Numerical simulation of fracture in plain and fibre reinforced concrete

Numerical simulation of fracture in plain and fibre reinforced concrete

as high-strength concrete, fibre-reinforced concrete and polymer composites sitates the use of fracture mechanics to effectively exploit their material propertiesfor reasons of safety and economy At present we are entering a period in whichthe introduction of fracture mechanics into concrete design is becoming possible(Mindess, 2002) This will help achieve more uniform safety margins, especially forstructures of different sizes This, in turn, will improve economy as well as structuralreliability It will make it possible to introduce new designs and utilise new concretematerials Applications of fracture mechanics are most urgent for structures such asconcrete dams, long span bridges, and nuclear reactor vessels or containments, forwhich the safety concerns are particularly high and the consequences of potentialdisaster enormous.

neces-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 earlydevelopment can be attributed to the independent works of Dugdale (1960) and

Barenblatt (1962) The cohesive crack models were developed to simulate the linear material behaviour near the crack tip In these models, the crack is assumed

non-to extend and non-to open while still transferring stress from one face non-to the other Thecohesive model proposed by Barenblatt (1959, 1962) aimed to relate the macroscopiccrack growth resistance to the atomic binding energy, while relieving the stress sin-gularity Barenblatt postulated that the cohesive forces were operative on only asmall region near the crack tip, and assumed that the shape of the crack profile

in this zone was independent of the body size and shape Dugdale (1960), in aninvestigation of yielding in steel sheets containing slits, formulated a model of a linecrack with a cohesive zone having constant yield stress Although formally close

to Barenblatt’s, this model was intended to represent a completely different ical situation: macroscopic plasticity rather than microscopic atomic interactions.Both models share a convenient picture in which the stress singularity is removed.Despite being very simplified, Dugdale’s approach to plasticity gave a good descrip-tion of ductile fracture for small plastic zone sizes However, it was not intended

phys-to describe fracture itself and, in Dugdale’s formulation, the plastic zone extendedforever without any actual crack extension The cohesive crack model came to theforefront in the mid 1970s with the work of Hillerborg and co-workers (Hillerborg

et al., 1976) The cohesive crack model served as a suitable nonlinear model formode I fracture Their research acted as a catalyst in rousing the interest of study-ing quasibrittle materials in fracture mechanics perspective Since then, a number

of fracture models have been introduced and used to predict and investigate fracturebehaviour of concrete-like materials In general, all the foregoing fracture mechanicstheories require a pre-existing crack to analyse the failure of a structure or compo-nent This is not so with Hillerborg’s fictitious crack model It is a cohesive crack

in the classical sense described above, but it is more than that because it includescrack initiation rules for any situation This means that it can be applied to initiallyuncracked concrete structures and describe all the fracture processes from no crack

at all to complete structural breakage It provides a continuous link between the

classical strength-based analysis of structures and the energy-based classical ture mechanics: cohesive cracks start to open as dictated by a strength criterionthat naturally and smoothly evolves towards an energetic criterion for large cracks.Other nonlinear models such as the two-parameter model by Jenq and Shah (1985)and effective crack model by Nallathambi and Karihaloo (1986) have also been pro-posed All these models use simplifying assumptions to reduce the computationalcomplexities inherent in fracture analysis.

frac-The cohesive crack model defines a relationship between normal crack openingand normal cohesive stresses, and assumes that there are neither sliding displace-ments nor shear stresses along the process zone This assumption is only partiallyvalid 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, andsignificant sliding displacements develop along the crack Therefore, it is desirable

to incorporate these shear effects Interface elements were first proposed by man et al (1968) to model nonlinear behavior of rock joints Since then, numerousinterface constitutive models have been proposed for a wide range of applicationssuch 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

Good-Cervenka, 1994) These models are basically the extension of Hillerborg’s cohesivecrack model for shear effects, and as such it can be also used to model interfacecracks

All fracture models are governed by a constitutive law The cohesive crack model,for instance, requires a tension-softening relation (softening law) to characterise thefracture behaviour of cementitious materials In the practical application of thecohesive crack model, the shape of the softening law is simplified and is assumed

Figure 1.1: Linear softening law and the cohesive crack model

to be known a priori Among the simplest softening relationships developed is the

linear softening law that was used by Hillerborg and co-workers (1976) to illustratethe applicability of their proposed fracture model As shown in Figure 1.1, onlytwo parameters need to be specified to sufficiently characterise the model One

can use any of the combinations of the tensile strength f t and fracture energy G F

or tensile strength f t and the critical crack width w c Petersson (1981) proposedthe two-branch law that is generally acknowledged to provide a better approxima-tion of the fracture behaviour of concrete The two-branch law, in general, is fullycharacterised by specifying four parameters, except if the breakpoint is known De-tails and the application of these models will be discussed further in the next section

τ

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

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 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

formu-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 interfacebehaviour Simple interface models of this type have been used by Roelfstra andSadouki (1986); Roelfstra (1989); Lorig and Cundall (1989); Vonk (1992) In thesemodels a tension cut-off criterion is added to the shear failure criterion A morecomplex model for the combination of tensile and shear loading including softeninghas 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 stressand strain components superposed, the plastic strain rate vector is normal to theactivated yield surface at the stress point Nonassociated constitutive law refers tocircumstances otherwise (Koiter, 1960; Maier, 1969)

The safety and durability of concrete structures are significantly influenced bythe fracture behaviour of the concrete There are many fracture formulations whichassume concrete as a homogeneous material or as a two-phase material composed

of aggregate particles dispersed in a cement paste matrix and provide reasonablesimulations However, such models do not include the effects of the transition zonebetween coarse aggregate and cement paste It is well known that this zone has asignificant effect on the elastic properties, but little is known on how it affects thesoftening process Therefore, it is necessary to access the adequacy of fracture mod-els considering the heterogeneous nature of concrete with three distinctive phases(Vonk, 1992; van Mier, 1997; Leite et al., 2004)

1.2 Aim and motivation of the research

The broad aim of the project is to develop novel methods apt to simulate fracturebehaviour and softening processes in plain and fibre-reinforced concrete as a quasi-brittle material Specifically, this study deals with identifying different modes offailure, i.e tension, shear, and compression with several questions about the in-teraction between shear and tension A mathematical programming based discreteinterface formulation is employed to achieve this goal Several benchmark problemsare tackled including the compressive softening of a concrete cube and crack inter-action in a beam

A composite model is used to represent the heterogeneity of plain concrete ing of coarse aggregates, mortar matrix and the mortar-aggregate interface Thecomposite elements of plain concrete are modelled using triangular finite elementunits which have six interface nodes along their sides Fracture is captured through

consist-a constitutive single brconsist-anch softening-frconsist-acture lconsist-aw consist-at the interfconsist-ace nodes, whichbounds the elastic domain inside each triangular unit The inelastic displacement

at an interface node represent the crack opening and/or sliding displacement and

is conjugate to the internodal forces The path-dependent softening behaviour isdeveloped within a quasi-prescribed displacement control formulation The crackprofile is restricted to the interface boundaries of the defined mesh No re-meshing

is carried out Solutions to the rate formulation are obtained using a mathematicalprogramming procedure in the form of a linear complementary problem Fibre par-ticles are modelled by introducing additional linear elements interconnecting distantinterface 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 allpossible combinations of distant nodes in the matrix which have a designated length

range and do not cross any present aggregate particles Limited experiments havebeen undertaken on plain and fiber-reinforced concrete specimens which are used toverify the analytical model developed.

This dissertation deals with the numerical simulation of fracture in plain concreteand fibre reinforced concrete and is organised into nine chapters and three appen-dices Each chapter starts with an introduction and ends with a summary Theintroduction provides an overview, and if necessary a brief review, of the topicscontained therein The summary highlights the important points discussed in thechapter Moreover, it also provides a smooth transition to the next chapter Thecontents of each chapter are briefly described in the following

The first chapter naturally constitutes the introduction to the thesis, aims, tivation of the research and objective scope of the work This chapter also containsseveral assumptions and common notations employed throughout the thesis

mo-Chapter 2 comprises the literature survey of topics related to this work, i.e ture mechanics in plain and fibre reinforced concrete and the cohesive crack model.Topics directly related to this thesis requiring more detailed discussion or derivationare separately covered in the subsequent chapters The literature survey provides

frac-a brief historicfrac-al overview of the efrac-arly development of frfrac-acture mechfrac-anics frac-and troduces the different fracture models developed over the years starting with linearelastic fracture mechanics (LEFM) The fundamental ideas underlying the concept

in-of the cohesive crack model are explained and the simplifying assumptions adoptedare discussed The tension-softening relationship required of the model is describedand the fracture parameters characterising this softening behaviour, and their sig-nificance to fracture mechanics, are also discussed

Chapter 3 deals with the formulation of the state problem expressed as a linearcomplementarity problem (LCP) It covers the mathematical descriptions of basicequations for elastic-plastic relations in structural mechanics The concepts andformulation of a structure into a finite number of six-node interface triangular unitseach consisting of nine constant strain triangle are then presented The implemen-tation of a piecewise linear inelastic failure surface and softening constitutive law isdescribed The single branch softening laws in tension and shear are formulated in

a complementarity format The structural relations are cast into a nonholonomic(irreversible) rate formulation Also introduced in this chapter is a review of thelinear complementarity problem and its applications in engineering mechanics aswell as some of the computational algorithms employed in the thesis, such as Lemke(Lemke, 1965) and the industry standard solver PATH (Dirkse and Ferris, 1995)

Chapter 4 discusses the methods and algorithms used in the automated meshgeneration and the composite model to include the heterogeneous nature of con-crete (modelling at meso-level) Concrete is modelled as a three-phase materialwith coarse aggregates, a mortar matrix and the mortar-aggregate interfaces Prop-erties of each constituent available in the literature is likewise mentioned

Chapter 5 analyses several verification examples using actual experimental data.One of them is the interacting crack problem The formulation developed in Chap-ter 3 is employed to report the investigation of multiple interacting cracks in thefour-point bending test of a simple plain concrete beam Material properties areassigned in a homogeneous manner The solution algorithm concentrates on theanalysis of various fracture modes in a plain concrete beam under four point bend-ing with several notches and examines the interacting crack itineraries by identifyingthe various equilibrium solutions available Next, two of the most cited problems inidentifying parameters of cohesive crack model in concrete, i.e the Brazilian test

and the three-point bending test, are numerically simulated using the same tion in conjunction with the composite model prepared in Chapter 4 The boundarycondition and factors that affect the outcome of these tests are examined.

formula-Chapter 6 deals with an articulated particle/interface model of concrete and theintroduction of a compression cap to the Mohr-Coulomb failure surface to furthertrack 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 ening behaviour in uniaxial compression - e.g the influence of size, the boundarycondition, etc - are alike discussed

soft-Chapter 7 presents experimental results on fracture in plain and fibre-reinforcedconcrete 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 andcritical 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 sheartest Lastly, the experimental results obtained in this study are simulated using theproposed model

Chapter 9 concludes the thesis with key summaries and recommendations forfuture research

1.5 Assumptions and notations

Where applicable, assumptions are stated immediately following the derivation andformulation 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 concreteconstituents; 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 earised (i.e., any given nonlinear load path is divided into a finite number ofproportional loading stages)

lin-The following conventions are used for general description throughout the thesiswhile specific ones are indicated where appropriate

1 Vectors and matrices are indicated by bold type symbols Column vectors areassumed throughout

2 A scalar quantity is denoted in italics

3 A real vector a of size n is indicated by a ∈ R n and a real m × n matrix A by

A ∈ R m×n 0 denotes a null vector.

4 Transpose of a vector or a matrix is indicated by the superscript T; the inverse

of a matrix by the superscript -1

5 The complementarity relationship between two nonnegative vectors f and z is written as fTz = 0 which implies the componentwise condition of f k z k= 0 for

all k Vector inequalities apply componentwise.

1.6 Abbreviations

The following will list some common abbreviations that are valid throughout thethesis, however many less common will be defined on a chapter by chapter basis

CMOD = Crack Mouth Opening Displacement

LEFM = Linear Elastic Fracture Mechanics

LVDT = Linear Variable Displacement Transducer

LCP = Linear Complementarity Problem

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

This section gives an introduction to fracture mechanics in general and deals withthe 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 whyearly attempts to use classical fracture methods failed to predict the behaviour ofconcrete and concrete-like materials The review then leads to a discussion of variousfracture 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 themodel are explained The formation and localisation of the fracture process zoneand its idealisation in the model are described Essential features and limitations

of the model are elaborated The relation of the cohesive crack model with otherproposed fracture models is likewise discussed.

The next four sections of this chapter will review various fracture mechanicsapproaches to quasibrittle materials The last part mentions the history of fiberreinforced concrete (FRC) and the modelling of its behaviour in light of fracturemechanics

quasibrit-tle models

The advent of fracture mechanics is generally attributed to the pioneering work ofInglis (1913) when he observed that stresses at the vertex of a degenerate ellipsoidalcavity tended to infinity Consequent studies by other researchers have, since then,led to a better and deeper understanding of fracture phenomena This in turn hasresulted in the development of theories to explain and quantify the observed physicalbehaviour of a structure in fracture

Among the earliest fracture theory developed was linear elastic fracture ics (LEFM) Its early development can be traced back to the work of Alan Griffith,

mechan-a British mechan-aeronmechan-auticmechan-al engineer, when he formulmechan-ated mechan-an energy equmechan-ation to describethe propagation of slit-like cracks using the concept of critical energy release rate

G c (Griffith, 1920) The theory began from a hypothesis that brittle materials tain elliptical microcracks, which introduce high stress concentrations near theirtips This fracture criterion, which is essentially a statement of the energy balanceprinciple, states that crack propagation initiates when the gain in surface energydue to the increase in surface area equals the reduction in strain energy due to thedisplacement of the boundaries and the change in the stored elastic energy

con-Griffith’s fracture theory, however, is applicable only to the failure analysis ofelastic homogeneous brittle materials such as glass and brittle ceramics Realis-ing this limitation, Orowan (1949) and Irwin (1957) proposed a modification of thetheory, which can be used for engineering materials exhibiting limited ductility Aflat line crack which presents two singularities at its extremes was introduced toconsider the friction developing between crack surfaces The model is an extension

of the energy formulation used by Griffith where the plastic strain energy rate forcrack propagation was added to the energy equation Both researchers recognisedthat the energy required to produce plastic strain at the crack tip is much greaterthan 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 calledfracture 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 toconventional 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

where E ∗ = E for plane stress and E ∗ = E

1− ν2 for plane strain E and ν are

Young’s modulus and Poisson’s ratio, respectively

Interest in the fracture mechanics of ductile materials arose out of the researchconducted by Dugdale (1960) and Barenblatt (1962) Dugdale proposed a simple

model, the strip-yield model, to deal with plasticity at the crack tip A key tion in the model states that the stress values at the crack tip are limited by theyield strength of the material and that yielding is confined to a narrow band alongthe crack line Although mathematically similar to Dugdale’s model, Barenblatt’swork, nonetheless, is conceptually different since it deals with the cohesive zone at amolecular level instead of macroscopic plasticity The independent work of Dugdaleand Barenblatt served as the foundation in the formulation of the cohesive crackmodel.

assump-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 wasgaining 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 thepredicted results obtained from theory differ significantly from experimental results.The reason for the discrepancy, which is essentially due to the microcracking of thetensile 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 quate for normal concrete structures This conclusion was supported by the results

inade-of Walsh (1972), who tested geometrically similar notched beams inade-of different sizesand plotted the results in a double logarithmic diagram of nominal strength versussize Without attempting a mathematical description, he made the point that thisdiagram deviates from a straight line of slope −12 predicted by LEFM

softening Nonlinear

Nonlinear zone

Nonlinear zone

(a) Brittle material (b) Ductile material (c) Quasibrittle material

Figure 2.1: Relative sizes of the fracture process zone (Baˇzant, 1985) Diagrams at

the top show the trends of the stress distribution along the crack line

The use of LEFM to model concrete fracture was borne out of ignorance of thematerial’s property at that time Concrete was then thought to be a brittle material

It was later realised that the material exhibits decreasing tensile carrying capacitywith increasing deformation after the peak stress is reached This response is known

as tension softening, and materials exhibiting such response are called quasibrittlematerials For quasibrittle materials, the physical processes occurring ahead of thecrack tip are quite different from those of brittle and ductile materials Figure 2.1shows a comparison of the relative sizes of the process zones occurring in the threetypes of materials mentioned

The original formulation of LEFM by Griffith (1920) and Irwin (1957) is cable only to materials where the size of the nonlinear region ahead of the crack tip

appli-is negligible Brittle materials (Figure 2.1) fall into thappli-is 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 plicability of LEFM due to the long length of the fracture process zone relative tothe dimension of the specimen It is evident then that a single fracture parametercriterion will not be sufficient to fully describe the complex behaviour of the fractureprocess zone

inap-For large structures, however, LEFM can be used as a valid fracture model vided a crack-like notch or flaw exists in the structure The applicability of thetheory 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 tothe size of a large structure In such cases, the nonlinear region can be lumped into

pro-a single point pro-and pro-a single ppro-arpro-ameter frpro-acture criterion is sufficient to describe thefracture 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 oflarge structures For instance, Saouma and Morris (1998) successfully used LEFMtheory 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 tic crack-tip zone espoused by Dugdale (1960) and Barenblatt (1959), Hillerborgproposed the cohesive crack model for analysing the physical behaviour of concrete

plas-in fracture Unlike LEFM-based models where their applicability depends on theprior existence of notch-like cracks, the cohesive crack model can be used to describethe behaviour of uncracked as well as cracked quasibrittle structures An essentialingredient of the proposed fracture model for analysis is the softening law For a

generic nonlinear softening law with the shape function σ = f (w) given (see Figure 2.2), the essential parameters include the tensile strength f t and fracture energy

G F The fracture energy G F is defined as the area under the stress-displacement

discontinuity (σ − w) curve, where σ is the tensile stress and w is the crack mouth

opening displacement A detailed discussion of the cohesive crack model is taken up

in the next section and topics relevant to the model are discussed therein

The introduction of the cohesive crack model as a suitable fracture mechanics

Figure 2.2: A generic nonlinear softening curve for cohesive crack model

model for concrete has ignited the interests of researchers working in fracture chanics of quasibrittle materials The last twenty years is characterised by a rapiddevelopment of the theory and experimental techniques employed in the investiga-tion of such fracture processes Within that period, numerous models have beendeveloped and proposed as suitable fracture mechanics tools for the analysis of qua-sibrittle fracture

me-Inspired by the success of the cohesive crack model, Baˇzant and Oh (1983) posed the crack-band model Also based on the concept of representing materialdamage by a cohesive zone, the formulation of the crack band model has somesimilarities to that of the cohesive crack model However, instead of idealising thefracture process zone as a line crack, the crack band model assumes that the fracture

pro-process zone forms within a band of finite width h c The width h c of this band isconsidered a constant A uniform distribution of microcracks is also assumed withinthis band

In the crack band model, a stress-strain curve is used to describe the materialbehaviour 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

Figure 2.3: Piecewise linear stress-strain curve for the crack band model

G f is evaluated as the product of the crack band width h c and the area under thestress-strain diagram given as:

E t



where E is Young’s modulus of elasticity and E tis the tangent strain-softening

mod-ulus The parameters f t , E, E t and h c are considered material properties Theseare the parameters required for use of the crack band model

Two inherent limitations of the crack band model are noted: (a) The tions of constant band width and uniform distribution of strain within the bandwidth appear to have no direct experimental evidence The value of the band width

assump-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) cal predictions show that the behaviour of the structure is essentially insensitive tothe band width within certain limits

Numeri-The numerical implementation of most nonlinear models for fracture analysis isquite computationally involved However, if only the maximum load and not thecomplete softening behaviour of the structure is required, approximate models maysuffice These models use the fracture criterion employed in LEFM Whereas, clas-sical LEFM requires only one fracture criterion, most approximate models use two

parameters to describe the process zone Among the more popularly known includeJeng and Shah’s two-parameter model (Jenq and Shah, 1985) and the effective crackmodel by Nallathambi and Karihaloo (1986) These models are often referred to asthe equivalent elastic crack model where a real structure is replaced by an equivalentelastic structure As a consequence, computations are simplified since only linear,instead of nonlinear, analysis is required.

Figure 2.4: Definition of the two-parameter fracture model (Jenq and Shah, 1985)

In the original formulation of the two-parameter model (Jenq and Shah, 1985),only the conditions for peak load are given The model assumes that at peak load

the stress intensity factor K I and crack tip opening displacement (CTOD) reachcritical values and the following relations hold:

stress intensity factor K Ic s is determined at the tip of the effective crack length a e

using the LEFM formula:

K Ic s = σ c √

πa e g1



a e d



(2.4)

in which, σ c is the stress at peak load; g1 is a function of the geometry of a specimen;

and d is the specimen height Diagrammatically, K Ic s is measured at the tip of the

effective crack length a e , CT OD c is determined at the notch (or real crack) tip The

effective crack length a eis obtained from the unloading compliance measured at thepeak load

The model parameters K Ic s and CT OD c are considered material constants, i.e.,the values are independent of specimen geometry and loading arrangements Theseparameters can be measured directly using three-point bending tests of a notchedspecimen It is not easy however to obtain accurate measurements of these param-eters

A conceptually similar approach to the two-parameter model is the effectivecrack model proposed by Nallathambi and Karihaloo (1986) However, a secantcompliance 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 Ic e 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 Ic e

Another widely used adaptation of LEFM for quasibrittle fracture is the sizeeffect law by Baˇzant (1984) Using dimensional analysis and similitude, Baˇzantproposed a scaling law that can predict the value of the failure stress using notchedgeometrically similar structures The equation for the scaling law is expressed as:

ber of geometrically similar notched beam specimens of different sizes B and d o are

related to the size effect model parameters G f and c f where the latter is defined as

Nonlinear fracture mechanics

Limit analysis

LEFM

2 1

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 parametersare defined at the critical state, these models can only predict the peak load and thecorresponding displacement of the structure As it is, the models cannot describethe softening response of the structure beyond the peak load A generalisation ofthe 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 sizespecimens to actual structural size was proposed by Carpinteri and Ferro (1994)and Carpinteri et al (1997) To quantify the degree of disorder present in themicrostructures of quasibrittle materials, fractal geometry was used instead of thetypical integer topological dimensions of Euclidean sets

There are other fracture models that were developed for quasibrittle fracture.Among these include local and nonlocal continuum damaged mechanics models, lat-tice models, stochastic methods, among many others Excellent monographs written

σy

σy

Crack opening displacement

Figure 2.6: Dugdale’s plastic zone model

by Karihaloo (1995), Baˇzant and Planas (1998), Shah et al (1995) and van Mier(1997) provide comprehensive discussions on these models

The concept of using a cohesive zone to model stress behaviour near the crack tipwas pioneered by Dugdale (1960) and Barenblatt (1962) In Dugdale’s model (Fig-ure 2.6), it is assumed that a stress, equal to the yield value of the material, actsuniformly across the cohesive zone Barenblatt’s model, which is mathematicallysimilar to Dugdale’s model, assumes that the stress varies across the cohesive zone

as a function of the cohesive crack width

The application of cohesive zones to study fracture nucleation and crack agation in concrete was first explored by Hillerborg and co-workers (1976) Thedeveloped model, which they implemented within a FEM to study the fracture be-haviour of an unreinforced beam in bending, was called the fictitious crack model(Hillerborg et al., 1976; Petersson, 1981) However, more recently, its semblancewith the cohesive model proposed by Barenblatt led many researchers to call it withthe former terminology of ”cohesive crack model” (Carpinteri et al., 2003; Carpin-teri, 1989), and the model has been used with this name by a number of researchers(for instances, Carpinteri and Valente (1988); Cen and Maier (1992); Elices et al

prop-(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 ics This is no doubt one reason for its popularity (Baˇzant, 2002) Yet it provides

mechan-an excellent description of the fracture processes in quasibrittle structures UnlikeLEFM models which can be applied only to initially cracked structures, the cohesivecrack 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 hasnevertheless wide ranging application in fracture problems since tension failure is

by far the most dominant mode of failure in quasibrittle structures Recently, someresearchers have attempted to extend the concept to mixed mode I and II situations(Carpinteri, 1989; Hassanzadeh and Hillerborg, 1989)

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

of the stress-deformation diagrams obtained from a simple tension test (Figure 2.7).Displacement control, which is monotonically increasing in time, is assumed in thetest to ensure stable crack propagation Moreover, since the level of analysis is

macroscopic, the specimen can be assumed homogeneous This illustration lights many of the assumptions used in defining the cohesive crack model.

high-The figure shows two curves, ABC and ABD high-These curves represent the deformation behaviour at two different locations of the tension test specimen ABCdescribes the behaviour at location x1 where a localised fracture zone (fracture pro-cess zone) develops ABD, on the other hand, is a representative behaviour of thematerial at a location other than the fracture process zone

stress-The fact that both curves have the same ascending branch A indicates that prior

to the attainment of the peak stress (tensile strength) f tthe whole specimen is

sub-jected to the same stress and deformation Therefore, a stress-strain (σ − ε) law can

be used to describe the material behaviour at this stage For concrete (and othercementitious materials) under tension, segment A deviates very little from a straightline It is not surprising then that in most applications, a linear stress-strain relation

is assumed

Right after the tensile strength f t of the material is reached, the fracture processzone is assumed to develop at location x1 Its formation is essentially due to mi-crocracking which ”softens” the material at this location The stress-displacement

discontinuity (σ − w) relation which describes the material behaviour in this zone

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 stresswith increasing deformation

Any increase in the deformation of the specimen at this stage is localised withinthe fracture process zone In fact, as the deformation increases, the more localisedthe 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

zone however, a stress-displacement discontinuity (σ −w) relation must be employed.

The cohesive crack model can therefore be thought of as a two-part constitutive law

model: for the bulk material a stress-strain (σ − ε) relation is applied, while for the

fracture process zone, a stress-displacement discontinuity (σ − w) relation is used.

The localisation phenomena of deformation in the fracture process zone leads tovarious simplifying assumptions when using the cohesive crack model For instance,since material damage is concentrated within the fracture process zone, it is rea-sonable to assume that the bulk material remains undamaged Consequently, thewhole material, excluding the fracture process zone, can be assumed to behave in alinear-elastic manner The localisation of the damage zone also allows the interpre-tation of the fracture process zone as a line crack of zero width

The distribution of tensile stress varies nonlinearly along the length of the ture process zone (Figure 2.8) At the tip of the fracture process zone, the stress anddeformation are equal to the tensile strength of the material and zero, respectively

frac-It is assumed that fracture initiates when the principal stress at a point attains the

value of the tensile strength f t of the material Crack propagates orthogonal tothe direction of the principal stress As Hillerborg et al (1976) have pointed out,this is not a real crack but only a fictitious one which is capable of transferringstresses between crack faces These stresses decrease with increasing displacement

discontinuity w The model is a convenient mathematical idealisation of the

mate-rial damage occurring in the fracture process zone of a quasibrittle matemate-rial It wasthis feature of the model which prompted Hillerborg and co-workers to call it the

”fictitious crack model” In the model, a true crack appears when the critical crack

width w c is reached At this point, the value of the tensile stress has just reachedzero

An essential ingredient of the cohesive crack model is the softening relation or

Undamaged material

Crack

w

Figure 2.8: Definition of the cohesive crack model

softening law (Figure 2.8) The softening law is the analytical description of the

variation of the tensile stress and displacement discontinuity w along the length of

the fracture process zone For as long as tension forces dominate in a structure andwhere the effects of lateral deformations and stresses can be neglected, the softeninglaw is considered a material property

The determination of the softening law is a necessary step before the cohesivecrack model can be used for fracture analysis Considerable research has been spenttowards its characterisation using direct (experimental) and indirect methods byinverse analysis and other techniques In the practical applications of the model,simplified softening curves are employed For instance, a two-branch law (Petersson,1981) is generally considered a good approximation of the mode I fracture behaviour

of concrete Nonlinear softening laws have also been used in the investigation of thefracture processes of cementitious materials (Foote et al., 1986; Planas and Elices,1991; Hu and Mai, 1992; Carpinteri and Massab`o, 1997; Reinhardt and Xu, 1999).These types of softening curves are a good compromise between computational sim-plicity and predictive accuracy

Depending on the chosen softening curve, as few as two parameters are required

to completely describe the softening law For example, a generic nonlinear

soften-ing law (Figure 2.8) with a known softensoften-ing function σ = f (w) requires only two parameters (e.g., f t and w c or f t and G F) to completely describe the curve

In the context of the cohesive crack model, these parameters are considered terial properties that are independent of specimen geometry and loading Often than

ma-not, however, only the tensile strength f t and the fracture energy G F are chosen due

to the difficulty of experimentally measuring the critical crack width w c Both the

tensile strength f t and the fracture energy G F can be determined by performingappropriate fracture mechanics tests It is worth noting that in some engineeringapplications, it may not even be necessary to completely characterise the softeninglaw For instance, if only the maximum load of a structure is required for design

purposes, then the tensile strength f t and the initial slope of the softening curve willprovide sufficient information (Alvaredo and Torrent, 1987; Guinea et al., 1997)

The fracture energy G F is considered an important property of the softeningcurve It is a measure of the energy absorbed to create and fully separate a unitsurface area of cohesive crack It can be evaluated as the area under the softeningdiagram and given by the expression:

 w c0

Another useful material parameter characterising the cohesive crack model is the

characteristic length l ch expressed as:

f2

t

(2.7)

The characteristic length l ch can be interpreted as a ratio of the fracture energy G F

to the strain energy density at failure f t2

E It is essentially a measure of the

brittle-ness of the material The lower the value of this parameter, the more brittle is the

material The cohesive crack model is a fracture formulation that is general enough

to encompass other models Its equivalence with the crack band model proposed byBaˇzant and Oh (1983) is well discussed in Chapter 8 of the monograph written byBaˇzant and Planas (1998) A study made by Elices and Planas (1996) showed thatwithin the practical range of laboratory size specimens, the size effect predictions

of the size effect model (Baˇzant, 1984) and the two-parameter model (Jenq andShah, 1985) compare very well with the cohesive crack model Elices and Planas(1996) proposed a correlation between the parameters of the different models Itwas also shown that the effective crack model (Nallathambi and Karihaloo, 1986)

is approximately equivalent to the two-parameter model, and thus correlates wellwith the size effect prediction of the cohesive crack model Moreover, more involvedcontinuum based models can also be related within the framework of the cohesivecrack model (Planas et al., 1993; Elices et al., 1992)

The cohesive crack model is subject to limitations which are extrinsic and trinsic in nature The extrinsic limitation deals essentially with the difficulty inexperimentally measuring the material parameters governing the model This prob-lem is appropriately discussed and treated by using inverse analysis to extract thematerial parameters (Que, 2003) Interested readers might refer to the work of Maierand other researchers, see e.g Bolzon et al (2004); Maier et al (2005) The intrinsiclimitations of the cohesive crack model are essentially due to the simplifying assump-tions made in the model such as a localised locus of discontinuity, non-dissipativecondition of the bulk material and non-inclusion of tangential cohesive forces Formode I fracture of quasibrittle homogenous materials, however, these assumptionsare reasonable and such effects as multiple cracking can be effectively minimised byusing notches

in-The simplicity of the cohesive crack model (Hillerborg et al., 1976) coupled with

its above-mentioned capability to satisfactorily predict and simulate fracture cesses in quasibrittle structures is one overriding reason for its popular use Theease with which the concept can be included into FEM, BEM or other particlemodels is another reason However, the complete characterisation of the cohesivecrack model, in spite of its conceptual simplicity, is not an easy task, as indicated

pro-in earlier on A short discussion of direct (stable fracture mechanics tests) and pro-direct approaches for the characterisation of the softening law will be presented next

in-The advancement of fracture theory was accompanied by the development oftesting methods for the observation of the physical behaviour of a material in frac-ture It can even be considered that the growth of quasibrittle fracture mechanics isstimulated by the introduction of new testing techniques, which are used to verifythe predictions made by the theory The advent of modern testing equipment andfast computers has led to a better and deeper understanding of the fracture pro-cesses in quasibrittle materials

Material properties, which are required of the different proposed models, areoften determined using fracture mechanics tests Different testing techniques havebeen developed to measure the key parameters An important consideration in thedesign of testing techniques is simplicity of the method and repeatability of the re-sults

The usefulness of data obtained from any stable fracture test is dependent on anumber of factors These factors are related not only to the tests but also to datainterpretation For instance, load-control tests cannot capture the complete postpeak behaviour of quasibrittle materials This type of setup is essentially designed

to obtain the elastic properties of a material and it cannot be used to capture thecomplete load-displacement diagram of a quasibrittle structure The realisation thatmaterials such as concrete retain some strength after the peak load is reached led to

the development of closed-loop test methods which use elongation as the controllingvariable In this set-up, the deformation is monotonically increased in time Withthis procedure, it is possible to observe stable crack propagation up to failure andmeasure the post peak softening curve.

The boundary conditions used in the test setup are known to have significanteffects on the results of materials testing Studies made by researchers indicate theimportance of the gripping system employed in mode I fracture tests (Cattaneo andRosati, 1999; van Mier and van Vliet, 2002; Rocco et al., 1999; Guinea et al., 1992).Boundary effects such as frictional influences and rotational effects can lead to spu-rious results

Specimen imperfections add to the difficulty in obtaining perfect results fromfracture tests Even with seemingly seamless preparation of a specimen, internalflaws in the material cannot be avoided Environmental factors also contribute tothe specimen imperfection prior to testing The composition of material itself maypose a problem Concrete, for example, is a heterogeneous material and its het-erogeneity can affect the distribution of the material stiffness over the specimencross-section This is one reason why it is difficult, if not impossible, to obtain uni-form stress distributions in a uniaxial tensile test

In principle, the most direct way to measure the softening curve is by means ofstable tensile tests (Petersson, 1981) This kind of test seems to provide the whole

stress-displacement (σ − w) curve as a direct output from the experiment However,

the experimental results have known that such an approach is extremely difficult ifnot impossible because of some major drawbacks:

(a) The location of the cohesive crack is not known a priori, and in most occasion

multiple cracking occurs due to the material heterogeneity (Guo and Zhang,

1987; Philips and Binsheng, 1993; Planas and Elices, 1986).

(b) For materials where the process zone is very small, uncracked specimens

un-dergo large amounts of yielding before the cohesive crack forms, which creases the specimen compliance and instability The strain energy stored inthe specimen makes the test unstable even under crack opening displacementcontrol (G´omez et al., 2000)

in-(c) When a small crack is introduced to initiate fracture and a single cohesive crack

is produced, the specimen tends to asymmetric modes of fracture, and thecrack opening is not uniform across the specimen Stiffer testing machines donot solve the problem due to elastic internal rotations in the specimen itself.Moreover, when the rotations are avoided using very short specimens and avery stiff machine, or by means of special servo-controlled systems, the twocracks formed on each side of the specimen tend to move away from each otherwhen they approach, and hence never constitute a true single crack (Carpinteriand Ferro, 1994; van Mier and Vervuurt, 1995)

Another test that has been regulated to measure the fracture energy mode I ofconcrete fracture is the three-point bending test Details of this test will be dis-cussed in length in Chapters 6 and 7

The inherent difficulties associated with a direct determination of softening haviour led researchers to apply indirect methods based on parametric fitting ofexperimental results These methods, known as inverse analysis or data reduction,usually do not yield identical results for the same given material due to the differentweight assigned to the various experimental data Inverse analysis procedures arebased on the simultaneous solving of functional equations The functionals must beselected with special care because the set of equations tends to be ill-posed Two

be-main groups of techniques can be distinguished regarding the selection of als: techniques that use the individual points of the load-displacement, or CMOD,curve for a specific test, and those that use characteristic points, or properties of thecurve, e.g the peak load, or the area under the curve, of different specimens Withinthe first group two main approaches have been published: those that use a soften-ing curve defined a priori by parameters, which are best fitted by an optimisationalgorithm to the experimental data (Mihashi, 1992; Roelfstra and Wittmann, 1986;Ulfkjr and Brincker, 1993), and those that use the load-displacement (or CMOD)curve to obtain a piece-wise softening curve based on a point-to-point correspondence(Bolzon et al., 1997b; Tin-Loi and Tseng, 2003; Kitsutaka et al., 1994; Kitsutaka,

function-1995, 1997; Uchida et al., 1995) The second group of inverse analysis techniques

is based on the use of relevant points or properties of the load-displacement curve

in specimens with different geometry Two different procedures were proposed byElices et al (1992) depending on the relative size of the cohesive zone developed

in the specimen In conclusion to the indirect approach, the essence of the inverseanalysis methodology, in the sense of Maier et al (2001), requires: (a) to carry outproper experiments on material samples and to record meaningful qualities; (b) toformulate a suitably accurate mathematical model of the experiment for computingthe quantities corresponding to the measured ones; (c) to select a function whichdefines the discrepency between measured and computed quantities; and (d) to min-imise this error function using the parameters as optimisation variables

The classical cohesive crack model, as referred to in the previous section, defines

a relationship between normal crack opening and normal cohesive stresses, and sumes that there are no sliding displacements nor shear stresses along the processzone This assumption is only partially valid for concrete materials Based on ex-perimental observations, it is indeed correct that a crack is usually initiated in pure

τ

~t

~u

Figure 2.9: Mixed mode crack propagation

mode I (i.e opening mode) in concrete, even for mixed mode loading However,during crack propagation, the crack may curve due to stress redistribution or non-proportional loading, and significant sliding displacements develop along the crack

as schematically shown in Figure 2.9 Therefore, it is desirable to incorporate theseshear effects, along both the fracture process zone and the true crack, into the crackmodel

The theory of interacting planes was initially suggested by Maier (1970) Inhis research, the inelastic failure surface can be approximated by a piecewise linearrepresentation of ”yield” surfaces, as in classical plasticity The yield surfaces are afunction of the normal and shear interface stresses Zienkiewicz and Pande (1977)introduced hyperbolic yield surfaces for applications in rock and soil mechanics.Since then, hyperbolas have been used extensively as loading surfaces in the con-stitutive modelling of interface behavior (Carol et al., 1992; ˇCervenka, 1994; Carol

et al., 1997) The failure surface is described by a failure function of normal andinterface traction components, as shown in Figure 2.10

Recent literature provides some examples of normal/shear coupled interface lawsincorporating fracture concepts used to simulate grouting joints in dynamic analyses

1tan(φf)

tan(φf)

c : cohesion

φf : friction angle

σt : tensile strength

Figure 2.10: Hyperbolic failure function after ˇCervenka (1994)

of arch dams (Hohberg, 1992), aggregate-mortar interface in microstructural analysis

of concrete (Vonk, 1992; Stankowski et al., 1993), mortar joints in masonry structures(Lotfi and Shing, 1994), fibre-reinforced composite interfaces (Weihe et al., 1994),concrete-steel bar bond behavior (Cox and Herrmann, 1995), three-dimensional con-stitutive modeling for reinforced concrete (Maekawa et al., 1997), and discrete cracks

in a fracture analysis of concrete gravity dams ( ˇCervenka, 1994) For example, Vonk(1992) formulated a linear model throughout Tensile failure is governed by a ten-sile strength and a reduction angle to take into account the presence of shear stress.Shear failure is governed by a Mohr-Coulomb criterion which is based on cohesionand a friction angle

The idea of modelling is that we would like to have a numerical tool that can scribe in sufficient detail the observations we have made With such a model wewould like to have the ability to make predictions outside the domain that was used

de-for tuning Of course the term ”sufficient detail” is rather vague In principle, wewould like to forecast the fracture response of structures that were not explored be-fore by means of experiments Important for making predictions is that the modelparameters can be determined independently of specimen geometry and boundaryconditions; otherwise they are not real material properties This is of course themain problem Many empirical formulations exist that are applicable only to therange of experiments to which they were fitted.

Computer analysis of concrete structures requires general and robust materialmodels for distributed cracking and other types of strain-softening damage such assoftening plastic-frictional slip The problem can be approached through two types

of models, i.e the discrete crack concept and the continuum construct with smearedcracking For example in the discrete crack model (Ngo and Scordelis, 1967), thecommon node between two elements is split into two when the tensile strength of thematerial is reached One of the disadvantages of this method is the constant change

of the nodal connectivity of the finite element mesh Another disadvantage is thatthe crack growth is along a pre-defined path, i.e along the element edges In or-der to overcome this problem, Ingraffea and Saouma (1985) proposed an interactiveapproach to discrete cracking Following this procedure, the mesh automaticallyadjusted in the vicinity of a crack to accommodate arbitrary crack trajectories

Generalised continuum theories such as (Cosserat and Cosserat, 1909) - or called micropolar - continuum and rate dependent continuum have been developed(Li and Cescotto, 1996) De Borst (2002) gives an overview of continuum damage-based approaches used to study fracture in quasibrittle materials He shows thatthe smeared-crack models can be cast into a damage format and can be conceived

so-as a special cso-ase of anisotropic damage models Evidently, the length scale which

is introduced into the model is the element size, which has a numerical nature,and therefore has to be tuned to the particular boundary value problem that is

analysed When ”smeared cracks” propagate at lines that are inclined to the gridlayout, or when quadratic or higher-order finite elements are used, the numericallyobtained crack band width normally no longer coincides with the element size Var-ious investigators have proposed formulas to estimate the numerical length scale,depending on the interpolation order of the polynominals, the spatial integrationscheme and the expected angle between the crack and the grid lines (Oliver, 1989;Feenstra and De Borst, 1995) Although experience shows that these approachesgenerally work well, they are of heuristic nature and, therefore, can result in grosserrors in particular cases Considering the mathematical difficulties that are inher-ent in smeared representations of cohesive-zone models, the search has continued for

a proper representation of the true discrete character of these models, while allowingfor an arbitrary direction of crack propagation, not hampered by the initial meshdesign Originally, meshless methods (Belytschko et al., 1994) were thought as theanswer to this problem, but they appear to be less robust than traditional finiteelement methods, they are computationally more demanding and the implementa-tion in three dimensions appears to be less straight forward However, out of thisresearch, recently a method has emerged, in which a discontinuity in the displace-ment field is captured exactly (De Borst et al., 2004) It has the added benefit that

it can be used advantageously at different scales, from microscopic to macroscopicanalyses The method makes use of the partition-of-unity property of finite elementshape functions

As discussed earlier, the importance of the cohesive-zone concept for describingfracture in a wide range of engineering materials has been recognised However,its proper numerical implementation has caused problem Essentially, the cohesive-zone concept is a discrete model and cannot be implemented readily in standard,continuum-based finite element methods While the development of special interfaceelements that are equipped with cohesive-zone models is fairly simple, the difficultyremains where to put these interface elements Crack paths are normally not known

(a) (b)

Figure 2.11: Discontinuity on a structured mesh (a); and on an unstructured mesh

(b) (Mo˙es and Belytschko, 2002)

in advance To partly circumvent this difficulty, proposals have been made to insert

interface elements between all continuum elements, to carry out remeshing

proce-dures or to use mesh-free methods Neither of these approaches is entirely

satisfac-tory and it seems that only recently by using partition-of-unity property of finite

element shape functions an elegant and powerful method has been developed to

fully exploit the potential of cohesive-zone models for arbitrary crack propagation

It is able to model delaminations independently of the underlying mesh structure,

to achieve a consistence formulation for large strains and to obtain a smooth

tran-sition from plastifying or damaging continuum to a traction free, macroscopically

observable crack

That extension of the finite element method called the extended finite element

method (X-FEM) (Mo˙es et al., 1999; Belytschko and Black, 1999) has been

devel-oped to model arbitrary discontinuities in meshes (see also Belytschko et al (2001))

This extension exploits the partition of unity property of finite elements (Melenk and

Babuˇska, 1996), which allows local enrichment functions to be easily incorporated

into a finite element approximation while preserving the classical displacement ational setting The X-FEM models the discontinuity in a displacement field alongthe crack path, wherever this path may be located with respect to the mesh Thisflexibility enables the method to simulate crack growth without remeshing In thepapers by Belytschko and Black (1999) and Mo˙es et al (1999), the crack surfaceswere considered free of tractions In the paper by Dolbow et al (2001) contactand friction was incorporated on the crack faces to simulate crack growth undercompression In a recent paper, Belytschko and his co-workers have extended themethod to the case of crack growth involving a cohesive law on the crack faces andstudy its performance The modelling of cohesive cracks using the partition-of-unityproperty of finite elements was also considered recently by Wells and Sluys (2001).

vari-All the constitutive models describing fracture exhibit properties such as peak strain softening and deviations from the normality rule (or Drucker’s pos-tulate) These difficulties, which are inevitable if the constitutive relation shoulddescribe cracking, friction, and loss of cohesion realistically, cause well-known math-ematical difficulties such as ill-posedness of the boundary value problem, spuriouslocalisation instabilities, and spurious mesh sensitivity ( ˇCervenka et al., 2005) Inorder to avoid these difficulties, the constitutive relations must be combined withsome kind of localisation limiter, a condition that prevents the strain from localisinginto a region of measure zero (Baˇzant and Belytschko, 1985) It is often thought thatthe continuum approach cannot be applied to the final stages of failure, in whichdamage localises into large continuous cracks However, the continuum approachcan provide a relatively good (albeit not perfect) model for the propagation of suchcracks The reasons can be explained in connection with the crack band model,which may be regarded as the simplest version of the nonlocal approach The width

post-of the localised damaged band has, in most cases, negligible influence on the results

of structural analysis A zero width, i.e a distinct crack, and a finite width oftenyield about the same results Forcing, through the nonlocal concept, the distinct

crack to spread over a width of several finite element sizes, or forcing a narrow age band to be wider than the real width, is usually admissible, provided that theenergy dissipation per unit length of advance of the band is adjusted to remain thesame It should be noted that such spreading of damage over a width of severalelement sizes is also a convenient way to avoid the directional bias of finite elementmesh (Baˇzant and Planas, 1998).

dam-A large amount of research, made favourable by the advent of powerful ers, has been devoted to the simulation of material behaviour based directly on arealistic modelling of the microstructure - its particles, phases, and the bond betweenthem A spectrum of diverse approaches will be discussed at length in Chapter 4

comput-An extreme example of the continuum approach - in view of the fineness of materialsubdivision - is the numerical concrete of Roelfstra et al (1985); Wittmann et al.(1988); Roelfstra (1988), in which the mortar, the aggregates, and the interfacesare independently modelled by finite elements This requires generation of the ge-ometry of the material (random placement of aggregates within the mortar) andthe detailed discretisation of the elements to adequately reproduce the geometry

of the interfaces With a completely different purpose, but with the same kind ofanalysis, Rossi and Richer (1987); Rossi and Wu (1992) developed a random finitemodel element in which the microstructure is not directly modelled, but is takeninto account by assigning random properties to the element interfaces The commonfeature of these approaches is that, before cracking starts, the displacement field isapproximated by a continuous function

The discrete crack methods, on the other hand, take account of cracking bydefining a boundary to the finite element (or boundary element) mesh along thecrack path, concentrating inelasticity along this boundary and allowing the crack toopen between the finite elements that remain linearly elastic Consideration of thefracture process zone ahead of the crack tip is also essential and can be incorporated