1. INTRODUCTION 1
Chapter 1
INTRODUCTION
1.1 Background
The rapid development of fracture mechanics of quasibrittle materials in the last
three decades was essentially dictated by the realisation that its application can
lead to a satisfactory simulation and prediction of the local damage phenomena and
the effect of structural size to fracture (Baˇzant and Planas, 1998). Moreover, it
offers a logical approach to structural analysis and design based on sound mathe-
matical and mechanics concepts. Furthermore, the advent of new materials such
as high-strength concrete, fibre-reinforced concrete and polymer composites neces-
sitates the use of fracture mechanics to effectively exploit their material properties
for reasons of safety and economy. At present we are entering a period in which
the introduction of fracture mechanics into concrete design is becoming possible
(Mindess, 2002). This will help achieve more uniform safety margins, especially for
structures of different sizes. This, in turn, will improve economy as well as structural
reliability. It will make it possible to introduce new designs and utilise new concrete
materials. Applications of fracture mechanics are most urgent for structures such as
concrete dams, long span bridges, and nuclear reactor vessels or containments, for
which the safety concerns are particularly high and the consequences of potential
disaster enormous.
The applicability of fracture mechanics to real engineering problems depends
on the availability of fracture models that can simulate satisfactorily the behaviour
of quasibrittle fracture. One such model is the cohesive crack model whose early
development can be attributed to the independent works of Dugdale (1960) and
1. INTRODUCTION 2
Barenblatt (1962). The cohesive crack models were developed to simulate the non-
linear material behaviour near the crack tip. In these models, the crack is assumed
to extend and to open while still transferring stress from one face to the other. The
cohesive model proposed by Barenblatt (1959, 1962) aimed to relate the macroscopic
crack growth resistance to the atomic binding energy, while relieving the stress sin-

posed. All these models use simplifying assumptions to reduce the computational
complexities inherent in fracture analysis.
The cohesive crack model defines a relationship between normal crack opening
and normal cohesive stresses, and assumes that there are neither sliding displace-
ments nor shear stresses along the process zone. This assumption is only partially
valid for concrete materials. Based on experimental observations, it is indeed cor-
rect that a crack is usually initiated in pure mode I (i.e. opening mode) in concrete,
even for mixed mode loading (Saouma, 2000). However, during crack propagation,
the crack may curve due to stress redistribution or non-proportional loading, and
significant sliding displacements develop along the crack. Therefore, it is desirable
to incorporate these shear effects. Interface elements were first proposed by Good-
man et al. (1968) to model nonlinear behavior of rock joints. Since then, numerous
interface constitutive models have been proposed for a wide range of applications
such as rock mechanics (Goodman et al., 1968), masonry structures (Lotfi, 1992)
and concrete fracture (Stankowski, 1990; Feenstra et al., 1991; Carol et al., 1992;
ˇ
Cervenka, 1994). These models are basically the extension of Hillerborg’s cohesive
crack model for shear effects, and as such it can be also used to model interface
cracks.
All fracture models are governed by a constitutive law. The cohesive crack model,
for instance, requires a tension-softening relation (softening law) to characterise the
fracture behaviour of cementitious materials. In the practical application of the
cohesive crack model, the shape of the softening law is simplified and is assumed
1. INTRODUCTION 4
f
t
w
c
w
σ


σ

ϕ

c
Figure 1.2: Mohr-Coulomb criterion and shear band predicted in principal stress
space (De Borst, 1986)
Up to now, the most practical failure models that incorporate shear have been
1. INTRODUCTION 5
the Mohr-Coulomb type models, which limit and control the shear stress at a plane
as a function of the normal stress on that plane (Figure 1.2). Though they are formu-
lated in principal stress space, they actually limit the shear stress on certain planes.
Figure 1.2 shows a shear band in a specimen loaded in compression as predicted
by the use of a Mohr-Coulomb continuum model. When a shear plane is known,
it is possible to use a Mohr-Coulomb type of model for the description of interface
behaviour. Simple interface models of this type have been used by Roelfstra and
Sadouki (1986); Roelfstra (1989); Lorig and Cundall (1989); Vonk (1992). In these
models a tension cut-off criterion is added to the shear failure criterion. A more
complex model for the combination of tensile and shear loading including softening
has been proposed by Stankowski (1990).
The normality rule and/or the association of the flow laws with the yield function
in classical plasticity refer to the following circumstance: in the space of the stress
and strain components superposed, the plastic strain rate vector is normal to the
activated yield surface at the stress point. Nonassociated constitutive law refers to
circumstances otherwise (Koiter, 1960; Maier, 1969).
The safety and durability of concrete structures are significantly influenced by
the fracture behaviour of the concrete. There are many fracture formulations which
assume concrete as a homogeneous material or as a two-phase material composed
of aggregate particles dispersed in a cement paste matrix and provide reasonable

interface nodes in the matrix media after the generation of matrix-aggregate struc-
ture. The allocation of fibres is associated with the mesh structure by choosing all
possible combinations of distant nodes in the matrix which have a designated length
1. INTRODUCTION 7
range and do not cross any present aggregate particles. Limited experiments have
been undertaken on plain and fiber-reinforced concrete specimens which are used to
verify the analytical model developed.
1.4 Organisation of the research
This dissertation deals with the numerical simulation of fracture in plain concrete
and fibre reinforced concrete and is organised into nine chapters and three appen-
dices. Each chapter starts with an introduction and ends with a summary. The
introduction provides an overview, and if necessary a brief review, of the topics
contained therein. The summary highlights the important points discussed in the
chapter. Moreover, it also provides a smooth transition to the next chapter. The
contents of each chapter are briefly described in the following.
The first chapter naturally constitutes the introduction to the thesis, aims, mo-
tivation of the research and objective scope of the work. This chapter also contains
several assumptions and common notations employed throughout the thesis.
Chapter 2 comprises the literature survey of topics related to this work, i.e frac-
ture mechanics in plain and fibre reinforced concrete and the cohesive crack model.
Topics directly related to this thesis requiring more detailed discussion or derivation
are separately covered in the subsequent chapters. The literature survey provides
a brief historical overview of the early development of fracture mechanics and in-
troduces the different fracture models developed over the years starting with linear
elastic fracture mechanics (LEFM). The fundamental ideas underlying the concept
of the cohesive crack model are explained and the simplifying assumptions adopted
are discussed. The tension-softening relationship required of the model is described
and the fracture parameters characterising this softening behaviour, and their sig-
nificance to fracture mechanics, are also discussed.
1. INTRODUCTION 8

Chapter 6 deals with an articulated particle/interface model of concrete and the
introduction of a compression cap to the Mohr-Coulomb failure surface to further
track compressive failure. As an example, results on the fracture process in a cube
of concrete under compression are studied. All major factors that affect the soft-
ening behaviour in uniaxial compression - e.g. the influence of size, the boundary
condition, etc. - are alike discussed.
Chapter 7 presents experimental results on fracture in plain and fibre-reinforced
concrete. Material tests, shear tests and three-point bending tests are in turn pre-
sented. Basically, all parameters in the particle/interface model are derived. Differ-
ent fibre dosage is used to verify how fibre content affects the fracture energy and
critical crack opening displacement of shear and beam specimens.
Chapter 8 is the further development of the presented model to include fibres.
Simulation of several tests in the literature are performed and compared with exper-
imental results. These consist of the three-point bending test and the push-off shear
test. Lastly, the experimental results obtained in this study are simulated using the
proposed model.
Chapter 9 concludes the thesis with key summaries and recommendations for
future research.
1. INTRODUCTION 10
1.5 Assumptions and notations
Where applicable, assumptions are stated immediately following the derivation and
formulation of mathematical expressions used in the thesis. The following are as-
sumed throughout:
1. The formulation is applied to quasibrittle materials.
2. Structural modes of failure are opening, shear and/or compression for concrete
constituents; tension and/or pullout failure for steel fibres.
3. Linear softening laws are employed for all modes of fracture.
4. Displacements are assumed to be small. The loading path is piecewise lin-
earised (i.e., any given nonlinear load path is divided into a finite number of
proportional loading stages).

FRC = Fibre Reinforced Concrete
SFRC = Steel Fibre Reinforced Concrete
NSC = Normal Strength Concrete
HSC = High Strength Concrete
TPB = Three-Point Bending
FEM = Finite Element Method
BEM = Boundary Element Method
2. PRELIMINARIES AND LITERATURE REVIEW 12
Chapter 2
PRELIMINARIES AND LITERATURE
REVIEW
2.1 Introduction
The cohesive crack model (Hillerborg et al., 1976) is undoubtedly one of the most
widely used nonlinear fracture models for quasibrittle materials to date. Its pop-
ularity stems from its conceptual simplicity coupled with its proven capability to
predict and simulate fracture processes satisfactorily. Moreover, the model can be
implemented quite easily using such numerical analysis tools as the finite element
method (FEM), the boundary element method (BEM) and/or other discrete crack
models. Numerous papers have been written regarding its application on fracture
problems, and in many instances the model has been used as a yardstick for other
fracture models.
This section gives an introduction to fracture mechanics in general and deals with
the cohesive crack model in particular. In the next section, a brief historical review
of the evolution of fracture mechanics is given which provides an insight into why
early attempts to use classical fracture methods failed to predict the behaviour of
concrete and concrete-like materials. The review then leads to a discussion of various
fracture models, which were inspired by the introduction of the cohesive crack model.
The development of the cohesive crack model, as formulated by Hillerborg et al.
(1976), is discussed at length in Section 2.3. The different assumptions used in the
model are explained. The formation and localisation of the fracture process zone

elastic homogeneous brittle materials such as glass and brittle ceramics. Realis-
ing this limitation, Orowan (1949) and Irwin (1957) proposed a modification of the
theory, which can be used for engineering materials exhibiting limited ductility. A
flat line crack which presents two singularities at its extremes was introduced to
consider the friction developing between crack surfaces. The model is an extension
of the energy formulation used by Griffith where the plastic strain energy rate for
crack propagation was added to the energy equation. Both researchers recognised
that the energy required to produce plastic strain at the crack tip is much greater
than the surface energy needed to create new crack surfaces. It is through this work
of Orowan (1949) and Irwin (1957) that LEFM was formally developed.
Irwin (1957) formulated a novel approach where the concept of the critical stress
intensity factor K
Ic
is used as a criterion for crack extension; the subscript ”
I
” refers
to mode I fracture or pure opening. The critical stress intensity factor K
Ic
is called
fracture toughness and it is a measure of the resistance of a material to fracture.
Known as the Irwin’s criterion, the formulation is appealing due to its proximity to
conventional stress analysis. Moreover, its application to linear elasticity allows the
stress intensity factor K
I
to be additive. Irwin (1957) also derived a relationship
that exists between the stress intensity factor K
I
and Griffith’s energy release rate
G
I

model.
Another notable contribution on elastoplastic fracture mechanics was the intro-
duction of the path independent integral known as the J -integral by Rice (1968).
By idealising plastic deformation within the deformation theory of plasticity, Rice
was able to show that the energy release rate G is equivalent to the J -integral. It
is worth noting that the path independence of the J-integral holds only for elastic
materials where unloading follows the path of loading.
In the 1960s, a study on the fracture behaviour of concrete using LEFM was
gaining interest. Attempts by researchers, such as Kaplan (1961), to apply the prin-
ciples of LEFM to specimen-size concrete were unfruitful. It was observed that the
predicted results obtained from theory differ significantly from experimental results.
The reason for the discrepancy, which is essentially due to the microcracking of the
tensile response of concrete-like materials, is now common knowledge. A study made
by Kesler et al. (1972) shows conclusively that LEFM of sharp cracks was inade-
quate for normal concrete structures. This conclusion was supported by the results
of Walsh (1972), who tested geometrically similar notched beams of different sizes
and plotted the results in a double logarithmic diagram of nominal strength versus
size. Without attempting a mathematical description, he made the point that this
diagram deviates from a straight line of slope −
1
2
predicted by LEFM.
2. PRELIMINARIES AND LITERATURE REVIEW 16
softening
Nonlinear
zone
softening
softening
Nonlinear
z

is negligible. Brittle materials (Figure 2.1) fall into this category where the state
of stress ahead of the crack tip can be described by a single parameter singularity
such as the critical values of the energy release rate G
Ic
or the stress intensity factor
K
Ic
. For quasibrittle materials such as concrete, Figure 2.1 clearly shows the inap-
plicability of LEFM due to the long length of the fracture process zone relative to
the dimension of the specimen. It is evident then that a single fracture parameter
criterion will not be sufficient to fully describe the complex behaviour of the fracture
process zone.
2. PRELIMINARIES AND LITERATURE REVIEW 17
For large structures, however, LEFM can be used as a valid fracture model pro-
vided a crack-like notch or flaw exists in the structure. The applicability of the
theory lies in the relative size of the fracture process zone compared to the dimen-
sion of the structure, i.e., when the length of the process zone is negligible relative to
the size of a large structure. In such cases, the nonlinear region can be lumped into
a single point and a single parameter fracture criterion is sufficient to describe the
fracture processes. Studies have shown that the value of the critical stress intensity
factor K
Ic
, a parameter used in LEFM, reaches a constant value for large structures.
A number of papers have been published on the use of LEFM for the analysis of
large structures. For instance, Saouma and Morris (1998) successfully used LEFM
theory in the safety evaluation of a concrete dam.
Hillerborg and co-workers (Hillerborg et al., 1976) were the first to introduce
a nonlinear fracture model for quasibrittle materials. Based on the idea of plas-
tic crack-tip zone espoused by Dugdale (1960) and Barenblatt (1959), Hillerborg
proposed the cohesive crack model for analysing the physical behaviour of concrete

development of the theory and experimental techniques employed in the investiga-
tion of such fracture processes. Within that period, numerous models have been
developed and proposed as suitable fracture mechanics tools for the analysis of qua-
sibrittle fracture.
Inspired by the success of the cohesive crack model, Baˇzant and Oh (1983) pro-
posed the crack-band model. Also based on the concept of representing material
damage by a cohesive zone, the formulation of the crack band model has some
similarities to that of the cohesive crack model. However, instead of idealising the
fracture process zone as a line crack, the crack band model assumes that the fracture
process zone forms within a band of finite width h
c
. The width h
c
of this band is
considered a constant. A uniform distribution of microcracks is also assumed within
this band.
In the crack band model, a stress-strain curve is used to describe the material
behaviour at the fracture process zone. The energy consumed in the formation and
opening of all the microcracks per unit area is known as the fracture energy G
f
.
For a piecewise linear stress-strain curve as shown in Figure 2.3, the fracture energy
2. PRELIMINARIES AND LITERATURE REVIEW 19
σ

f
t
ε
E
G

c
(2.2)
where E is Young’s modulus of elasticity and E
t
is the tangent strain-softening mod-
ulus. The parameters f
t
, E, E
t
and h
c
are considered material properties. These
are the parameters required for use of the crack band model.
Two inherent limitations of the crack band model are noted: (a) The assump-
tions of constant band width and uniform distribution of strain within the band
width appear to have no direct experimental evidence. The value of the band width
h
c
equal to 3d
max
, where d
max
is the largest aggregate size used in the concrete mix,
as suggested by Baˇzant, was indirectly determined by inverse analysis; (b) Numeri-
cal predictions show that the behaviour of the structure is essentially insensitive to
the band width within certain limits.
The numerical implementation of most nonlinear models for fracture analysis is
quite computationally involved. However, if only the maximum load and not the
complete softening behaviour of the structure is required, approximate models may
suffice. These models use the fracture criterion employed in LEFM. Whereas, clas-

the stress intensity factor K
I
and crack tip opening displacement (CTOD) reach
critical values and the following relations hold:
K
I
= K
s
Ic
CTOD = CTOD
c
(2.3)
where K
s
Ic
and CTOD
c
are the critical values of the stress intensity factor and crack
tip opening displacement, respectively. A graphical representation of the model is
shown in Figure 2.4. Evidently, as Figure 2.4 illustrates, the critical value of the
stress intensity factor K
s
Ic
is determined at the tip of the effective crack length a
e
using the LEFM formula:
K
s
Ic
= σ

peak load.
The model parameters K
s
Ic
and CTOD
c
are considered material constants, i.e.,
the values are independent of specimen geometry and loading arrangements. These
parameters can be measured directly using three-point bending tests of a notched
specimen. It is not easy however to obtain accurate measurements of these param-
eters.
A conceptually similar approach to the two-parameter model is the effective
crack model proposed by Nallathambi and Karihaloo (1986). However, a secant
compliance at peak load is used in the determination of the effective crack length
a
e
. Moreover, the key parameters which indicate the onset of fracture are K
e
Ic
and
the effective crack length a
e
. The model assumes that the critical fracture state is
reached when stress intensity factor K
I
corresponding to the effective crack length
a
e
takes the critical value K
e

and c
f
where the latter is defined as
2. PRELIMINARIES AND LITERATURE REVIEW 22
Nonlinear fracture
mechanics
Limit
analysis
LEFM
2
1
log (d)
log (
c
)
Figure 2.5: Size effect law as defined in Equation (2.5) after Baˇzant (1984)
the critical crack extension for infinite sizes. A definition of Baˇzant’s size effect law
is graphically illustrated in Figure 2.5.
It is worth mentioning that the two-parameter model, the effective crack model
as well as the size effect model are predictive models. Since the required parameters
are defined at the critical state, these models can only predict the peak load and the
corresponding displacement of the structure. As it is, the models cannot describe
the softening response of the structure beyond the peak load. A generalisation of
the models to allow full analysis of the fracture processes can be achieved through
the use of R-curves (Karihaloo, 1995).
A multi-fractal scaling law capable of extrapolating results from laboratory size
specimens to actual structural size was proposed by Carpinteri and Ferro (1994)
and Carpinteri et al. (1997). To quantify the degree of disorder present in the
microstructures of quasibrittle materials, fractal geometry was used instead of the
typical integer topological dimensions of Euclidean sets.

(for instances, Carpinteri and Valente (1988); Cen and Maier (1992); Elices et al.
2. PRELIMINARIES AND LITERATURE REVIEW 24
(2002), among others). From this point forward, the term ”cohesive crack model”
might be used to refer to the fictitious crack model as formulated by Hillerborg et al.
(1976) and Petersson (1981).
Hillerborg’s cohesive crack model is conceptually simple, and it is simple enough
to be understood even by someone who has little knowledge of fracture mechan-
ics. This is no doubt one reason for its popularity (Baˇzant, 2002). Yet it provides
an excellent description of the fracture processes in quasibrittle structures. Unlike
LEFM models which can be applied only to initially cracked structures, the cohesive
crack model can capture the behaviour of a structure from crack initiation to fail-
ure. Although the model was developed for mode I fracture (tension failure), it has
nevertheless wide ranging application in fracture problems since tension failure is
by far the most dominant mode of failure in quasibrittle structures. Recently, some
researchers have attempted to extend the concept to mixed mode I and II situations
(Carpinteri, 1989; Hassanzadeh and Hillerborg, 1989).
σ

f
t
δ
A
D
C
B
w
x
1
Figure 2.7: Stress-deformation behaviour of a quasibrittle specimen in tension
The fundamental idea of the cohesive crack model is best described from a study

is known as the tension-softening relation, or simply softening relation. As segment
C of Figure 2.7 shows, the softening relation is characterised by a decreasing stress
with increasing deformation.
Any increase in the deformation of the specimen at this stage is localised within
the fracture process zone. In fact, as the deformation increases, the more localised
the damage zone becomes. As a consequence, outside the process zone, the whole
specimen can still be described by a stress-strain (σ − ε) relation. At the damaged