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Full text of "Strength and Durability of Concrete: Effects of Cement Paste-Aggregate Interfaces, Part 1 - Theoretical Study on Influence of Interfacial Transition Zone on Properties of Concrete Materials : Final Report"

Joint 

Transportation 

Research 

Program 



FHWA/IN/JTRP-97/4- o \ 



Final Report 

STRENGTH AND DURABILITY OF CONCRETE: 
EFFECTS OF CEMENT PASTE-AGGREGATE 
INTERFACES 

PARTI 

THEORETICAL STUDY ON INFLUENCE OF 
INTERFACIAL TRANSITION ZONE ON 
PROPERTIES OF CONCRETE MATERIALS 



Yiguo Zhang 
Wai-Fah Chen 



August 1998 



Indiana 

Department 

of Transportation 



Purdue 
University 



FINAL REPORT 

FHWA/IN/JHRP-97/4 

STRENGTH AND DURABILITY OF CONCRETE: EFFECTS OF 
CEMENT PASTE-AGGREGATE INTERFACES 

PARTI : 

THEORETICAL STUDY ON INFLUENCE OF INTERFACIAL 

TRANSITION ZONE ON PROPERTIES OF CONCRETE MATERIALS 



by 

Yiguo Zhang 

Research Assistant 

and 

Wai-Fah Chen 

Research Engineer 

Purdue University 
School of Civil Engineering 

Joint Transportation Research Program 

Project No.: C-36-37EE 
File No.: 5-8-31 

Prepared in Cooperation with the 

Indiana Department of Transportation and 

the U.S. Department of Transportation 

Federal Highway Administration 



The contents of this report reflect the views of the authors who are responsible for the facts and 
the accuracy of the data presented herein. The contents do not necessarily reflect the official 
views of or the Federal Highway Administration and the Indiana Department of Transportation. 
This report does not constitute a standard, a specification, or a regulation. 

Purdue University 
West Lafayette, IN 47907 

August 1998 



Digitized by the Internet Archive 

in 2011 with funding from 

LYRASIS members and Sloan Foundation; Indiana Department of Transportation 



http://www.archive.org/details/strengthdurabiliOOzhan 



11 



TECHNICAL REPORT STANDARD TITLE PAGE 



1. Report No. 

FHWA/IN/JTRP-97/4 



2. Government Accession No. 



3. Recipient's Catalog No. 



4. Title and Subtitle 

Strength and Durability of Concrete: Effects of Cement Paste-Aggregate Interfaces 

Parti : Theoretical Study on Influence of Interfacial Transition Zone on Properties of 
Concrete Materials 

Part II : Significance of Transition Zones on Phyiscal and Mechanical Properties of Portland 
Cement Mortar. 



5. Report Date 

August, 1998 



6. Performing Organization Code 



7. Authors) 

Part I: Yiguo Zhang and Wai-Fah Chen 

Part II: Turng-Fan F. Lee and Menashi D. Cohen 



8. Performing Organization Report No. 

FHWA/IN/JTRP-97/4 



9. Performing Organization Name and Address 

Joint Transportation Research Program 
12S4 Civil Engineering Building 
Purdue University 
West Lafayette, Indiana 47907-1284 



10. Work Unit No. 



11. Contract or Grant No. 

HPR-2071 



12. Sponsoring Agency Name and Address 

Indiana Department of Transportation 
State Office Building 
100 North Senate Avenue 
Indianapolis, IN 46204 



13. Type or Report and Period Covered 

Final Report 



14. Sponsoring Agency Code 



15. Supplementary Notes 

Prepared in cooperation with the Indiana Department of Transportation and Federal Highway Administration. 



16. Abstract 

This research was based on a two-part basic research investigation studying the effects of cement paste-aggregate interfaces (or interfacial 
transition zones-ITZ) on strength and durability of concrete. Part I dealt with the theoretical study and Part II dealt with the experimental. 

Part I. the theoretical part, illustrates the effect of ITZ on the concrete properties by assuming its elastic moduli to be varied continuously in the 
region. A four-phase composite model is employed and three functions are chosen to model the moduli variation in the ITZ. A theoretical 
solution for an n-layered spherical inclusion model is used to estimate the overall effective moduli of the modified four-phase model. The 
influence of material and geometric characteristics of the TTZ, as well as that of the aggregate on the overall effective moduli is investigated. 
The effects of three different moduli variations in ITZ on the overall moduli are compared. Their potential application is discussed. Finally, by 
comparing the prediction of the proposed models to a set of data on mortar, it is found that the elastic modulus at the interface is about 20-70% 
lower than that in the bulk paste for port] and cement mortar, and 10-40% lower for silica fume mortar. 

Part II the experimental part, illustrates the relationship between the ITZ microstructure and the mechanical properties of the concrete. The 
mechanical properties studied included the dynamic modulus of elasticity, dynamic shear modulus, logarithmic decrement of damping, flexural 
tensile strength, and compressive strength. In addition, the effects of changing the water-tc-cementitious material ratio by mass, aggregate type, 
volume fraction of aggregate, and silica fume substitution, on these properties were investigated. A criterion based on water quantity and the 
specific surface area of aggregate by mass in a mixture was developed to eliminate biased date from the analysis process. This criterion was 
used to detect mixing and compaction problems that may have resulted in erroneous values of mechanical properties of specimen. In order to 
realize the compaction condition of the fresh mixture, an index of compaction (called gross porosity) was introduced. The three-phase model of 
Hashin-Shtrikman bounds was employed, tested, and validated with the experimental data from this research. A modification of this model 
linked the theory of Hashin-Shtrikman bounds to the results of this research on dynamic moduli of the transition zone. A form of optimal water 
content is recommended. This optimal water content may be used for a mixture to gain its possibly highest moduli, strengths and density. 
Thus, the rule of the optimal water content may potentially be applied to optimize the mixture design for conventional and high-strength 
concrete with consideration of LTZ. 



17. KeyWords 

Interface transition zone (TTZ), four-phase composite model, elasticity, 
effective moduli, elastic moduli, dynamic modulus, shear modulus, 
logarithmic decrement, damping, flexural tensile strength, compressive 
strength, porosity, microstructure, Hashin-Shtrikman model. 



18. Distribution Statement 



No restrictions. This document is available to the public through the 
National Technical Information Service, Springfield, VA 22161 



19. Security Classif. (of this report) 

Unclassified 



20. Security Classif (of this page) 

Unclassified 



21. No. of Pages 

Parti - 43 
Part H- 261 



22. Price 



Form DOT F 1700.7 (8-69) 



Ill 
TABLE OF CONTENTS 

LIST OF TABLES 

LIST OF FIGURES 

IMPLEMENTATION REPORT 

1 . Introduction 1 

1 . 1 Concrete as a three-phase composite material 1 

1 .2 Microstructure and micromechanical properties of ITZ 2 

2. Effective moduli of the modified four-phase model 4 

2. 1 Modified four-phase model with variation of interface moduli 4 

2.2 Application of n-layered solution to present model 6 

3. Numerical results 8 

3.1 Sensitivity of overall moduli to different types of function 8 

3.2 The effects of the interfacial transition zone 9 

3.2. 1 The influence of local damage parameter D 10 

3.2.2 The influence of the interface volume fraction 11 

3.3 The effects of aggregate type 12 

3.3. 1 The effect of aggregate concentration 12 

3.3.2 The effects of aggregate stiffness 13 

3.4 Application to the data of Zimmerman et al. (1986) 14 

3.5 Application to the data of Cohen et al. (1995) 16 

4. Conclusions and discussions 18 

Acknowledgements 19 

References 18 



IV 

LIST OF TABLES 

Table Page 

1 The effective Young's modulus with different types of n 2 -functions 9 

2 The effective Young's modulus with various D values 10 

3 The effective Young's modulus with various volume fraction v 2 11 

4 The effective Young's modulus with different aggregate concentration v, 13 

5 The effective Young's modulus with various aggregate stiffness E, 14 

6 Material parameters of mortar specimens tested by Zimmerman et al. (1986) 15 

7 The estimated local damage parameter D based on Zimmerman et al. tests (1986) .... 16 

8 The estimated local damage parameter D based on Cohen et al. tests (1995) 17 



LIST OF FIGURES 

Figure Page 

1 Modified four-phase composite model 23 

2 n-layered spherical inclusion model (Herve and Zaoui, 1993) 24 

3 Influence of n 2 25 

4 Effects of local damage factor D 26 

5 Effects of the volume fraction v 2 27 

6 Effects of the aggregate concentration v, 28 

7 Effect of the aggregate stiffness E, 29 

8 Comparison with the test results by Zimmerman et al. (1986) 31 

9 Comparison of different models 37 

10 Comparison with the test results by Cohen et al. (1995) 38 



1. Introduction 

1.1 Concrete as a three-phase composite material 

It has been recognized for many years that in cement composites (mortar, concrete, etc.) the 
microstructure of paste in the vicinity of an inclusion (sand particle, coarse aggregate, etc.) is 
significantly different from that of bulk cement paste. However, the interfacial effect has not 
received special attention until recent study on the role of silica fume (SF) in the 
high-performance concrete. Using scanning electron microscopy (SEM), Bentur and Cohen 
(1987) demonstrated that in mortar the paste located at 50 urn and less from the sand particle 
surface is a relatively porous region containing large calcium hydroxide crystals which have little 
cementitious properties, forming the weakest link in mortar. This particular region is termed as 
the interfacial transition zone (ITZ). While, with 15% condensed silica fume added, the ITZ has 
a homogeneous and dense microstructure, similar to that of the bulk paste. In the work of Cohen, 
Goldman and Chen (1993), an experimental program was successfully developed to verify the 
ability of silica fume to modify the microstructure of the interfacial transition zone (i.e. transition 
zone modification), and thus the significant influence of the interfacial transition zone on the 
overall behavior of mortar is exposed. These experimental events require that concrete materials 
should be considered as a three phase composite, consisting of inclusion, interfacial transition 
zone and bulk cement paste. 

As for the inclusion problem, there is a considerable literature available in the field of 
composite mechanics. Hashin (1962) first proposed a two-phase concentric-spheres model and 
derived the bounds and expressions for its effective elastic moduli by an approximate method 
based on the variational theorems of the theory of elasticity. A more general bounds approach 
applicable to any macroscopically isotropic two-phase composite material was derived by Hashin 
and Shtrikman (1963). To calculate the effective properties, Christinsen and Lo (1979) centered 
their interest around macroscopically homogeneous, isotropic or transversely isotropic two phase 
material, and proposed a model referred to as the three-phase model. This model consists of the 



single composite sphere embedded in the infinite medium of unknown effective properties. A 
generalized self consistent method was formalized to derive the exact expression for the effective 
shear modulus of this model. Herve and Zaoui (1993) extended Christinsen-Lo's work, and 
obtained a general analytical solution for the effective bulk and shear modulus for an n-layered 
spherical inclusion model. This n-layer model will be employed in the present work and more 
details will be given later. 

Zhao and Chen (1996a, 1998a) proposed a dual-layer inclusion model to model concrete 
material as a three-phase composite material, and derived a close-form solution for the stress and 
displacement fields of this model. The influence of the ITZ was exclusively investigated (Zhao 
and Chen, 1996b). Then, the effective elastic moduli of this model was obtained (Zhao and 
Chen, 198b,c), and its practical application and the significance of the ITZ was discussed by 
Chen and Zhao (1995). Meanwhile, following Christensen and Lo's work, Ramesh, Sotelino and 
Chen (1995, 1996a,b) proposed a four-phase model for a three-phase composite, and derived the 
effective moduli for the four-phase model. Application of this model is also made to estimate the 
overall moduli of concrete or mortar. However, the ITZ in all these models is assumed to be 
isotropic and homogeneous with a lower elastic moduli than that of bulk cement paste. Thus 
there is. a jump in material properties between the ITZ and the bulk cement paste. 

1.2 Microstructure and micromechanical properties of ITZ 

According to the work of Scrivener and Gartner (1988), there is a porosity gradient in the 
transition zone, with increasing porosity as one approaches the aggregate particle. When the 
computerized imagine analysis was used to quantify the porosity gradient in the transition zone, 
it shows that there is no clear demarcation between the ITZ and the bulk cement paste. 
Therefore, the material properties are not expected to have a jump between these two phases. 

Some experimental tests indicate that the Young's modulus E is related to the porosity r\ 
by E-ClvC , where a and C are material constants. Hence, it is believed that the mechanical 



properties in the ITZ are not uniform, but vary gradually as a function of distance from the 
inclusion, approaching continuously to that of the bulk cement paste. 

To this end, Lutz and Monteiro (1995) propose a spherical model considering the 
variation of interface moduli. They assume the elastic moduli in the ITZ vary with radius 
according to a power law. The primary reason for such an assumption is because a closed-form 
expression for the overall effective bulk modulus of this model has been found by Lutz and 
Zimmerman (1996). Using this model, the predicted overall bulk modulus is compared to the 
data of Zimmerman et al. (1986), where mortar specimens were measured to obtain the effective 
bulk modulus. They came to the conclusion that the modulus in the ITZ is about 15-50% lower 
than that in the bulk cement paste. This conclusion is somewhat closer to the prediction of 
Cohen, Lee and Goldman (1995) for silica fume (SF) mortar, where the average elastic moduli is 
indicated about 12-52% less than that of SF paste, but does not match very well with that for 
Portland cement (PC) mortar, where 26-85% is indicated. This fact gives us an impression that 
the power law model, which by itself is a very localized damage model, can model the interface 
in SF mortar quite well, but may be improper to model the interface in PC mortar. The objective 
of this work is to propose several types of variation for the elastic moduli in the ITZ. Their 
effects on the overall moduli are investigated and the behavior of the proposed models is 
discussed for their applications to concrete materials. 

In this work, the four-phase model proposed by Ramesh et al. (1995, 1996a,b) is modified 
by considering the variation of the elastic moduli through the interface layer. Three types of 
function (linear, quadratic and power law) are assumed to model the elastic moduli in the ITZ. A 
general analytical solution for an ^-layered inclusion model obtained by Herve and Zaoui (1993) 
is utilized to predict the overall effective moduli of the modified four-phase model. Therefore, 
the influence of different types of variation for the interface moduli as well as other parameters of 
the proposed model on the overall behavior can be revealed. The inverse application of this 
procedure can be used to estimate the elastic moduli at the interface if the overall moduli are 
given. This serves as a quantitative and non-destructive means of estimating the properties in the 



rrz. 



2. Effective moduli of the modified four-phase model 

2.1 Modified four-phase model with variation of interface moduli 

We propose a micromechanical model for concrete material based on the four-phase model by 
Ramesh et al. (1995, 1996a,b) as shown in Fig. 1. The aggregate particle is assumed to be 
spherical with radius a. Outside the aggregate, there is an interfacial transition zone from inner 
layer with radius a to outer layer with radius b. Beyond the ITZ, there is the bulk cement paste 
which limited in a sphere with radius c. This individual composite sphere is then embedded in 
an equivalent homogeneous medium. The material property for each constituent phase is also 
shown in Fig. 1 . The aggregate and bulk cement paste are taken to be isotropic homogeneous 
with constant elastic moduli £, {k { ,\n v \ { ) and £ 3 (& 3 ,p 3 ,v 3 ) respectively. The interfacial 
transition zone is considered as a radially-inhomogeneous region where the elastic modulus 
E 2 (k 2 ,\i 2 ,v 2 ) is taken as a function of the distance t from the inclusion, with E 2Q (k 2Q ,\x 2Q ,\ 2( ^ as 
the modulus at t = 0. In the present work, three types of function are proposed, i.e. linear, 
quadratic and power law. They are described below respectively. 



1). Linear model: 



E 2 (t)=E 20 +kt (i) 



Let E 2 (b-a)=E 3 , we have 



k-^ (2) 

b-a 



The linear function is the simplest zero-order continuous model, where 
E 2 (t) does not smoothly approach E v i.e. E 2 (b -a) * E^(b -a) =0. 



2). Quadratic model: 

E 2 {f)=E 20 +k x t+k 2 t 2 (3) 



Let E 2 (b-a)=E^ and E 2 (b-a) =0, we have 

, _ 2 ( £ 3~ £ 2o) , __ £ 3 - £ 20 

fc-fl (6-a) 2 



The quadratic function is the simplest first-order continuous model, where 
E 2 (t) approaches £ 3 smoothly. 

3). Power law model 

Following the work of Lutz and Monteiro (1995), we assume the moduli 
vary according to 

v a+t p 



E 2 (t)=E 3 -(E,-E 20 ) 



\ a J 



in which E 3 -E 2 (b-a) = \%(E ? .-E 20 ) , we have 

ln(100) 



P : 



In 



V 



(5) 



(6) 



E 2 {t) in this model can approach continuously and smoothly in an 
approximate manner, i.e. E 2 (b-a)->E 3 , E 2 (b-a)~0. Generally, P is around 8 for 
concrete materials and this model is actually representing the most localized 
damage zone among the three models and referred to as Power law model in the 
present work. 

Herein, the unknown parameter for the three models reduces to the elastic modulus E 20 . 



6 

Due to the incomplete process of hydration especially around the aggregate, E 2Q is expected to lie 
between and E v We define £' 20 =(1 -D)E Ji , where D is a local damage parameter. When D=Q, 
we have E 20 -E^ , there is no damage in the ITZ or the model can be taken here as a two-phase 
composite model. The three proposed models will converge to the same one. When D = 1, we 
have E 20 = 0, there is a complete damage around the aggregate. We will show later that there is a 
so-called "hole-effect" for this case. 

2.2 Application of n -layered solution to present model 

Herve and Zaoui (1993) obtained a general solution for the effective moduli of an n-layered 
spherical inclusion model, which can be utilized to approximate the proposed models stepwisely. 
The n-layered spherical inclusion model is shown in Fig. 2. Each phase is assumed to be 
homogeneous and isotropic with elastic constants (p i ,v i ,k i ) representing shear modulus, 
Poisson's ratio and bulk modulus respectively for phase /, which lies within the shell limited by 
spheres with the radii R._ x and R.(l€[l, n+\], R ~0, R n+l ~°°). They derived the elastic strain 
and stress fields subjected to hydrostatic pressure and pure shear conditions at infinity. Then, 
using the average strain method, which has been shown to be equivalent to Christinsen-Lo's 
energy condition, they obtained the effective bulk modulus k for the equivalent homogeneous 
medium as 

k 3^. 3 g,r"-4 M ,e 2 'r" 



where Qfj 1 (/ = 1 ,2; j = 1 ,2) are the elements of a 2x2 matrix Q (n !) given by 

n-\ 



g(«-i) = riiv< /) (8) 



The effective shear modulus can be expressed by the following second-order equation: 



N U) 



3k 



y + i +4 H; +1 



3k j +4 ^ —^j+rVj) 



r; 



3(k j+r k j )R^ 3/:, 1+ 4u ; 



(9) 



( \ 2 



+ B 



V 



+ c=o 



(10) 



where A, B and C are constants in terms of the elastic moduli as well as the dimension 
parameters of each phase. 



This general solution for (w-t-l)-phase model can be reduced to the classical two-phase 
model of Hashin (1962) and three-phase model of Christinsen and Lo (1979) or the recent 
four-phase model of Ramesh et al. (1995, 1996a,b) as well. With the bulk modulus and shear 
modulus known, one can easily obtain the commonly-used engineering parameters E and v as 
follow: 



, 9k\x 2>k-2\x 

9&+u 6fc+2u 



(11) 



When the (n+l)-phase composite model is incorporated into the present models, we take 
phase 1 as the aggregate, phase n as the bulk cement, and leave phase 2 to phase (n-1) with 
stepwisely increased constant moduli to model the interfacial transition zone. The overall 
effective moduli of the present model are then predicated from the solution of the ^-layered 
inclusion model as that of the equivalent homogeneous medium — phase (n+1). Theoretically, 
the smooth variation of moduli can be modeled to any required accuracy by simply increasing n. 



8 

We should point out here that the proposed models as well as the n-layered inclusion 
model require constant ratios of alb and blc for each individual composite sphere, independent of 
its absolute size. This assumption requires particle size down to infinitesimal, while still having 
a volume filling configuration. These models would be expected to predict reasonable results for 
actual systems that have a rather fine gradation of sizes, but it should not be expected to provide 
reasonable results for systems containing single size particles at high concentrations (Christinsen, 
1979). In other words, no overlap of the associated cement paste of any single aggregate is 
considered, i.e. the interaction effects between particles are not accounted for in the present 
models. It is important to keep this limitation in mind in estimating the possible error when 
applying these models to experimental data. 



3. Numerical results 

In this section, we shall provide some numerical results of the application of the n-layer solution 
to the modified four-phase models. Since the present work is limited to the stress level within 
elastic stage, for the sake of simplicity, we take c- 1 . The relationship between the volume 
fraction of each constituent component and the dimension size of each phase is the same as that 
of the four-phase model of Ramesh et al. (1995, 1996a,b). If we define v, , v 2 and v 3 as the 
volume fraction of the aggregate, interface and bulk cement paste respectively, then we have 

3, 3, 

a = Jv l ,b = J v,+v 2 and v,+v 2 +v 3 = l . 

In the following examples, we first investigate the sensitivity of the overall moduli to n 
for the three models, then the influence of material and geometric characteristics of the inclusion 
as well as that of the ITZ on the overall properties is studied. Finally, we apply the present 
models to the data of Zimmerman et al. (1986) and that of Cohen et al. (1995). 



3.1 Sensitivity of overall moduli to different types of function 

The three types of function have been briefly described in the preceding section. To investigate 
the sensitivity of the overall moduli predicted by these models to the number of n 2 = (n-2), we 
choose the most critical damage case, i.e. D = 1, where the sensitivity can be greatly amplified. 
Taking volume fraction v, = 0.5, v 2 = 0.3 and Young's modulus £, = 5.0, E 3 = 1.0 with the 
Poisson's ratio v, = 0.3 (/ = 1, 2, 3) as the known parameters, we can obtain the effective Young's 
modulus as listed in Table 1 and shown in Fig. 3. 

It is clear from Fig.3 that as n 2 increases, the three models predict smaller effective 
moduli. There is an obvious drop for the effective moduli when n 2 increases from 1 to 10, then 
turns to a rather slow decrease when n 2 increases beyond 10. The power law model exhibits a 
less sensitivity to n 2 , with the predicated overall Young's modulus only 22% less when n 2 
increases from 1 to 100. However, the linear model is very sensitive to n 2 , with the overall 
modulus up to 92% less. The quadratic model lies somewhere in between with a moderate 
sensitivity. 

Table 1 The effective Young's modulus with different types of n 2 functions 



«2 


1 


2 


3 


4 


5 


10 


15 


20 


50 


100 


Linear 


1.474 


1.273 


1.180 


1.122 


1.083 


0.979 


0.930 


0.899 


0.817 


0.768 


Quadratic 


1.804 


1.626 


1.539 


1.483 


1.444 


1.336 


1.282 


1.248 


1.153 


1.094 


Power law 


2.024 


1.969 


1.932 


1.906 


1.886 


1.826 


1.793 


1.771 


1.705 


1.660 



It is also important to notice that power law model, which can be considered as the most 
localized damage model among the threes, always predicts the highest overall Young's modulus, 
and the linear model predicts the lowest value. This fact indicates that the less the damage zone, 
the higher the overall moduli. This observation agrees well with that of Zhao et al. (1994b) and 
Chen et al. (1995). In the following, we will take n 2 =100 unless otherwise indicated. 



10 
3.2 The effects of the interfacial transition zone 

In the proposed models, the interfacial transition zone is controlled by two factors: the local 
damage parameter D and the volume fraction v 2 . In this section, we shall study the influence of 
these two factors on the overall moduli. The other parameters are taken to be the same as those 
in Section 3.1. 

3.2.1 The influence of local damage parameter D 

In this study, we shall investigate the influence of the local damage parameter D on the Young's 
modulus. The predicted effective Young modulus with various D for the three models is listed in 
Table 2 and also plotted in Fig. 4a, where v 2 = 0.3. 

Table 2 The effective Young's modulus with various D values 



D 


Linear 


Quadratic 


Power law 


0.0 


2.047 


2.047 


2.047 


0.1 


2.003 


2.018 


2.035 


0.2 


1.955 


1.985 


2.022 


0.3 


1.901 


1.950 


2.007 


0.4 


1.842 


1.909 


1.990 


0.5 


1.774 


1.863 


1.971 


0.6 


1.694 


1.808 


1.948 


0.7 


1.599 


1.741 


1.920 


0.8 


1.477 


1.653 


1.883 


0.9 


1.300 


1.521 


1.824 


1.0 


0.768 


1.094 


1.660 



It is observed that as D increases from to 0.9, the overall modulus decreases slowly, 



11 

although the power law model always predicts the highest value. Then when D increases beyond 
0.9 and up to 1 .0, there is an obvious drop of the overall modulus, especially for the linear model. 
The less sensitivity of the power law model to the local damage factor D is due to the fact that 
the model itself has a extremely localized damage zone. This observation indicates that if the 
interface is extremely localized, its influence on the overall modulus is negligible. 

Figure 4b shows the results of the four-phase model, which can be reproduced by the 
present models with n 2 = 1 (thus the term "four-phase model" in this paper specially refers to the 
triple-layered inclusion model with the interface considered to be homogeneous). It can be seen 
that there is no obvious drop of the overall modulus even when the local damage parameter D 
approaches its maximum value, as revealed by the present models. This observation implies that 
the four-phase model is not a suitable one for the seriously damaged case, i.e. D > 0.9. 

3.2.2 The influence of the interface volume fraction 

In this study, the effect of the interface thickness or the volume fraction, i.e. v 2 = (P-a 3 )/c 3 , on the 
overall modulus of the model is addressed. The predicated effective Young's modulus is listed in 
Table 3 and also shown in Fig. 5a, where D = 1 . 



Table 3 The effective Young's modulus with various volume fraction v 2 



V 2 


Linear 


Quadratic 


Power law 


0.0 


2.047 


2.047 


2.047 


0.1 


1.219 


1.515 


1.870 


0.2 


0.933 


1.257 


1.749 


0.3 


0.768 


1.094 


1.660 


0.4 


0.654 


0.976 


1.589 


0.5 


0.568 


0.885 


1.532 



12 

It is observed that the overall modulus has a large reduction when the interface volume 
fraction increases from to 0.2, as revealed by the linear and quadratic models, then turns to a 
moderate reduction for further increase of the interface volume fraction. Again, the power law 
model turns out to be less sensitive to the effects of the interface volume fraction due to its 
localized damage zone feature. 

From the result of the four-phase model, plotted in Fig. 5b, it shows that the four-phase 
model cannot detect the obvious drop of the overall moduli if the interface fraction increases 
from to 0.2, and always predicts a much higher value for the overall modulus than that from the 
present models if we take an arithmatic average value for the interface modulus. 

3.3 The effects of aggregate type 

It is well known that the aggregate type has a considerable influence on the properties of concrete 
materials. In this section, we shall investigate the effects of the inclusion concentration v, and 
stiffness £, on the effective moduli of the proposed models. The other parameters are taken to be 
the same as those in Section 3.1. 

3.3.1 The effect of aggregate concentration 

The effect of aggregate concentration, i.e. v, = a 3 /c\ on the overall modulus of the proposed 
models is investigated here. The predicted effective Young's modulus is listed in Table 4 and 
also shown in Fig. 6a, where E [ = 5.0. 



13 



Table 4 The effective Young's modulus with different aggregate concentration v, 



v l 


Linear 


Quadratic 


Power law 


0.0 


0.904 


0.962 


0.999 


0.1 


0.745 


0.861 


1.042 


0.2 


0.716 


0.880 


1.148 


0.3 


0.717 


0.930 


1.286 


0.4 


0.736 


1.002 


1.455 


0.5 


0.768 


1.094 


1.660 


0.6 


0.811 


1.206 


1.906 


0.7 


0.866 


1.341 


2.201 



In the case of D = 1 , there is a totally damaged zone around the aggregate, the increase of 
the aggregate concentration has almost no contribution to the overall moduli as revealed by the 
linear and quadratic models. However, the power law model fails to demonstrate this 
phenomenon clearly. Similarly, the prediction of the four-phase model, as shown in Fig. 6b, 
cannot reveal this special phenomenon. 

3.3.2 The effects of aggregate stiffness 

In this study, we first investigate the effects of aggregate stiffness with D = 0.5. Fig. 7a shows 
the influence of the aggregate stiffness E x increasing from to 10£ 3 . It can be seen that the 
prediction from the three models agree very well for this case. 



However, when we take the critical case, i.e. D = 1 , the difference of these models is 
revealed as shown in Fig. 7b with the results listed in Table 5. For this case there is a complete 
damage around the aggregate, the increase of the aggregate stiffness has no obvious effects on the 
overall moduli except that this increase is made from zero to that of the bulk cement paste E 3 . 
This phenomenon is the so-called "hole-effects" and revealed quite well by the linear and 



14 



quadratic models, where two horizontal lines are predicted. The power law model again fails in 
this special case. 

Table 5 The effective Young's modulus with various aggregate stiffness E, 



£. 


Linear 


Quadratic 


Power law 


0.0 


0.187 


0.222 


0.265 


1.0 


0.585 


0.734 


0.912 


2.0 


0.680 


0.906 


1.226 


3.0 


0.725 


0.998 


1.423 


4.0 


0.751 


1.055 


1.559 


5.0 


0.768 


1.094 


1.660 


6.0 


0.780 


1.122 


1.737 


7.0 


0.789 


1.144 


1.798 


8.0 


0.796 


1.161 


1.848 


9.0 


0.801 


1.174 


1.890 


10.0 


0.806 


1.186 


1.925 



From Fig. 7c where the prediction from the four-phase model is plotted, we can see that 
the four-phase model cannot reveal the hole-effect very well. 

3.4 Application to the data of Zimmerman et al. (1986) 

In the previous sections, we have studied the behavior of the proposed models for predicting the 
overall moduli in terms of that of the constituent components. In this section, we shall apply 
these models to the experimental data reported by Zimmerman et al. (1986). 



In their work, acoustic wave measurements were performed on mortar specimens with 0- 
60% sand concentration. The effective bulk modulus can be found from the measured 



15 

wavespeeds using the relationship established in their work. However, these experimental results 
are found to lie below the Hashin-Shtrikman lower bound, which can be reproduced by the 
proposed models with D = 0. Since the Hashin-Shtrikman bounds are developed for a two-phase 
composite material, such violation can be accepted by considering concrete material to be of a 
three-phase composite material (Nilsen and Monteiro, 1993). 

In order to rationalize the test fact, Lutz and Monteiro (1995) apply the power law model 
to these data, using the closed-form expression obtained by Lutz and Zimmerman (1996) to 
predict the bulk modulus. Their application leads to the conclusion that the modulus at the 
interface is about 15-50% lower than in bulk cement paste (i.e. D = 0.15-0.5). 

The sand grains used in the mortar specimens tested by Zimmerman et al. (1986) had 
radii of about 50 um, according to the SEM photographs. While it is always difficult to 
determine the volume fraction v 2 for ITZ, not to mention that there is no information about the 
interfacial transition zone reported in their work at that time. To be consistent to the assumption 
made in the proposed models, the volume fraction v 2 for the ITZ, in the present models should be 
chosen not exceeding 0.4, since the aggregate concentration v, will approach 0.6. The material 
parameters for the mortar specimens are listed in Table 6 for reference. 

Table 6 Material parameters of mortar specimens tested by Zimmerman et al. (1986) 



Material 


k (GPa) 


u (GPa) 


E (GPa) 


v 


P (kg/m 3 ) 


Sand inclusion 
Cement paste 


44.0 
20.8 


37.0 
11.3 


86.7 
28.7 


0.17 
0.27 


2700 
2120 



In Fig. 8a-l, we present the prediction of the proposed three models for the effective bulk 
modulus as functions of the aggregate concentration v, for various values of D, with v 2 ranging 
from 0. 1 to 0.4. By comparing the experimental results of Zimmerman et al. (1986), we can 
locate the range as well as the best fitting value for the local damage parameter D. They are 



16 



summarized in Table 7, where the best fitting values are indicated in parentheses. 



Table 7 The estimated local damage parameter D based on Zimmerman et al. tests (1986) 


v 2 


Linear 


Quadratic 


Power law 


0.1 


0.3 - 0.9 (0.8) 


0.5 - 0.95 (0.9) 


0.8- 1.0+ (1.0+) 


0.2 


0.2 - 0.75 (0.55) 


0.3-0.8 (0.7) 


0.6- 1.0+ (0.95) 


0.3 


0. 1 - 0.6 (0.4) 


0.2-0.8 (0.6) 


0.5- 1.0 (0.9) 


0.4 


0.1 -0.5(0.35) 


0.2-0.7 (0.5) 


0.4 - 0.95 (0.85) 



From Table 7, we can see that the linear model predicts the smallest value for local 
damage parameter D, the quadratic predicts a moderate value, while the power law model 
predicts the largest value. In other words, a more localized interface layer can bear a more 
serious local damage to remain a same overll moduli, as demonstrated in Fig. 9. 

The obviously different prediction between the present power law model and that in Lutz- 
Monteiro's work might be due to the choice of the interface thickness. In their work, a = 50 um, 
b-a = 40 um. Such a choice must lead to the overlap of the associated cement paste of each 
particle when applied to the case with sand concentration up to 0.6. Since the work of Lutz and 
Zimmerman (1996) is not yet accessible to the authors, we cannot verify if such a choice meets 
the requirement of their model. In the present work, we prefer to keep the consistency in the 
proposed models and leave the error to the difference between the model and the actual case. 

3.5 Application to the data of Cohen et al. (1995) 



In the previous section, the proposed models are applied to predict the effective bulk modulus, In 
this study, we shall apply the models to predict the effective elastic modulus and to compare it 
with the data of Cohen et al. (1995). 



17 

In their work, the average values of the dynamic moduli at the interface are estimated 
using the logarithmic mixture rule. By assuming the interface volume fractions, it indicates that 
the elastic modulus at interface is 26-85% less than that of bulk cement paste for PC mortar, and 
12-52% less for SF mortar. 

The volume fraction for the sand particles is kept to be constant at 37%. The dynamic 
modulus of elasticity is measured for PC mortar (4.2xl0 6 psi), PC paste (2.7xl0 6 psi), SF 
mortar (4.4xl0 6 psi), SF paste (2.5xl0 6 psi) and sand particle ( 14xl0 6 psi). In this study, we 
take the Poisson's ratio v = 0.17 for sand particle, v = 0.27 for PC paste and v = 0.37 for SF paste. 

In Fig. 10, we present the prediction of the proposed three models for the effective elastic 
modulus as functions of the local damage parameter D, with the interface fraction v 2 = 0.54 for 
PC mortar and v 2 = 0.1 for SF mortar. By comparing the experimental results, i.e. 4.2x1 6 psi for 
PC mortar and 4.4xl0 6 psi for SF mortar, we can determine the local damage parameters for the 
three models as listed in Table 8. 

From the prediction of the proposed models, we can conclude that the elastic modulus at 
the interface is 23-69% lower than that of bulk paste for PC mortar, and 12-36% lower for SF 
mortar. This conclusion agrees reasonably well with that of Cohen et al. (1995), in which the 
logarithmic mixture rule is applied to estimate the average elastic modulus at the interface. We 
can also see that silica fume in mortar reduces the interface volume fraction and the extent of 
damage (thus the lower of the local damage parameter). These experimental events can be 
explained here by the proposed models. 



Table 8 The estimated local damage parameter D based on Cohen et al. tests (1995) 




Linear 


Quadratic 


Power law 


PC mortar 
SF mortar 


0.23 
0.12 


0.33 
0.17 


0.69 
0.36 



18 
4. Conclusions and discussions 

In this work, we have proposed three models to represent different variation of the interface 
moduli. Taking the advantage of the available n-layer solution and incorporating it into the 
present models, the behavior of the three models is investigated and compared. The effects of 
the interface revealed by these models can be used to explain some experimental events. Some 
characteristics of this investigation are summarized as follows: 

1 ) The more sensitivity of the linear model to the number n 2 than the power law 
model indicates that when the thickness of the interface layer is appreciable, the 
effects of the variation of the elastic moduli within the ITZ become important, 
otherwise large error may rise. 

2) With the same initial damage parameter D, the power law model always 
predicts the highest overall modulus and the linear model predicts the lowest 
value. On the other hand, if the overall modulus is given, the power law model 
predicts the most serious initial damage at the interface while the linear and 
quadratic models predict a moderate initial damage around the aggregate. 

3) When the damage parameter D increases from to 0.9, the overall modulus 
decreases slowly in all three models. When D increases beyond 0.9 and up to 1.0, 
there is an obvious drop of the overall modulus, especially for the linear model. 
The less sensitivity of the power law model, which by itself a localized damage 
model, to the damage parameter D implies that if the interface layer is extremely 
thin, its influence on the overall modulus is negligible. 

4) When D = 1 , which means a complete damage around the aggregate, the 
increase of aggregate concentration and stiffness has almost no contribution to the 
overall modulus, the so-called hole-effect. This phenomenon is revealed quite 



19 

well by the linear and quadratic models, while the power law model fails in this 
special case. 

5) When comparing the prediction of the proposed model to the experimental data 
by Cohen, Lee and Goldman (1995), we find that the elastic modulus at the 
interface is about 20-70% lower than that in bulk paste for Portland cement, and 
10-40% lower for silica fume cement. This procedure serves as a quantitative and 
non-destructive means of estimating the properties in the interfacial transition 
zone. 

From the present work, we have realized that much more remains to be learned about the 
microstructure of the concrete materials, especially the interfacial transition zone, in order to 
propose a reasonable micromechanical model, which can in turn help to predict the overall 
mechanical properties. 



Acknowledgments 

This research has been supported jointly by the Joint Transportation Research Program at Purdue 
University and the National Science Foundation under the grant number 9202134-CMS. We are 
grateful for the helpful discussion with Prof. Xinghua Zhao when he was a visiting professor at 
Purdue University. 



References 

1 . Bentur, A. and Cohen, M. D. (1987), "Effect of condensed silica fume on the 

microstructure of the interfacial zone in Portland cement mortars", Journal of the 
American Ceramic Society, Vol. 70, No. 10, pp. 738-743. 



20 

2. Chen, W. F. (1994), "Concrete Plasticity: Recent Developments," ASME Reprint No. 
AMR 146, Part of "Mechanics USA 1994", edited by A.S. Kobayash, Applied Mechanics 
Review, Vol. 47, No. 6, Part 2, June, 1994, pp. 586-590. 

3. Chen, W. F. and Zhao, X. H. (1995), "Influence of interface layer on mechanical behavior 
of concrete", Technical Report CE-STR-95-14, School of Civil Engineering, Purdue 
University, West Lafayette, IN. 

4. Christinsen, R. M. (1979), Mechanics of Composite Materials, John Wiley & Sons, New 
York. 

5. Christinsen, R. M. and Lo, K. H. (1979), "Solutions for effective shear properties in three 
phase sphere and cylinder models", Journal of Mechanics and Physics of Solids, Vol. 27, 
No. 4, pp. 315-330. 

6. Cohen, M. D., Goldman, A. and Chen, W. F. (1993), "The role of silica fume in mortar: 
transition zone versus bulk paste modification", Cement and Concrete Research, Vol. 24, 
pp. 95-98. 

7. Cohen, M. D., Lee, T. F. F. and Goldman, A. (1995), "A method for estimating the 
dynamic moduli of cement paste-aggregate interfacial zones in mortar", in Microstructure 
of cement-based systems/bonding and interfaces in cementitious materials, Vol. 370, 
Material Research Symposium Proceedings, 1995, pp. 407-412. 

8. Hashin, Z. (1962), "The elastic moduli of heterogeneous materials", Journal of Applied 
Mechanics, Vol. 29, pp. 143-150. 

9. Hashin, Z. and Shtrikman, S. (1963), "A variational approach to the theory of the elastic 
behavior of multiphase materials", Journal of Mechanics and Physics of Solids, Vol. 11, 



21 
pp. 127-140. 

10. Herve, E. and Zaoui, A. (1993), "n-layered inclusion-based micromechanical modeling", 
Int. J. Engrg. Sci., Vol. 31, No. 1, pp. 1-10. 

1 1 . Lutz, M. P. and Monteiro, P. J. M. (1995), "Effect of the transition zone on the bulk 
modulus of concrete", in Microstructure of cement-based systems/bonding and interfaces 
in cementitious materials, Vol. 370, Material Research Symposium Proceedings, 1995, 
pp. 413-418. 

12. Lutz, M. P. and Zimmerman, R. W. (1996), Private communication. 

13. Nilsen, A. U. and Monteiro, P. J. M., "Concrete: a three phase material", Cement and 
Concrete Research, Vol. 23, 1993, pp. 147-151. 

14. Ramesh. G., Sotelino, E. D. and Chen, W. F. (1995), "Analytical solutions for effective 
elastic moduli of a four phase composite model", Technical Report CE-STR-95-9, School 
of Civil Engineering, Purdue University, West Lafayette, IN. 

15. Ramesh. G., Sotelino, E. D. and Chen, W. F. (1996a), "Effect of interface on elastic 
moduli of concrete materials", Cement and Concrete Research, vol. 26, pp. 61 1-622. 

16. Ramesh, G., Sotelino, E. D. and Chen, W. F. (1996b), "Effect of Transition Zone on the 
Pre-Peak Mechanical Behavior of Mortar, Proceedings of the Materials Engineering 
Conference, ASCE Annual Convention and Exposition, Washington, D.C., November 
11-13. 

17. Scrivener, K. L and Gartner, E. M. (1988), Bonding in Cementitious Composites, edited 
by S. Mindess and S. P. Shah, Material Research Society Proceedings. Vol. 1 14, 



22 
Pittsburgh, PA, 1988, pp. 77-85. 

18. Zhao, X. H. and Chen, W. F. (1996a), "Stress analysis of a sand particle with interface in 
cement paste under uniaxial loading", Short communications, International Journal for 
Numerical and Analytical Methods in Geomechanics, vol. 20, pp. 275-285. 

19. Zhao, X. H. and Chen, W. F. (1996b), "The influence of interface layer on microstructural 
stresses in mortar", International Journal for Numerical and Analytical Methods in 
Geomechanics, vol. 20, pp. 215-228. 

20. Zhao, X. H. and Chen, W. F. (1998a), "Solutions of Multi-Layer Inclusion Problems 
Under Uniform Field," Journal of Engineering Mechanics, Vol. 124, No. 2, February, pp. 
209-216. 

21. Zhao, X. H. and Chen, W. F. (1998b), "Effective elastic moduli of concrete with interface 
layer", Computers and Structures, Vol. 66, Nos.2-3, pp. 275-288. 

22. Zhao, X. H. and Chen, W. F. (1998c), "The effective elastic moduli of concrete and 
composite materials", Journal of composites Engineering, Part B, 29B, pp. 31-40. 

23. Zimmerman, R. W., King, M. S. and Monteiro, P. J. M. (1986), "The elastic moduli of 
mortar as a porous-granular material", Cement and Concrete Research, Vol. 16, pp. 239- 
245. 




(1) Linear 

(2) Quadratic 

(3) Power law 



*■ r 



*■ r 



Figure 1 Modified four-phase composite model 

23 



Phase n-i 




Figure 2 ^-layered spherical inclusion model (Herve and Zaoui, 1993) 



24 




Linear 
Quadratic 
Power law 



Figure 3 Influence of n 2 



25 



c 
c 

v. 

"ex 
c 

c 
> 

g 




Linear 
Quadratic 
Power law 



Local damage factor D 

(a) n 2 = 100 



2.5 



■= 2.0 



| 1.0 - 



0.5 



■TL_— -J* A & it tc 



it it A. 




I 1 1 1 

0.0 0.2 0.4 0.6 0.8 1.0 



Local damage factor D 

(b)w 2 =l 

Figure 4 Effects of local damage factor D 



26 




Volume fraction v 2 
(a) 722=100 



Linear 
Quadratic 
Power law 




Volume fraction v 2 

(b) n 2 = 1 
Figure 5 Effects of the volume fraction v 2 

27 




3.0 



3 


2.5 






C 




r m 




c 




to 


2.0 


ex 




e 




c 


1.5 


« 




> 




CI 


1.0 


t— 




K 






0.5 



Aggregate concentration Vj 

(a) n 2 = 100 



Linear 
Quadratic 
Power law 




Aggregate concentration Vj 

(b) n, = 1 

Figure 6 Effects of the aggregate concentration v t 



28 




4 6 

Aggregate stiffness E { 

(a)Z) = 0.5,/? 2 =100 



Linear 
Quadratic 
Power law 



10 



2.5 




2 4 6 8 

Aggregate stiffness E l 
(b) D = 1.0, n 2 = 100 

Figure 7 EfFect of the aggregate stiffness £, 

29 



10 




-o— Linear 
--— Quadratic 
-a— Power law 



Aggregate stiffness E x 

(c)Z)=1.0, n 2 =\ 
Figure 7 (continued) 



30 





35 


<5 




c_ 




n 






30 


• , . 




<u 




^ 




yT 


25 



a 




0.0 0.1 



t — i — i — i — r 

0.2 0.3 0.4 0.5 0.6 0.7 



-D-D = 0.0 
<- D = 0. 1 
-*-D = 0.2 

D = 0.3 
■=- D = 0.4 

D = 0.5 

D = 0.6 
■=- D = 0.7 
— D = 0.S 
-=^D = 0.9 
-=-D = 1.0 
O Zimmerman et al. 



Aggregate concentration, v, 
(a) Linear model (v 2 = 0.1) 





35 


^— V 




C5 




CU 




U 


30 






r^^ 




I— - 




u 




id 


25 


on 




~j 




3 


20 


<J 




C 




F 




^ 


15 


^ 




CO 






10 




t — i — i — i — i — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 

Aggregate concentration, v, 

(b) Linear model (v 2 = 0.2) 

Figure 8 Comparison with the test results by Zimmerman et al. (1986) 

31 





35 


*Zi 




C 




O 


10 


■s.^ 




!j_ 




t4_ 




OJ 




^ 


25 


c/: 




— 


20 


«-* 




c 




F 




.^ 


lb 



10 




i — i — i — i — r 

0.0 0.1 0.2 0.3 0.4 0.5 

Aggregate concentration, v, 
(c) Linear model (v 2 = 0.3) 



35 



c 




O 


30 






u_ 




u_ 




^ 


25 


en 




3 




s 


20 






o 




E 




-^ 


15 



ca 



10 




T 1 1 1 1 1 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 

Aggregate concentration, v, 

(d) Linear model (v, = 0.4) 

Figure 8 (continued) 



32 



C3 

a. 
O 



T3 
C 

£ 




t — i — i — i — i — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Zimmerman et al. 

Aggregate concentration, v, 
(e) Quadratic model (v 2 = 0. 1) 



C- 
O 






c 

E 




t — i — i — i — i — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 

Aggregate concentration, v, 

(f) Quadratic model (v 2 = 0.2) 

Figure 8 (continued) 



33 



C- 

G 



o 
E 

en 




t — T 1 1 T — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 o Zimmerman « al. 
Aggregate concentration, Vj 
(g) Quadratic model (v 2 = 0.3) 



« 
O 






c 
S 

M 

5 

pa 




t 1 1 1 r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 

Aggregate concentration, v x 

(h) Quadratic model (v, = 0.4) 

Figure 8 (continued) 



34 



c- 
O 



<u 



c 
£ 




i i i r 

0.0 0.1 0.2 0.3 0.4 

Aggregate concentration, v, 
(i) Power law model (v 2 = 0. 1) 



O 



id 

o 
E 



.« 




i — i — i — t — r — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 

Aggregate concentration, v 2 

(j) Power law model (v, = 0.2) 

Figure 8 (continued) 



35 






U 



c 
£ 



u 




20 -f 



i — i — i — i — i — r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 o Zimmerman et ai. 

Aggregate concentration, w x 
(k) Power law model (v 2 = 0.3) 



C3 

0- 
O 



<u 



3 

3 

c 
£ 



C2 




t 1 1 1 r 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 

Aggregate concentration, v t 

(1) Power law model (v 2 = 0.4) 

Figure 8 (continued) 

36 



1.0 



m 

K 



63 



0.5 - 




Linear 
Quadratic 
Power law 



0.0 -[- 



i i i i 



i i i r 



0.0 



0.5 



1.0 



t/(b-a) 



Figure 9 Comparison of different models 



37 



V3 

c 



'ZJ 

a 

~v 

«— 
c 



c 
E 

u 

£ 

es 
s 
>> 

Q 




Linear 
Quadratic 
Power law 
■ Cohen et al. 



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
Local damage parameter D 

(a) Prediction for PC mortar 



>« 



U5 

c 



C 4-5 
■3 4.4 



TOT 




= 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

Q 

Local damage parameter D 

(b) Prediction for SF mortar 
Figure 10 Comparison with the test results by Cohen et al. (1995) 



38