding energy of
one coarse aggregate divided by the projected area of the aggregate on a plane

perpendicular to the pull-out direction (see equations 4.29 and 4.30).

_ 3.51mi

cad 7272 (1 _ 2'I)h ) (4.29)

Cd

 

G

=> Gm, = 3510- ZR) (4,,

4. 2. 2.2 Energy Released During Pull-out of Coarse Aggregatesﬁ'om Cement Mortar in
the Interfacial Transition Zone of Concrete

The surface area of coarse aggregate subjected to ﬁictional pull-out (A leap) is the shaded

area of the cylinder in Fig. 4.7, excluding the bottom surface.

162

A'wp = 1.25 m»; (1 — 210,)

(4.31)

When the dendrites of the cement mortar shear off due to aggregate pull-out, the

fractured diameter of the dendrites is about 1.414 d In (due to the diagonal shear effect), as

shown in Fig. 4.6. Due to the assumption that the dendrites of both cement mortar and
coarse aggregate are interlocked (Fig 4.5), the total surface area of the cement mortar

subjected to shear stress (A '60”) is about 1.414 times A 'cap. Therefore,

A =1..3414x125m~ (1— 2P)

capm (4.32)

:> A' =1 .777rr2 0(1- 213),) (4.33)

capm

The energy released due to CH pull-out when A,capm ﬁ’actures (Uchcapl) can be calculated
as the product of the modiﬁed GP", and A 'capm (see equation 4.34). Gm is the energy per
unit surface area released due to CH pull-out in a compact cement mortar paste. It is

calculated in the previous chapter, and is equal to 91.78 J /m2.

U = 91.78A'

chcap l capm (4.34)

163

=> U = 162.227sz (1 — 213,)

cheap 1 a (4.35)

Let Uchcapz represent the energy released due to CH pull-out at the bottom surface of the
aggregate particle. It can be calculated as the product of the modiﬁed GP and the base

area of the coarse aggregate:

U =91.78m3(1-2P,,) (4,3,,

cheap 2 a

The total energy released due to CH pull-out during the pull-out of the aggregate particle

from the cement mortar (Uchcap) is, therefore, the sum of Uchcap) and Uchcapz, that is:

U =U +U

cheap cheap] cheap 2 (4.37)
:>U =254nr2(1—2P)
cheap ea h (4.38)

The energy released per unit ﬁ‘actured area of the interfacial transition zone (dep, in
J/mz), due to CH pull-out, is the total energy released as one coarse aggregate pulls out
divided by the projected area of the aggregate on a plane perpendicular to the pull-out

direction (see equations 4.39 and 4.40).

164

_3254m (1— 2P)

WW— 727.2 (4.39)

Cd

G

 

: G = 254 (1 _ 2Ph) (4.40)

cheap

The total energy release rate per unit fractured area of the interfacial transition zone in

concrete (Gic) can be calculated as the sum of Gchmp and Gc , as shown below:

G =G

eheap+ +G cad (4.41)

=> 0,, = 257 .5(1 — 2P,,) (4.4,,

As was discussed for cement mortar, the other factor which enhances the ﬁ'acture
toughness of concrete is the shielding of crack growth. Coarse aggregates render crack
shielding or arresting effect by bridging between the two crack surfaces near a crack tip.
This phenomenon causes multiple microcracking ahead of the crack tip (in the so-called
process zone) within the cement mortar. For the volume ﬁ'actions of coarse aggregate
commonly used in concrete, formation of the process zone increases the fracture
toughness of cement mortar by about 50%. Therefore, the modiﬁed fracture toughness of

mortar is: 1.5 x 91.78 = 137.67 J/m2.3“5

165

The fracture toughness of concrete (G) can be calculated as the sum of the ﬁ'acture
toughness associated with the debonding/pull-out at the interfacial transition zone and
that of the cement mortar, accounting for the coarse aggregate and mortar volume

fractions. Therefore, G can be expressed as follows:

Ge = G V, + GmVn'r (4.43)

ie ea

where, V“, is the volume fraction of coarse aggregates (including interfacial transition
zone) in concrete; and V", is volume ﬁaction of the cement mortar in concrete. Using the
example (typical) values of Vc;= 0.455 and V,,',= 0.545, as explained in the previous

sections, and substituting for Go and Gm, in equation 4.43 yield:

G,=0.455x2575(1—213,)+0.545x137.7(1—13) (4,4,
G, =117.21(1— 2P,)+ 75.05(1— P,) (4,4,,

:3 e, =192.2(1—1.61P,,) (4,...)

The resulting values of Gc are plotted in Fig. 4.8 versus the capillary porosity of the

hydrated cement paste.

166

The ﬁ'acture toughness from this model for reasonable levels of capillary porosity ranges
from 105 to 161 J/mz, which is in conformance with the experimental values of the

fracture toughness of concrete, which range from 100 to 200 J/mz.”

 

 

 

 

 

180
"g 1604
3
c) 140—
9'

Q)

‘6 120~

C

O

0

0 1003

(D

U)

0

E

a, 80-

3

.9.

m _

g 60

O

9

LL 40 I I I

0.1 0.15 0.2 0.25 0.3

Capillary Porosity (Ph)

 

 

 

Fig. 4.8 - Fracture toughness of concrete as a fmetion of the capillary porosity of
hydrated cement paste

4. 2.3 Mechanistic Model for the Tensile Strength of Concrete

The tensile strength model of concrete was developed using the Grifﬁth equation, as was

done in the case of cement mortar and hydrated cement paste. Since the tensile behavior

of concrete is nearly linearly elastic at lower porosities, Grifﬁth’s equation was employed

to develop the strength-porosity relationship for concrete:

167

(4.47)

 

where, E, is the modulus of elasticity of concrete; Gc is fracture toughness of concrete;
and 1c is half the critical crack length at which an unstable crack propagates in concrete

under tension. Several microcracks form in concrete during its hardening due to
differential shrinkage stresses prior to application of any external loads. Microcracks are
expected to initiate at the interfacial transition zones and propagate into the cement
mortar. The size of these cracks may vary depending on several factors, including the
homogeneity of the distribution of coarse aggregates in mortar. It was assumed here, as
done in the case of mortar, that the largest existing microcrack bridges between two
coarse aggregateslo as shown in Fig. 4.9.

Fig. 4.9 - The largest inherent shrinkage crack in cement mortar
between two coarse aggregate particles

 

I‘er "Zrm

i

mortar

 

agg.

i

mortar I

mortar

 

 

 

168

From Fig. 4.9, the initial crack length in concrete was determined as the sum of the

center-to-center distance between two coarse aggregates (1:) and size of coarse
aggregate (2rca):

[e = 05(1: + 2’31) (4.48)

Shrinkage cracks are more likely to occur around larger aggregates due to the fact that
they block more of the bleeding water and also generate larger differential shrinkage
stresses in their vicinity. For this reason, the maximum aggregate size in concrete was
considered in computation of the critical crack length. If we assume a uniform

distribution of coarse aggregates in cement mortar, the center-to-center spacing between
two aggregates (I: ) using the following procedures for the typical (example) case with

coarse aggregates comprising 45% of the concrete volume:

Vca = 0~45Vc (4.49)

where, Va, is the volume of a (cylindrical) coarse aggregate; and Vc is volume of concrete.

Substituting their approximate values (refer to Fig. 4.9) yields:

:> 2.572;»,3,=0.451”3 (4,0)

169

3 )3
:> 1:: 257502..) =2.6r,,
0.45 (4-51)

 

Substituting equation 4.5] into equation 4.48 yields 1,: 2.3rca. This means that the

critical crack length is about 2.3 times the maximum aggregate size.

The tensile strength model of concrete can then be derived by substituting equations 4.24,

4.46 and 4.48, for EC, Gc and 10, respectively, into equation 4.47. The resulting model is:

(1—1.61P,)(E; —11.2E,) %
4.. (E3, + 85 .215, — 1060 ) (4.52)

 

0",, = 96 .4

where, P), is capillary porosity of the hydrated cement paste; dd, is the maximum

aggregate size (2rca); and Em is modulus of elasticity of cement mortar that is expressed

as follows:

~ 5.785,? — 28.3513,
’" 0.04Ef + 2.4E, —12.46 (4.53)

 

where, Eh, the modulus of elasticity of hydrated cement paste is given below as :

170

Eh

_ 5.852

 

ph

0.37

(1" P1X” — 4197.)”?

(4.54)

Table 4.8 shows the tensile strength of concrete computed for different maximum

aggregate sizes. The maximum aggregate sizes considered are, 20, 16 and 10 m.

Table 4.8 - Tensile strength of concrete with different maximum aggregate sizes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P, E, E, E, G, 0,, 0., 0.,
(MAS = 20 mm) (MAS =16 mm) (MAS =10 mm)
(GPa) (GPa) (GPa) (J/Mz) (Mpa) (Mpa) (Mpa)
0.05 25.11 40.15 51.10 176.73 11.18 12.50 15.81
0.10 17.93 31.08 42.24 161.26 9.71 10.86 13.73
0.15 14.17 25.66 36.29 145.78 8.56 9.57 12.10
0.20 11.63 21.68 31.52 130.31 7.54 8.43 10.66
0.25 9.72 18.46 27.37 114.84 6.60 7.37 9.33
0.30 8.17 15.71 23.48 99.37 5.68 6.35 8.04
0.35 6.89 13.23 19.20 83.90 4.72 5.28 6.68
0.37 6.43 12.25 16.51 77.71 4.21 4.71 5.96

 

 

The relationship between tensile strength (0",) and capillary porosity (Pk) of concrete

(equation 4.52) is shown in Fig. 4.10.

171

 

 

 

"Iv-20mm -¢—16mm —I—10mm

 

 

 

 

18.00
16.00

 

 

A
P
C
O

 
   
   

 

 

12.00
10.00 $7.77.. .,_ g

8.00 ""“--:‘.‘..*~=
6.00
4.00
2.00

0.00 I r . I
0.10 0.15 0.20 0.25 0.30 0.35

Capillary Porosity (Ph)

 

 

 

 

 

Tensile strength (MPa)

 

 

 

 

 

 

 

 

 

 

Fig. 4.10 - Tensile strength-capillary porosity relationship for concretes with different
maximum aggregates sizes

The measured values of tensile strength for concrete materials with the typical fracture

toughness range of the model (105-161 J/mz) are from 4.9 to 7.6 MPa, for a maximum

aggregate size of 20 mm. ‘5 The theoretical range of tensile strength derived here is from

4.5 to 9.5 MPa for a maximum aggregate size of 20 mm (Fig. 4.10), which is comparable

to the experimental range.

172

4. 2.4 Compressive Strength Model of Concrete

The mechanism of failure of concrete under compression is different from that of
hydrated cement paste and cement mortar. Failure of hydrated cement paste and cement
mortar under compression involves formation of straight cracks parallel to the direction
of the applied compressive stress. Therefore, the compressive strengths of hydrated
cement paste or cement mortar could be estimated by comparing the lateral tensile strains
caused by the poison’s effect under compression against the cracking strain. In the case
of concrete, the lateral tensile stress resulting from the poison’s effect under compression
produces cracks at the interfacial transition zones around the coarse aggregates, and the
combined effects of compressive stresses and lateral strains extend these cracks in such a
way that they form conical shaped hats on opposite sides of the aggregates (Fig. 4.11).
The schematic and dimensions of Fig. 4.11, representing the cones formed on opposite
sides of coarse aggregates, are based on experimental observations.49 The conical hats act
as wedges to further crack the concrete as the compressive stress increases toward the
peak (failure) stress. Therefore, determination of the compressive strength of concrete
involves computation of the ﬁictional energy loss at conical surfaces associated with the

wedging effect that leads to failure of concrete.

173

I dw I
A T

340
morta
0.4drz +1

aggregate

 

2.35dca

 

 

r

0.55dca

i i

Fig. 4.11 Schematic of conical wedges forming above and below coarse aggregates as

 

concrete fails under compression

The surface area of one of the mortar conical wedges covering the coarse aggregate was

calculated as follows:

 

Se : ”(0°2dea )\/(O°55dea )2 + (O'Zdea )2 4.55)

2) Se : 0°36d3a (4.56)

174

The energy dissipated as the mortar surrounding a coarse aggregate shears along the
surfaces of conical wedges was considered as an addition to the ﬁ’acture toughness of
concrete (calculated originally under tension). Since this energy dissipation is totally
related to the mortar phase of concrete, the additional energy is calculated as the product
of the fracture toughness of mortar and the ratio of the surface area of the wedge to the

projected area of aggregate:

 

5.
GM" : 0,257,213“ (Gm) (4.57)

2
— 036d“ 91.8(1—1.75P,,)

“dd _ 0.25mi;

 

(4.58)

:3 G,,, = 42.08(1—1.7510,) (4.59)

The fracture toughness of concrete under compression (G,,) was calculated by adding the
contribution to fracture toughness of the conical mortar wedges (Gadd) to the fracture

toughness of concrete under tension (Gc):

Gee = Gadd +Gc (4-59)

175

:5 G, = 234(1—1.64P,,) (4.44)

The compressive strength of concrete was then calculated using the Griffith’s equation

based on the tensile implications of compression associated with the Poisson’s effect:

 

EG

0. = C CC
CC ”lcV 2 (4.61)

 

where, 0' is the compressive strength of concrete; E c is the modulus of elasticity of

CC

concrete; G,, is the fracture toughness of concrete; 1, is half the critical crack length in

concrete; and V is the Poisson’s ratio of concrete.

Expressions for the parameters of equation 4.61 have been determined in the previous
section, and their substitution into equation 4.61 yields the following expression for the

compressive strength of concrete:

(1-1.64P,,)(E,2, -11.22E,) %
41,, (E3, + 85 .22 E, — 1060 ) (4.42)

 

0,, = 532

176

where, Em and E h are the elastic modulus of mortar and hydrated cement paste as given

by equations 4.53 and 4.54, respectively.

Table 4.9 shows the computed values of the compressive strength of concrete for

different maximum aggregate sizes (10, 16 and 20 m).

Table 4.9 -Computed values of the compressive strength of concrete materials with
different maximum aggregate sizes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P, E, Em E, 6,, 0,, 0,, 0,,
(MAS = 20 mm) (MAS =16 mm) (MAS =10 mm)

(GPa) (GPa) (GPa) (J/MZ) (Mpa) (Mpa) (Mpa)
0.05 25.11 40.15 51.10 214.81 61.63 68.90 87.16
0.10 17.93 31.08 42.24 195.62 53.47 59.78 75.62
0.15 14.17 25.66 36.29 176.44 47.06 52.62 66.56
0.20 11.63 21.68 31.52 157.25 41.41 46.30 58.57
0.25 9.72 18.46 27.37 138.06 36.16 40.43 51.14
0.30 8.17 15.71 23.48 118.87 31 .08 34.75 43.95
0.35 6.89 13.23 19.20 99.68 25.73 28.77 36.39
0.37 6.43 12.25 16.51 92.01 22.92 25.63 32.42

 

 

The theoretically derived relationship between the compressive strength (“44) and

capillary porosity (Pk) of concrete (equation 4.62) is plotted in Fig. 4.12.

177

 

 

--*-- 20 mm

 

—o- 16mm +10 mm

 

 

 

 

 

 

 

 

 

 

 

Compressive Strength (MPa)

 

 

 

 

 

0.00 (

 

0.15 0.20 0.25 0.30 0.35

l T

Capillary Porosity (Ph)

 

 

Fig. 4.12 Theoretically derived compressive strength-capillary porosity relationships for

concretes with different maximum aggregates sizes

4.2.5 Comparing the Predictions of the Compressive Strength Model of Concrete with

those of Empirical Models

The compressive strength model

of concrete presented above was evaluated by

comparing its predictions with the experimental results reported in the literatures and also

with empirical models. The empirical models developed by Balshin‘ (Eqn. 4.63) and

Ryshkevitch5 (Eqn. 4.64) were used for evaluating the predictions of the theoretical

model for the compressive strength of concrete.

178

0'44 = 6874(1— Pym. (4.63)

_ —8.96 p
0",, — 74.4e (4.64)

The comparisons between theoretical predictions of compressive strength, experimental
results and empirical models is presented in Fig. 4.13. It can be observed in Fig. 4.13 that
the predictions of the mechanistic compressive strength model compare favorably with

experimental values and predictions of the empirical models, though the theoretical

predictions are slightly on the higher side.

 

 

 

 

 

 

 

 

 

I Theoretical A Balshin o Ryskovitch o Experrnimental
70
E 60 ~
22',
5 50 4 I
8’ 8
0 4O 4 I
8 2 . I
e ’I
2 30 r C O .-
3 9 9 III
. I
g 20 4 . 0 89992;
E , 0
8 10 -
O l T l l l l
0.05 0.07 0.09 0.11 0.13 0.15 0.17
Porosity

 

 

 

Fig. 4.13 Predications of the of compressive strength model of concrete versus
experimental results and predictions of empirical models.

179

CHAPTERS

MECHANICAL MODELS OF HIGH-VOLUME FLY ASH CONCRETE

5.1 Introduction

Fly ash is the most commonly used mineral admixture in concrete. The use of coal ﬂy ash
as a pozzolanic admixture partially replacing cement in concrete started in late 19408. "’2
Coal ﬂy ash is commonly used to replace about 25% (by weight) of Portland cement in
concrete. Recent research has demonstrated the potential to further improve the
performance, cost, environmental and energy beneﬁts of ﬂy ash use in concrete by
increasing the replacement level of cement with ﬂy ash up to 50% (or more). In order to
make optimum use of ﬂy ash in concrete, one needs models which relate the structure of
ﬂy ash concrete to its properties. So far, various empirical formulas relating ﬂy ash
concrete strength to ﬂy ash content have been developed by simply ﬁtting curves to
experimental data. These formulas, however, apply to limited ranges of material types
and proportions, and do not provide a comprehensive basis to optimize the use of coal ﬂy
ash in concrete. In this chapter, an attempt has been made to model the mechanical
properties of high-volume ﬂy ash concrete. In this investigation, 50% replacement of

cement with ﬂy ash is considered to represent hi gh-volume ﬂy ash concrete.

180

The mechanistic model developed here is based on the microstructure of hydrated cement
paste, cement mortar, and concrete. The main inﬂuence of ﬂy ash results from its reaction
with the hydration product CH (calcium hydroxide) at later ages, and involves reﬁnement
of the capillary pore system. This affects the size and shape of capillary pores, which in
turn alter the strength and other mechanical properties of cementitious materials. During
pozolanic reaction, there is a reduction of CH crystals, which alters the fracture toughness

of cementitious materials.

5.2 Development of :1 Strength Model for High-Volume Fly Ash Concrete

High-volume ﬂy ash concrete generally incorporates more than about 30% ﬂy ash by
weight of cement. As explained in previous chapters, parameters inﬂuencing the strength
of concrete include: capillary porosity, pore size distribution, intrinsic crack length, and
CH size. The approach to development of the model for high-volume ﬂy ash concrete
was based on the inﬂuence of ﬂy ash on these parameters. The major effect of ﬂy ash on
the strength of concrete is due to the capillary pore-size reﬁnement and the reduction of
CH content and size. In this research, the effect of 50% replacement of Portland cement
with ﬂy ash was analyzed. The 50% ﬂy ash content is a practical level targeted for

commercial use in the United States.

181

5.3 Concrete Containing 50% Replacement of Cement with Fly Ash

As the concentration of CH in the hydration solution reaches the critical mass for
pozolanic reaction to occur (which tends to take longer with increasing ﬂy ash conctent),
new C-S-H gels start to form due the reaction between the CH and the silicates from ﬂy

ash. This pozolanic reaction reduces the amount and size of CH over time.

In developing the mechanical model of high-volume ﬂy ash concrete, the relatively slow
rate of pozzolanic reactions make it important to consider the factor of time. This requires
a model for the rate of pozzolanic reactions in cementitious materials. The percent
reduction with time of ﬂy ash during pozolanic reaction for concrete containing 50% ﬂy

ash as replacement for cement was obtained from literatures as: ”3

12,, = 13.177(10gt)+ 5.07 (5,,

where, R F, is percent reduction of ﬂy ash after t day of hydration.

Since the ﬂy ash content is expressed as the percent by weight replacing cement, an
equivalent percent by volume was calculated by taking the speciﬁc gravities of ﬂy ash
and Portland cement to be 2.3 and 3.15, respectively. ”3 For a given water-to-binder ratio

of w/ b and 50% ﬂy ash replacement of cement, the volume of hydration products that

could be obtained from 1 gram of cementitious binder can be calculated as follows:

182

0.5 + 0.5
GFA SGPC (5'2)

 

 

V,,=w/b+

where, V}. is the total volume of mixture; w/ b is water-to-binder ratio; SGFA is the
speciﬁc gravity of ﬂy ash (SGFA =23); and SGPC is the speciﬁc gravity of Portland

cement (SGPC =3.15).

Substituting the values of SGFA and $0,, (2.3 and 3.15, respectively) into Equation 5.2

yields an expression for the volume of hcp that can be obtained from 1 gram of binder:

V, = w/b+9°—5-+—9—°§-= w/b+0.376
2.3 3.15

(5.3)

The volume fraction of ﬂy ash in hydrated cement paste was then determined as the ratio

of the volume of ﬂy ash to the total volume of hydrated cement paste for a given w/ b

ratio:

_ 0.5/2.3
f“ w/b+0.376 (54>

 

where, V, is the volume fraction of ﬂy ash in hydrated cement paste.

183

The volume fractions of ﬂy ash (V'f, ) in hydrated cement paste for w/ b ratios of 0.3 and

0.4 were calculated (for 50 wt.% replacement level of cement with coal ﬂy ash) using

Equation 5.4, and the corresponding values were 0.32 and 0.28, respectively.

5. 3.1 Fracture Toughness of Concrete Containing 5 0% Fly Ash as Replacement for
Cement

The unreacted ﬂy ash particles were treated as spherical micro-aggregates in hydrated
cement paste. Cracks in hydrated cement paste are expected to deﬂect and pass around
these particles. This implies that ﬂy ash particles pull out upon ﬁ'acture of hydrated
cement paste. The same logic applies to unreacted cement grains, which are irregular in
shape and are expected to dissipate more energy during pull-out. The ﬁrst task in
development of a time-dependent model for high-volume ﬂy ash concrete was to
determine the signiﬁcance of unreacted ﬂy ash particles and cement grains in determining

the fracture toughness of hydrated cement paste, and thereby that of concrete.

The contribution of unreacted ﬂy ash particles to the fracture toughness of hydrated
cement paste can be approximated by estimating the average additional surface area
created as the crack deﬂects around ﬂy ash particles. The maximum possible pull-out
distance is half the particle diameter, and the minimum is zero. Therefore, half of the
radius was assumed as a reasonable average pull-out distance, as was the case with CH

pull-out. The contribution of ﬂy ash pull-out to fracture toughness can be estimated by

184

considering the ratio of the fractured surface area of the spherical cap (Fig. 5.1) to its

projected plane area:

_ 2
S fa — n- r a (5.5)
_ 2
where, Sf, is the pulled-out surface area of ﬂy ash, which is the shaded surface in Fig.

5.1 (spherical cap); Af, is the projected plane area of the shaded region; and r f, is the

radius of the ﬂy ash particle.

—-—- dFA‘———j

r“ $ 0. rFA

 

Fig. 5.1 - Schematic presentation of the average surface area of ﬂy ash particle subjected
to pull-out upon cracking of the hydrated cement paste

185

Therefore, the inﬂuence of the pull-out of ﬂy ash particles on the ﬁ'acture toughness of
hydrated cement paste can be estimated by considering the additional ﬁactured area due
to the pull-out of ﬂy ash particles along the crack path which tends to be smooth in the
case of plain cement paste. Because ﬂy ash particles are spherical in shape, their effect on
the fracture toughness results from the debonding of C-S-H globules, and not from
ﬁictional pull-out. Hence, the contribution of the pull-out of unreacted ﬂy ash particles to

the ﬁacture toughness of hep was determined as follows:

 

(5.7)

where, G f, is the contribution to fracture toughness of a single ﬂy ash particle; and G, is

fracture toughness of hydrated cement paste associated with the debonding of C-S-H

globules, as discussed in Chapter 2. Substituting the values of S F, and A 12 into Equation

5.4 and simplifying it yields the following relationship between G f, and G, :

2

”f“ G -133G
O.7572r2 "_ ° "

a

 

f“ — (5.8)

186

Since G, is the minor contributor to the fracture toughness of hcp (it constitutes only
about 5% of the hop fracture toughness), the effect of unreacted ﬂy ash particles on the

fracture toughens of hcp is not signiﬁcant, and could be reasonably ignored.

The other major contributor to fracture toughness of hydrated cement paste is the pull-out
of unreacted cement grains. These grains are irregular in shape and have rough surfaces.
To simplify the model, they were assumed to have a ﬂat surface. Since their mean size is
almost the same as that of CH, their effect on fracture toughness was almost similar to

that of CH.

To determine the amount of ﬂy and cement grains reacted at a given age of hydration, it
was necessary to develop a relationship between the degree of hydration of hcp and its
age. The degree of hydration of hcp for a given w/b ratio can be estimated ﬁ'om the

empirical relationship given below: ‘°‘

-—0.l35

_ W/b
“4:28 — 96 e (5.9)

 

where, a,=28 is degree of hydration at the age of 28—days.

The above equation was further reﬁned for estimating the degree of hydration at any

hydration time:

187

—0.135
“1:28 =96e w/b (1.0.16)

 

(5.10)

It was necessary to express the effect of the unreacted cement particles and CH on the
fracture toughness of hop in an equation format so that it could be incorporated into the
concrete model. The procedure followed was to determine the relevant parameters shown
in Tables 5.1 and 5.2. The values of RM and 01 were computed ﬁom Equations 5.1 and
5.10, respectively. The amount CH produced (CH,) at a given time, as a percentage of
hcp, was calculated as the product of the degree of hydration and the total amount of CH
as a percentage of hep (which is 22% upon full hydration). "’5 The value of CH, (in grams)
was obtained by multiplying the percent CH, by the volume of hcp (V11 from Equation
5.3) and the unit weight of CH (which is 2.21). During pozolanic reaction, 0.85 grams of
CH is needed to react with one gram of ﬂy ash. "’6 Therefore, the amount of CH in grams
reacted (CH,) at a given time was computed using this relationship, with the results

presented in Table 5.1.

Table 5.1 - Reduction of CH and ﬂy ash in hcp during pozolanic reactions

 

 

 

 

 

 

 

t R” 0 CHp CH, CHr

(day) %FA %hcp (g)— (g)
28 24.19 0.630 0.139 0.223 0.103
90 30.85 0.760 0.167 0.268 0.131
180 34.80 0.849 0.187 0.300 0.148
270 37.12 0.906 0.199 0.320 0.158
360 38.76 0.949 0.209 0.335 0.165

 

 

 

 

 

 

188

 

After determining the amount of reacted ﬂy ash, cement and CH, the amounts of
unreacted cement grains (UCG) and unreacted CH crystals (UCH) were computed. These
are the two major contributors to the fracture toughness of hcp, which affect G,. The
value of G,( at a given time was determined by varying the amounts of unhydrated
particles (cement and CH) with time, as shown in Table 5.2. The maximum G, that can
be obtained under full hydration is about 29.44 J/mz, when CH is 22% of hcp. The effect
of unhydrated particles on the ﬁacture toughness of hcp at different hydration times is
shown as a ratio of the G, to the maximum value of 29.44 J/mz. It can be observed ﬁom

Table 5.2 that ﬂy ash reduces the fracture toughness of concrete over time.

Table 5.2 -Effect of unreacted ﬂy ash and cement particles on fracture toughness of

 

 

 

 

 

 

 

 

 

 

 

 

hydrated cement paste
1 ”cc UCH UCG+UCH Got Got, Go
(day) %hcp %hcp %hcp J/m2

28 8.079 7.464 15.543 23.93 0.813
90 5.248 8.548 13.796 23.72 0.806
180 3.300 9.461 12.761 23.65 0.803
270 2.056 10.100 12.156 23.56 0.800
360 1.123 10.604 11.728 23.49 0.798

 

 

A graph shown in Fig. 5.2 was plotted using the data presented in Table 5.2 in order to
show the trends in development of the ﬁacture toughness of cementitious paste with high

ﬂy ash volume over time. The following relationship expresses this trend:

189

G, = 1.0274 G, 44-0069 (,1,

where, G,f is intrinsic fracture toughness of high-volume ﬂy ash concrete.

 

 

0.814
0.812 ~
0.810 4
0.808 , G,./G, = 1.02741410069
0.806 ~
0.804 -
0.802 -
0.800 .
0.798 ~ o

0.796 4 ( . 4 ( ( .
0 50 100 150 200 250 300 350 400

t(days)

Got/Go

 

 

 

 

 

 

 

Fig. 5.2 - Effect of age on fracture toughness of high-volume ﬂy ash hep

Since the G, has a direct relation with fracture toughness of hep (G1,), the fracture
toughness of hcp containing high-volumes of ﬂy ash (th) can be determined based on

Equation 5.11 as follows:

G, = 1.0274 G, 1410069 (5,,

190

The above relationship indicates that the effect of ﬂy ash on the fracture toughness of
hydrated cement paste is relatively small (Fig. 5.3). The reason for this could be that the
pozolanic reaction consumes relatively small amounts of CH to produce new C-S-H
particles; therefore, much of the CH crystals remain unreacted to almost preserve the

original fracture toughness of the hydrated cement paste.

 

 

--ax---Gh +th-28 +th-90 +th-180

 

 

 

 

30.00

N

9‘

o

o
l

15.00 a

Fracture Toughness of hep (J / rn 2)
N
O
O
O

 

 

 

10.00 ( ( (

0.00 0.10 0.20 0.30
Capillary Porosity (Ph)

 

 

 

Fig 5.3 — Comparison between the fracture toughness of normal hep with that of high-
volume ﬂy ash cement pastes at the ages of 28, 90 and 180 days.

191

5.3.2 Modulus of Elasticity of Concrete Containing 50% Fly Ash As Replacement for
Cement

In determining the effect of ﬂy ash on the elastic modulus of concrete, it was necessary to
evaluate the change in the structure of concrete phases resulting from pozolanic reaction.
During pozolanic reaction, the newly produced C-S-H gels grow in capillary pores and
thus reﬁne the pore sizes. The reduction in capillary pore sizes is considered to be
proportional to the reduction in the aspect ratios (a/b) of capillary pores. An important
task is to determine the reduction in capillary pore size with time. A relationship was
drawn by plotting experimental results collected from the literatures addressing the effect
of time on capillary pore size in high-volume ﬂy ash concrete. This relationship was as

follows: ‘07

% = 8.41411“8 (,1,

Substituting the above equation into Equation 2.29 (from Chapter 2) yields an expression

for the elastic modulus of hydrated cement paste at time t:

_ 077154 (1 -p)(ﬂ -4Ph)°'37

A

(1+2(8.4t-°-38))p,‘3-37 (5.14)

hf

where, E ,, is the modulus of elasticity at time t of hep containing hi gh-volume of ﬂy ash.

192

Stating E ,, as a function of the modulus of elasticity of normal hep (E ,, ) yields:

E _ 5E,
hf "' 1+ 16.8t_0'38 (5.15)

 

Fly ash reﬁnes the capillary pore size, and lowers the aspect ratio of the elliptical
capillary pores which undermine the modulus of elasticity. As can be observed from Fig.
5.4, hi gh-volume ﬂy ash paste offers improved elastic moduli when compared with
normal Portland cement paste at later ages due to the relatively slow pace of pozzolanic

reactions.

 

 

--ar:---Eh +Ehf-28 +Ehf-90 +Ehf-180

 

 

 

8

 

8 8
_L l

N
01

A
0'1
1

_s
o

 

Modulus of Elasticity of hcp (GPa)
N
0

0|
1

 

 

 

o F I T I l l I
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Capillary Porosity (Ph)

 

 

 

Fig 5.4 — Comparison of the modulus of elasticity model of normal hep with that of high-
volurne ﬂy ash cement pastes at 28, 90 and 180 days.

193

In the same manner, the modulus of elasticity of mortar (Em) was determined by
substituting Eh, in Equation 3.25 (from Chapter 3) for E), to get an expression for Em, as

follows:

E _ 5.78Eff—2835E,
"'f _ 0.04E3f + 2.413,, —12.46 (5.16)

 

The effect of ﬂy ash on the elastic modulus of cement mortar is results from the
corresponding effects of ﬂy ash on cement paste. As shown in Fig. 5.4, the elastic
modulus of high-volume ﬂy ash mortar exceeds that of cement paste at later ages. At 28
days of age, however, the elastic modulus of cement mortar is still higher than that of the

hi gh-volume ﬂy ash mortar, which is due to the slow pace of pozzolanic reactions.

194

 

 

"ax-“Em —I—Emf-28 +Emf-90 —o——Emf—180

 

O)
O

 

0'1
O
1

A
O
1

N
O
l

 

Modulus of Elasticity of Mortar (GPa)
-x 00
O O

 

 

0 1 l r l l I l
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Capillary Porosity (Ph)

 

 

 

 

Fig 5.4 — Comparison of the modulus of elasticity of cement mortar with that of high—
volume ﬂy ash mortar at the ages of 28, 90 and 180 days.

The elastic modulus of concrete containing high ﬂy ash volume (Ed) was also determined

by substituting Emf for Em in Equation 4.24:

_ 4.02ng —45.09E,,,
“f _ 0.02383, +1.96Emf —24.38 (4.17)

 

The comparison between the elastric modulus of normal concrete and high-volume ﬂy

ash concrete depicted in Fig. 5.5 indicates that, at later ages, high-volume ﬂy ash concrete

195

offers higher elastic moduli when compared with normal concrete. At 28 days of age,
however, normal concrete still provides a higher elastic modulus than high-volume ﬂy

ash concrete, which is due to the relatively slow pace of pozzolanic reactions.

 

 

- - ax- - - Ec + ch-28 + ch-90 —O— ch-180

 

 

 

 

N
O
1

 

Modulus of Elasticity of Concrete
3 89°38

 

 

o I I 1 I I I I
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Capillary Porosity (Ph)

 

 

 

 

Fig 5.5 — Comparison of the modulus of elasticity model of normal concrete with that of
high-volume ﬂy ash concrete at the ages of 28, 90 and 180 days.

196

5. 3. 3. Tensile Strength of Concrete Containing 5 0% Fly Ash

The tensile strength model of concrete containing 50% replacement of cement with ﬂy
ash was determined by upgrading the tensile strength model for Portland cement concrete
developed in Chapter 4. In this process, the effect of ﬂy ash on the properties of hydrated
cement paste, cement mortar and concrete were considered. The hypothesis is that the
major effect of ﬂy ash on cementitious materials involves their fracture toughness and
capillary porosity, and thus the tensile strength model of concrete was developed by

modifying Equation 4.52 (from Chapter 4) as follows:

 

E 6f GCIf

ﬁle (5.18)

 

06¢:

Substituting equation 5.12 and 5.17 for GM and E,,, respectively, into equation 5.18

yields:

'(1 —1.611r>,,)(E,3,f —11.22E,,)z:j°-007 ”
(1,,(133, + 85.22Emf —1060) (5.19)

 

6,, = 96.4

 

 

197

 

 

where, 0',“ is the tensile strength of concrete with 50% replacement of cement with ﬂy

ash; Em, is the modulus of elasticity of mortar as deﬁned in equation 5.15; t is (hydration)

age of concrete in days; and (1,, is the maximum aggregate size.

The changes in elastic modulus of hi gh-volume ﬂy ash hep, cement mortar and concrete,
also tensile strength of concrete (50% ﬂy ash) with time, at the age of 28, 90 and 180

days, is shown in Tables 5.3, 5.4 and 5.5, respectively, for maximum aggregate size of 20

m.

Table 5.3 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and
tensile strength of concrete, containing 50% ﬂy ash, in 28-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E, 15.,f Em. Ecf 6,, 0.4

P), (GPa) (GPa) (GPa) (GPa) J/m2 MPa
0.05 25.11 21.89 36.27 47.46 48.11 6.30
0.10 17.93 15.63 27.83 38.73 44.35 5.47
0.15 14.17 12.35 22.84 32.95 40.59 4.82
0.20 11.63 10.14 19.19 28.35 36.83 4.26
0.25 9.72 8.47 16.25 24.29 33.07 3.74
0.30 8.17 7.13 13.71 20.17 29.31 3.20
0.34 7.13 6.21 11.76 13.89 26.30 2.51

 

198

 

Table 5.4 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and

tensile strength of concrete, containing 50% ﬂy ash, in 90-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eh Em Emf Ed Gag ow

Ph (GPa) (GPa) (GPa) (GPa) J/m2 MPa
0.05 25.11 31 .08 46.66 56.75 44.40 6.89
0.10 17.93 22.20 36.66 47.84 40.93 6.08
0.15 14.17 17.55 30.55 41.68 37.46 5.43
0.20 11.63 14.40 26.01 36.69 33.99 4.85
0.25 9.72 12.03 22.31 32.31 30.52 4.31
0.30 8.17 10.12 19.16 28.30 27.05 3.80
0.35 6.89 8.53 16.36 24.44 23.58 3.30

 

Table 5.5 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and

tensile strength of concrete, containing 50% ﬂy ash, in ISO-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eh Ehf 15..., Ed G... 01:11
Ph (GPa) (GPa) (GPa) (GPa) J/m2 MPa
0.05 25.11 37.64 52.93 61.72 42.33 7.19
0.1 17.93 26.88 42.18 52.92 39.02 6.39
0.15 14.17 21.25 35.47 46.68 35.71 5.74
0.2 11.63 17.44 30.40 41.53 32.40 5.16
0.25 9.72 14.57 26.25 36.97 29.10 4.61
0.3 8.17 12.26 22.68 32.76 25.79 4.09
0.35 6.89 10.33 19.51 28.76 22.48 3.58

 

199

 

 

The effect of ﬂy ash on the tensile strength of normal concrete is related to the
corresponding effects of ﬂy ash on the ﬁ'acture toughness and modulus of elasticity of
concrete. The gain in the the strength of concrete with high-volumes introduction of ﬂy
ash becomes evident at later ages, which is due to the relatively slow pace of pozzolanic
reacitons. A comparisons between the tensile strength of normal concrete and that of
high-volume ﬂy ash concrete at 28, 90 and 180 days of age is shown in Fig. 5.6. At the
age of 28 days, the strength of normal concrete is higher than that of high-volume ﬂy ash
concrete. With time, the strengths of hi gh-volume ﬂy ash concrete increases beyond that

of normal concrete.

The relationship between tensile strength (0“,) and capillary porosity (Pk) of concrete

containing 50% replacement of cement with ﬂy ash (Equation 5.19) is shown in Fig. 5.6.

200

 

 

 

 

 

 

 

 

 

"it-“oct +octf-28 +octf-90 —O—octf-180
A14
(U
0.
E12—
.9
93 10 —
O
8
o 8*
“5
.C
a 6‘
E
03 4‘
9
E 2‘
.‘l’
0 I I l l I l
0.02 0.07 0.12 0.17 0.22 0.27 0.32 0.37
Capillary Porosity (Ph)

 

 

 

Fig 5.6 — Comparison of the tensile strength of normal concrete with that of high-volume
ﬂy ash concrete at the ages of 28, 90 and 180 days.

5. 3.4 Compressive Strength of Concrete Containing 5 0% Fly Ash

After determining the modulus of elasticity and fracture toughness of concrete, the next
step was to develop a model for the compressive strength of concrete. The same concept
and approach as in the case of normal concrete (Chapter 4) was applied for modeling the
compressive strength of high-volume ﬂy ash concrete. For this purpose, the elastic

modulus and fracture toughness models of normal concrete were replaced with the

201.

corresponding models of high-volume ﬂy ash concrete introduced above. The resulting

model for the compressive strength of high-volume ﬂy ash concrete is presented below:

(1 —1.54P,)(E,§f — 11 .22 Emf )t-0.0069 %
0,, = 532 - 2 _
dc, (Emf + 85 .22 Emf —106O ) (5.20)

 

Tables 5.6, 5.7 and 5.8 show the calculated modulus of elasticity and fracture toughness
at various ages of concrete. The resulting compressive strength of concrete, as computed
from Equation 5.20, is shown in the tables and also plotted in Fig. 5.4. It can be observed

from the graph that the strength of high-volume ﬂy ash concrete gains strength slowly

with time, as stated in several literatures.

Table 5.6 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and
compressive strength of concrete, containing 50% ﬂy ash, in 28-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eh Em Emf ch Gccf Occf

P, (GPa) (GPa) (GPa) (GPa) Jim2 MPa
0.05 25.11 21.89 36.27 47.46 143.20 57.35
0.10 17.93 15.63 27.83 38.73 132.01 49.74
0.15 14.17 12.35 22.84 32.95 120.82 43.89
0.20 11.63 10.14 19.19 28.35 109.62 38.77
0.25 9.72 8.47 16.25 24.29 98.43 34.00
0.30 8.17 7.13 13.71 20.17 87.23 29.16
0.34 7.13 6.21 11.76 13.89 78.28 22.86

 

202

 

Table 5.7 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and

compressive strength of concrete, containing 50% ﬂy ash, in 90-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E, E,f E... E... ' Gccf Occf
P, (GPa) (GPa) (GPa) (GPa) J/m2 MP3
0.05 25.11 31 .08 46.66 56.75 132.15 62.70
0.10 17.93 22.20 36.66 47.84 121.82 55.27
0.15 14.17 17.55 30.55 41.68 111.49 49.35
0.20 11.63 14.40 26.01 36.69 101.16 44.10
0.25 9.72 12.03 22.31 32.31 90.83 39.22
0.30 8.17 10.12 19.16 28.30 80.50 34.55
0.35 6.89 8.53 16.36 24.44 70.17 29.97

 

Table 5.8 - Modulus of elasticity of hep, mortar and concrete and fracture toughness and

compressive strength of concrete, containing 50% ﬂy ash, in l80-days.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eh Ehf Emf ch Gccf accf

P, (GPa) (GPa) (GPa) (GPa) J/m2 MPa
0.05 25.11 37.64 52.93 61.72 125.99 65.40
0.1 17.93 26.88 42.18 52.92 116.15 58.14
0.15 14.17 21.25 35.47 46.68 106.30 52.24
0.2 11.63 17.44 30.40 41.53 96.45 46.93
0.25 9.72 14.57 26.25 36.97 86.60 41.96
0.3 8.17 12.26 22.68 32.76 76.75 37.18
0.35 6.89 10.33 19.51 28.76 66.90 32.53

 

203

 

 

Fig. 5.7 compares the compressive strength of normal concrete versus those of high-
volume ﬂy ash concrete at different ages. While normal concrete provides a higher
compressive strength at 28 days of age, the 90- and l80-day compressive strengths of
high-volume ﬂy ash concrete exceed the compressive strength of normal concrete. The
relatively slow pace of pozzolanic reactions explain why high-volume ﬂy ash concrete

exhibits improved performance at later ages.

 

Fau- - - occ + (reef-28 + occf-90 + occf-180

 

 

70

60-

50-

40-

30-

 

20-

Compressive Strength of Concrete (MPa)

 

 

 

10 l l l l l I T
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Capillary Porosity (Ph)

 

 

 

Fig 5.7 — Comparison of the compressive strength model of normal concrete with that of
high-volume ﬂy ash concrete at the ages of 28, 90 and 180 days.

204

5. 3.5 Comparing the Predictions of the Mechanistic Model of the Strength of Concrete
Containing 5 0% Fly Ash Cement with Experimental Results

Veriﬁcation of the mechanistic models involves comparison of their predictions with the
corresponding experimental results. Noting that the experimental data on high-volume ﬂy
ash concrete are relatively limited, an attempt was made to verify the mechanistic models
of high-volume ﬂy ash concrete using the available experimental data. ‘°‘ Fig. 5.8 presents
a comparison of the mechanistic model predictions of the 28-days compressive strength
of concrete containing 50% ﬂy ash versus experimental results. The experimental data
are observed to cover a narrow range of porosity, which could be the reasonable porosity
of high-volume ﬂy ash concrete concrete at 28 days of age. The mechanistic model
predictions appears to compare reasonable with experimental results, especially at the
lower capillary porosities measured in this experimental study. The mercury intrusion
technique used here to measure capillary porosity tends to overestimate the actual
porosity. Removal of this effect could improve the comparison between the predictions of

the mechanistic model and the experimental results.

205

 

 

 

 

I Mechanistic A Experimental

 

 

 

 

 

70
A 60 1
(u I
n
E 50 - I
E.
U) I
E) 40 " I
5 n

A

.3 30— 5 ‘-
3 s
9 20 ‘ '
Q - A A
E f M
8

10 -

0 I I I I
0 0.03 0.06 0.09 0.12 0.15

 

 

Porosity (P)

 

Fig. 5.8 - Comparison of the predictions of the mechanistic model for the 28-day
compressive strength of concrete containing 50% ﬂy ash versus experimental
results

Fig. 5.9 shows a comparison of the 90-day compressive strength of high-volume ﬂy ash
concrete versus experimental results. The mechanistic model is observed to predict
experimental results with reasonable accuracy. The general over-estimation of the
experimental results could be partly attributed to the over-estimation of capillary porosity

measurements by the mercury intrusion technique.

206

 

 

 

 

 

 

 

 

 

I Mechanistic A Experimental
70

E 60 ~ '
g I
5 50 ‘ I ‘
2’ I
2 40 1 ‘ ‘ I
a ‘ A A l
g 30 - A “A I
(I)
(D
2 20 -
D.
E
o 10 "i

0 l T I I

0 0.03 0.06 0.09 0.12 0.15
Porosity (P)

 

 

 

Fig. 5.9 - Comparison of the predictions of the mechanistic model for the 90-day
compressive strength of concrete containing 50% ﬂy ash versus experimental results

207

SUMMARY AND CONCLUSIONS

Theoretical models, based on structure-property relationships, were developed
sequentially for prediction of the mechanical properties of hydrated cement paste, mortar
and concrete. Experimental results and empirical models were used to verify the
mechanistic models. The models were used to predict the effects of high-volume

replacement of Portland cement with coal ﬂy ash on mechanical properties of concrete.

For the purpose of mechanistic modeling, hydrated cement paste was treated as a multi-
phase material with calcium silicate hydrate C-S-H) particles acting as the binder (glue)
rendering cementing effects. The model starts with simulation of the structure of C-S-H
at nano-scale. The nano-scale C-S-H particles offer effective adhesive qualities capable
of developing physical bonds (based on van der Waals interactions) between themselves
and also with other hydration products as well as aggregates. This adhesive force
provided the fundamental basis for the development of the mechanistic models. The
intrinsic elastic modulus was modeled based on the energy buildup associated with bond
deformations. The energy required to break van der Waals bonds was accounted for in
development of the intrinsic fracture toughness model of hydrated cement paste; the
energy associated with pullout of calcium hydroxide (CH) crystals was also considered in
development of the fracture toughness model of hydrated cement paste. The phononic

frictional pullout of CH crystals from the C-S-H matrix was found to be the major

208

contributor to the intrinsic fracture toughness of hydrated cement paste. The elastic
modulus and fracture toughness models of hydrated cement paste were developed based
on the above intrinsic models by introducing the effects of capillary pores and
microcracks. The strength model of hydrated cement paste was developed using fracture
mechanics concepts considering the effects of restrained shrinkage microcracks.
Experimental data and empirical models were used to verify the mechanistic models at

different stages of development.

The mechanistic models of cement mortar were developed using the corresponding
models of hydrated cement paste, accounting for the effects of ﬁne aggregates and their
interfacial transition zone. The elastic modulus of cement mortar was modeled by
simulating mortar as a three-phase solid comprising the hydrated cement paste, ﬁne
aggregates, and the interfacial transition zone. While the high-modulus ﬁne aggregates
positively affected the elastic modulus of mortar, the corresponding effects of the
transition zone were negative. The ﬁacture toughness model of cement mortar was
developed by introducing the energy dissipation associated with pullout of ﬁne
aggregates to the fracture toughness model of hydrated cement paste, and also by
considering the positive effects of multiple crack formation in cement mortar in the
presence of aggregates. The tensile strength model of cement mortar was developed using
ﬁacture mechanics concepts, with due consideration given to the effect of ﬁne aggregates
on the restrained shrinkage microcrack size in mortar. Experimental results and empirical
models were used to verify the mechanistic models developed for the mechanical

properties of cement mortar.

209

The mechanistic models of concrete were developed based on the corresponding models
of mortar by considering the effects of coarse aggregates and their interfacial transition
zone. Concrete was analyzed as a three-phase material comprising cement mortar, coarse
aggregates, and the interfacial transition zone. The presence of the interfacial zone
negatively affected the elastic modulus and strength of concrete. The relatively high-
modulus coarse aggregates, on the other hand, made positive contributions to the elastic
modulus of concrete. The fracture toughness model of concrete was developed by
accounting for the energy dissipation associated with the pullout of coarse aggregates
with rough surfaces from cement mortar, and also the crack diverging effects of coarse
aggregates. The tensile strength model of concrete was developed based on fracture
mechanics principles, using procedures similar to those used for mortar, considering the
effect of coarse aggregates on restrained shrinkage microcrack size in concrete. A
compressive strength model was also developed for concrete considering the tensile
stress development in concrete under compression and also the shear ﬁiction effects
associated with the peculiar failure modes of concrete under compression. Experimental
results and empirical models were used to verify the mechanistic models developed for

various aspects of the mechanical performance of concrete.

The mechanistic models developed for concrete were reﬁned for application to high-
volume ﬂy ash concrete with 50% replacement of Portland cement with coal ﬂy ash.
Given the slow rate of hydration of high-volume ﬂy ash concrete, the models were

developed as a function of time. The effect of ﬂy ash on conversion of CH to C-S-H and

210

on reﬁnement of the capillary pore size provided a basis to develop the mechanistic
models of hi gh-volume ﬂy ash concrete. Further, the effects of unhydrated cement grains
and unreacted ﬂy ash particles were considered in development of time-dependent
mechanistic models. The predictions of mechanistic models were compared with the

limited test data available for hi gh-‘volume ﬂy ash concrete.

The following conclusions can be drawn from the research conducted on mechanistic

modeling of hydrated cement paste, mortar and concrete:

1. The physical interatomic bonds between calcium silicate hydrate (C-S-H)
globules together with the phononic frictional pullout of calcium hydroxide (CH)
crystals govern the intrinsic mechanical properties of hydrated cement paste. The
predicted intrinsic strength, elastic modulus and fracture toughness of the compact
hydrated cement paste were in conformance with the corresponding

experimentally measured values.

2. The introduction of defects (capillary pores and microcracks) signiﬁcantly alters
the engineering properties of hydrated cement paste when compared with the
corresponding intrinsic properties. The geometry and volume fraction of capillary
pores and the size of microcracks are important factors governing the mechanical

performance of hydrated cement paste and also cement mortar and concrete.

211

3. The fracture toughness of hydrated cement paste receives contributions from
debonding of the C-S-H globules and ﬁictional pullout of CH crystals, with the
latter mechanism playing the dominant role. The fracture toughness of cement
mortar and concrete exceed that of hydrated cement paste due to the energy
dissipation associated with the ﬁictional pullout of ﬁne and coarse aggregates as
well as the crack arresting and diverting effects of aggregates. Predictions of the

ﬁacture toughness models compared well with experimental results.

4. The elastic modulus model of hydrated cement paste was developed to account
for the effect of the elliptical geometry of capillary pores on local stress rises that
undermine the stiffness of the hydrated cement paste. Predictions of the elastic

modulus model compared well with experimental results.

5. The tensile strength model of hydrated cement paste was developed based on
fracture principles using the elastic modulus and ﬁacture toughness models, and
compared favorably with experimental results when considering the presence of

intrinsic restrained shrinkage microcracks.

6. The energy absorption capacity of concrete in compression beneﬁts from the
wedging action of conical cement mortar attached to the coarse aggregates in the
direction of loading. The corresponding energy dissipation is due to the shear

friction effects on surfaces of these conical wedges.

212

7.

10.

The elastic modulus model of cement mortar was developed based on the
corresponding model of hydrated cement paste and considering the effects of the
elastic modulus of ﬁne aggregates as well as the negative effects of the interfacial
transition zone. The predictions of the model were evaluated by comparing them

with experimental results.

The ﬁ'acture toughness models of cement mortar and concrete reﬂect the
signiﬁcance of the contributions made by ﬁne and coarse aggregates through
frictional pullout, crack arresting and diverting effects, and its predictions

compare favorably with experimental results.

The strength models of cement mortar and concrete are based on the
corresponding elastic modulus and fracture toughness models, considering the
effects of aggregates on restrained shrinkage microcracking of concrete. The

predictions of these models occur within the range of experimental results.

In high-volume ﬂy ash concrete, the slow pace of pozzolanic reactions necessitate
development of time-dependent models. Pozzolanic reactions consume calcium
hydroxide to generate the calcium silicate hydrate binder and thus reﬁne the
capillary pore system of hydrated cement paste. Given the slow pace of
pozzolanic reactions, one should also account for the effects of unhdrated cement
grains and ﬂy ash particles on the mechanical performance of high-volume ﬂy ash

paste, mortar and concrete. The theoretical models developed for normal cement,

213

11.

mortar and concrete were reﬁned to account for the above time-dependent effects
of ﬂy ash (when 50% of cement is replaced with ﬂy ash). The models indicated
that high-volume ﬂy ash concrete can surpass normal concrete in terms of elastic
modulus, and tensile and compressive strengths over longer time periods; the
ﬁacture toughness of high-volume ﬂy ash concrete is comparable to that of
normal concrete. At the age of 28 days normal concrete still offers better elastic
modulus and tensile and compressive strengths, and comparable fracture
toughness when compared with high-volume ﬂy ash concrete; the advantages of

high-volume ﬂy ash concrete are apparent at 90 (or more) days of age.

The mechanistic models developed in this research provide insight into the
mechanical performance and failure mechanisms of cement-based materials. The
fundamental nature of these models makes them of value in guiding development
efforts toward further improvement of the mechanical performance of cement-
based materials and/or use of new ingredients in cementitious mixtures. The
favorable comparisons of the predictions of these models with experimental
results suggest that the complex structure of cement-based materials should not be
viewed as an unsurmountable obstacle against development of fundamental
models and structure-property relationships for concrete. The mechanistic models
developed in this project provide a basis for development of more reﬁned

structure-property relationships for hydrated cement paste, mortar and concrete.

214

 

12. The models developed in this research express concrete properties in terms of
interatomic interactions and the concrete structure at nano-, micro- and macro-
scales. These models highlight the key aspects of concrete structure which
critically inﬂuence its mechanical performance, and reveal the sensitivity of
various mechanical attributes of concrete to its structure at different scales. The
models can guide new developments in concrete technology aimed at achieving

enhanced mechanical properties through reﬁnement of concrete structure.

Future research needs:

 

13. The research reported herein has highlighted some key aspects of the structure of
cement-based materials which govern their mechanical performance. These
structural attributes, however, are largely unexplored. Development of
fundamental models for cement-based material thus requires making assumptions
on some basic aspects of their structure. The models developed in this research
could be further improved (and veriﬁed) through efforts devoted to better
understanding of the structure of calcium silicate hydrate at nano-scale, and
calcium silicate hydrate, capillary pores and microcracks (at micro-scale). More
detailed information is also needed on the structure, composition and failure
mechanisms of the interface zone. Finally, the interaction of calcium hydroxide

crystals with cracks is a topic which requires further investigation.

14. Certain aspects of the models developed in this research also require

improvement. In particular, the complex interactions of aggregates with the crack

215

15.

16.

propagation process have been considered in this research in a simpliﬁed way.
More reﬁned modeling procedures are needed to capture the effects of aggregates

on the tortuosity of cracks and also on multiple cracking.

The models developed in this research express the mechanical properties of
cementitious materials in terms of capillary porosity. These models could be
developed into more practical models which predict properties in terms of mix
proportions and curing conditions/time. For this purpose, relationships are needed
for predicting capillary porosity in terms of the mix proportions and curing
conditions/time. The models can be even more beneﬁcial towards new
developments in the ﬁeld if predictive models could be developed to predict the
effects of (chemical, mineral and polymeric) admixture on the structure of
cementitious materials (e. g., the nano-strueture of calcium silicate hydrate or the
micro-structure of capillary pores). Such predictive models would complement
the models developed in this research to predict the effects of various admixtures

on the mechanical properties of cementitious materials.

Finally, efforts to further validate the developed models require availability of
experimental results on the mechanical properties and structure (at least capillary
porosity) of cementitious materials. The available test results on mechanical
properties generally do not accompany measurements of capillary porosity (or

other structural aspects).

216

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