Mixed mode i II III fracture criterion and its application to cement mortar 1

1.1.1.2 Traditional Linear Elastic Fracture Mechanics 5 1.1.2 Laboratory Experiments of Fracture under Mixed Mode Loading 9 1.1.2.1 Traditional Fracture Test Specimens under Mixed 2.1 Cr

MIXED MODE I – II – III FRACTURE CRITERION

AND ITS APPLICATION TO CEMENT MORTAR

ZHONG KUI

(B Eng., Tongji U.; M Eng., NUS)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2002

ACKNOWLEDGEMENTS

The author wishes to express his most sincere gratitude to his Supervisor, Associate Professor Lo Kwang Wei for his invaluable, patient and constructive guidance throughout the research period His in-depth knowledge and expertise in the field of Fracture Mechanics had been a great contribution to this work

The author is furthermore indebted to Research Fellow, Dr T Tamilselvan, and all technicians and colleagues in the Geotechnical Laboratory and Concrete and Structural Engineering Laboratory for their kind co-operation and assistance in the research work

Also, the author would like to take this opportunity to express his deep thanks

to his family for their endless support, encouragement and love

Last but not least, the author gratefully acknowledges the financial assistance offered by the National University of Singapore through grants from university research project RP 960643 and a research scholarship

1.1.1.2 Traditional Linear Elastic Fracture Mechanics 5

1.1.2 Laboratory Experiments of Fracture under Mixed Mode

Loading 9 1.1.2.1 Traditional Fracture Test Specimens under Mixed

2.1 Crack Closure Analysis in θ0 Plane under Pure Mode I or II Loading 24

CHAPTER 3 EVALUATION OF STRESS INTENSITY FACTORS

4.2.3 Determination of Stress Intensity Factors by Finite Element

Analysis 80

4.3.3 Determination of Stress Intensity Factors by Finite Element

Analysis 99

4.4.3 Determination of Stress Intensity Factors by Finite Element

Analysis 120

APPENDIX NUMERICAL AND EXPERIMENTAL RESULTS 156

SUMMARY

In the linear elastic fracture mechanics (LEFM) of brittle and slightly ductile materials, there are, in general, three modes of deformation near a crack tip The combination of modes I and II deformation has constituted the major portion of investigation in traditional fracture mechanics To study crack behaviour under pure mode III loading, the closure analysis of an infinitesimal crack extension has been carried out, and relationship between the stress intensity factor and energy release rate established After that, the closure analysis of a small crack extension along the generalized θ plane, under mixed mode I–II–III loading, has been conducted On the

basis of the unified model (Lo et al., 1996a), which specifies the in-plane crack

behaviour under mixed mode I-II loading, a three-dimensional mixed mode I–II–III fracture criterion has been derived herein, by adopting the simple conversion of pure to mixed mode loading energy, in the ratio of their corresponding fracture energies (Lo et al., 2001)

In verifying the proposed fracture criterion, two experimental methods have been designed using cement mortar specimens, in order to conduct three-dimensional mixed mode fracture tests, as well as the corresponding pure mode fracture tests In the first case, a traditional compact tension specimen has been adopted, which was subjected to mixed mode I–III loading, as well as the respective pure mode loading, through a loading fixture An advantage of the method adopted is that the full range of

mixed mode I–III loading combinations may be readily applied to the same specimen configuration

As to the second case, a beam specimen has been tested under mixed mode I–II–III loading A groove, which was rotated vertically as well as horizontally with respect to the beam section, leaving a truncated V-shaped throat segment, was formed

in the specimen Each test specimen had different orientations of the groove, so as to provide differing mixed mode loading conditions at the crack front In doing so, a crack was initiated and hence guided to extend along the V-shaped throat segment as a mixed mode fracture

The stress intensity factors of the respective modes of deformation were evaluated numerically by finite element analysis As a result of the numerical analyses and measurements made in laboratory tests, reasonable agreement was obtained between the proposed 3-D fracture criterion and the experimental results obtained for its verification

The essential findings of the proposed fracture criterion and experimental results have been published in an invited keynote lecture at the Third International

Conference on Micro Materials (MicroMat 2000) (Lo, et al., 2000), the 10th

International Conference on Fracture (10 ICF) (Zhong et al., 2001) and an international refereed journal (Lo et al., 2002) These publications collectively constitute the contents of this thesis, where they are dealt with in greater detail

NOMENCLATURE

δa small crack extension

εxx, εyy, εzz normal strain components

γxz, γyz shear strain components

ν Poisson’s ratio

θ direction of plane of interest with respect to existing crack plane

θ0 self-similar direction, that is θ = 0

θ0C crack extension in self-similar direction

θC general direction of crack extension

σθθ in-plane circumferential stress

τθz, τxz, τyz out-of-plane shear stress referred to parent crack tip

τrθ in-plane shear stress referred to parent crack tip

Aλ, Bλ constants

a0 pre-crack length

E Young’s modulus

FC fracture load measured in mixed mode fracture test

FIC fracture load measured in pure mode I fracture test

FIIC fracture load measured in pure mode II fracture test

FIIIC fracture load measured in pure mode III fracture test

f(θ) function of θ

G rate of energy release

Gθ mixed mode rate of energy release rate along θ plane

G q strain energy release rate for non-elastic materials

GI mode I rate of energy release

GII mode II rate of energy release

GIII mode III rate of energy release

GC critical rate of energy release

GIC mode I critical rate of energy release

GIIC mode II critical rate of energy release

GIIIC mode III critical rate of energy release

GIθ mode I rate of energy release with respect to θ plane

GIIθ mode II rate of energy release with respect to θ plane

KI mode I stress intensity factor

KII mode II stress intensity factor K′I mode I stress intensity factor referred to the extended crack tip K′II mode II stress intensity factor referred to the extended crack tip K′III mode III stress intensity factor referred to the extended crack tip

KIθ unified pure mode I stress intensity factor

KIIθ unified pure mode II stress intensity factor

KI0 mode I stress intensity factor with respect to θ0 plane

KII0 mode II stress intensity factor with respect to θ0 plane

KIII0 mode III stress intensity factor with respect to θ0 plane

KIC mode I fracture toughness

KIIC mode II fracture toughness

KIIIC mode III fracture toughness

L length of beam specimen

r radial distance from crack tip

uθθ circumferential displacement

u′θθ circumferential displacement referred to extended crack tip

u′rr radial displacement referred to extended crack tip

u xx , u yy , u zz displacements along x, y and z directions

u′zz displacement along z direction referred to extended crack tip

LIST OF FIGURES

(1989) 11

loading 41

Figure 3.3 Determination of stress intensity factor at crack tip by

extrapolation 62

Figure 3.4 Formation of triangular prismatic quarter-point crack tip

Figure 4.2 Typical compressive load – axial strain curves for evaluation of

tests 79

fracture 89

Figure 4.14 Failure of modes I and II fracture test specimens 90

Figure 4.18 Geometrical configuration of partial loading fixture and

Figure 4.22 Mid-sectional view showing details at crack front (left half of

Figure 4.23 Sectional view of twenty layers of quarter-point triangular

prismatic elements around crack front with cut-out view (left

Figure 4.24 Finite element modelling of left half of loading fixture and

coupling 104

Figure 4.26 Distributions of stress intensity factors across throat of

Figure 4.29 Comparison of unified fracture criterion with mixed mode I–III

test 116

Figure 4.35 Sectional view showing details of crack front (right half of test

specimen) 124

Figure 4.36 Sectional view of twenty layers of quarter-point triangular

prismatic elements around crack front with cut-out view (right

Figure 4.37 Distributions of stress intensity factors across crack fronts of

loading 130

LIST OF TABLES

CHAPTER 1 INTRODUCTION

Most materials contain some cracks, or they have to be assumed to contain cracks, for life assessment purposes at the design stage Propagation of these initial cracks results in failure of a structure Fracture mechanics is thereby employed to study the response and failure of structures as a consequence of crack initiation and propagation It provides engineers with a method of characterizing fracture behaviour

in terms of the structural parameters of stress and flaw size

A crack in a solid can be stressed in three different modes, as illustrated in Figure 1.1 Normal stress give rise to the “opening mode” or mode I loading The displacements of the crack surfaces are perpendicular to the plane of the crack In-plane shear results in the mode II or “sliding mode”, in which the displacement of crack surfaces is in the plane of crack, and perpendicular to the leading edge of crack The “tearing mode” or mode III is caused by out-of-plane shear Crack surface displacements are in the plane of the crack and parallel to the leading edge of the crack The superposition of the effects of all the three modes describes the general case of loading, which is referred to as mixed mode loading

(a) mode I

z x y

(c) mode III (b) mode II

Figure 1.1 Three modes of loading

Traditionally, the main focus of research of so-called linear elastic fracture mechanics (LEFM), with which the present work is concerned, has been the in-plane problem, that is the pure modes I and II, and mixed mode I – II, loading In practice, however, engineering structures undergo mixed mode loading involving in-plane and/or out-of-plane bending and torsion and/or tension or compression A combination

of this loading can lead to mixed mode I–II–III fracture Theoretical and experimental studies have only just begun to deal with the combination of all three fracture modes as they occur in three-dimensional crack problems

In this chapter, a brief historical account of basic fracture concepts as well as existing methods of laboratory fracture testing will, firstly, be reviewed Thereafter, the scope and objectives of the present work will be set out

1.1 Literature Review

1.1.1 Conventional Fracture Mechanics and Mixed Mode Fracture Criterion

1.1.1.1 The Pioneering Work of Griffith and Irwin

The founding of fracture mechanics is generally attributed to Griffith (1920), who deduced the pure modes I and II critical rates of energy release, GIC and GIIC

respectively, as material constants for determining crack extension Griffith postulated that the reduction in strain energy upon crack formation would transform into surface energy at the crack face Furthermore, he showed that the surface energy of glass

obtained from fracture testing was in good agreement with the surface energy of glass

obtained by way of measuring its surface tension experimentally However, his theory

was considered to have limited application since it was applicable only to brittle

materials, whereas most of the structural materials exhibit ductile properties during

failure

The next significant contribution to fracture mechanics was Irwin’s (1957)

quasi-brittle fracture theory, in which he rationalized that Griffith’s theory would be

applicable to most materials, if the surface energy were replaced by irreversible energy

dissipated in a thin layer of plastic strain beneath the surface of the crack He also

introduced the concept of pure modes I and II stress intensity factors, KI and KII, and

showed, by closure analysis, that their critical values at a crack tip, which are the

fracture toughness KIC and KIIC, respectively, correspond directly to Griffith’s critical

rates of energy release, GIC and GIIC respectively, namely

'

KG

2 I I

2 II II

E

C

where E′ represents Young’s modulus E in plane stress, and E/(1-ν2) in plane strain, in

which ν is Poisson’s ratio Equations (1.1) and (1.2) establish the fracture toughness as

a material constant on an equal footing with the critical energy release rate for

determining crack extension

1.1.1.2 Traditional Linear Elastic Fracture Mechanics

Since Irwin’s work of the preceding discussion, the development of LEFM

has developed in various directions, as follows

It has been demonstrated (Lo et al., 1996a) that Irwin’s concept of the modes I

and II stress intensity factors may be appropriately specified in terms of the

0

and

r r

limK

0

respectively, where σθθ is the circumferential stress and τrθ the shear stress, in polar

coordinates, and r is the radial distance from the crack tip However, Irwin's

interpretation of the stress intensity factor is strictly pertinent to the θ0 plane only, so

that its application to some other θ plane, which would be untenable, has led to various

inconsistencies in the subsequent development of fracture mechanics The oversight of

the sole emphasis placed by Irwin's derivation, on the θ0 plane, seems to have been

perpetrated, in turn, by two basic assumptions Firstly, that the stress intensity factors,

KI and KII, are absolute constants of a given boundary value problem, and secondly, that a crack will extend when either KI or KII reach their respective critical values of

KIC and KIIC However, as borne out by the following discussion, not only are the factors, KI and KII, strictly pertinent to the θ0 plane only, and therefore more explicitly denoted as KI0 and KII0 (where the subscript "0" would refer to the same plane), but also the stress intensity factors KIθ and KIIθ, which arise from Panasyuk (1968) and

the formulation of the unified model (Lo et al., 1996a), would vary with angle θ (where

-π≤θ≤π), and in so doing, constitute the appropriate parameters for predicting fracture along the generalised θC plane The unified factors are as specified by equations (2.70) and (2.71), of subsequent §2.3

In a later development, crack kink analysis (Lo, 1978; Cotterell and Rice, 1980; Hayashi and Nemat-Nasser, 1981; He and Hutchinson 1989) apparently evolved out of a reaction to the above lack of consensus in traditional fracture mechanics, particularly with regard to the need to determine non self-similar crack extension Accordingly, whereas the notion of an Irwin-type stress intensity factor (which is referred to the current crack plane) is maintained as the basis for determining crack extension, it is, in contrast with the traditional approach, applied to both self-similar

and non self-similar crack extension In doing so, however, the closure analysis of a

non self-similar crack extension becomes unnecessarily elaborate, in that it is

necessary to provide an à priori kinked branch crack, thereby distinguishing it from

that of a self-similar crack extension, which according to Irwin effectively takes place directly from the existing crack tip On the contrary, non self-similar crack extension should be addressed at source, that is on the basis of the near field stresses of the

existing crack tip, rather than from the standpoint of the considerably more complex boundary value problem of a crack in its kinked state (Lo et al., 1996b)

1.1.1.3 Existing Mixed Mode Fracture Criteria

In the past few decades, extensive efforts had been directed at modelling crack extension and fracture subjected to mixed mode loading, and various criteria for crack extension under mixed mode loading have been proposed

The maximum tangential stress criterion was proposed by Erdogan and Sih (1963) It states that a crack would extend at an angle where the tangential stress became a maximum, and fracture would start when the maximum tangential stress reached a critical value which equalled the fracture stress in uniaxial tension This criterion had been widely used because of its simplicity and support by experimental observations It has been proven by Nuismer (1975) to be equivalent to the maximum strain energy release rate criterion of Ichikawa and Tanaka (Hussain et al., 1974) The application of this criterion may be found in the work of several authors, such as Gdoutos (1984), Pook (1985a), Nayeb-Hashemi et al (1987), Hyde and Chambers (1988), and Mahajan and Ravi-Chandar (1989) However, the work of Tanaka (1974), Royer (1986) and Abdel-Mageed and Pandey (1992) do not support the criterion

The minimum strain energy density criterion was proposed by Sih (1974a, 1991) The crack is assumed to grow in a direction, along which the strain energy density factor reaches a minimum value, and fracture occurs when this factor reaches a critical value It has been experimentally investigated by a number of researchers,

including Sih (1974a, 1974b), Sih and Barthelemy (1980) and Badaliance (1980) A main advantage of the criterion is its ease and simplicity of application, as well as its ability to handle various combinations of loading conditions However, several authors have reported contradictory observations to it (Tanaka, 1974; Abdel-Mageed and Pandey, 1991; 1992), and questioned its theoretical basis, since the physical meaning

of the criterion is not clear (Theocaris and Andrianopoulos, 1984)

Other criteria have also been proposed, such as those based on the J-integral (Hellen and Blackburn, 1975), dilatational strain energy density (Theocaris and Andrianopoulos, 1982; Theocaris et al., 1982), maximum strain energy release rate at the propagation and initiation of a small kinked crack (Ueda et al., 1983), maximum circumferential strain (Maiti and Smith, 1984), vector crack tip displacement (Li, 1989), tangential stress and strain factors (Wu and Li, 1989), and maximum tangential strain (Chambers et al., 1991)

Among the aforementioned criteria, the maximum tangential stress and minimum strain energy density criteria have been the most widely used in mixed mode crack growth studies The application of these two criteria have, moreover, been extended to mixed mode I – II – III loading by Chen et al (1986), where it was found that the latter criterion resulted in better predictions than the former However, it was reported that none of the abovementioned criteria was able to give satisfactory results under all loading conditions (Qian and Fatemi, 1996) It is noteworthy that most of the conventional fracture mechanics parameters have been used to predict crack growth under mixed mode loading, including the stress intensity factor, strain energy density, J-integrals and crack tip opening displacement (CTOD) However, the use of these

parameters has been limited to LEFM problems

1.1.2 Laboratory Experiments of Fracture under Mixed Mode Loading

1.1.2.1 Traditional Fracture Test Specimens under Mixed Mode Loading

An assortment of specimen geometries, which have been used to produce different combinations of mixed mode loading, under varying test conditions, have been investigated by Richard (1989) In this connection, an evaluation of corresponding fracture criteria and suitable specimens for studies of crack growth under mixed mode loading was made The investigation included the ability to apply the full range of mixed mode load combinations on the specimen, and its compactness, ease of manufacture, and ease of clamping and loading

Richard (1989) proposed nine different specimens which are often used in mixed mode fracture studies, namely, the plate specimen with inclined crack under tension (S1), plate specimen with inclined edge crack under tension (S2), disc specimen with inclined central crack (S3), cruciform specimen with inclined central crack (S4), shear specimen with inclined central crack (S5), tubular specimen with inclined crack under torsion (S6), tubular specimen with transverse crack under combined tensile and torsional stress (S7), three- or four-point bending and shear specimen with offset edge crack (S8), and compact tension and shear specimen (S9) ,

as shown in Figure 1.2

Other specimen geometries, such as shown in Figure 1.3, have also been used

in mixed mode crack growth studies, for example, the: (a) double oblique edge crack specimen (Lal, 1970); (b) general, compact tension specimen with inclined loads (Mahanty and Maiti, 1990); (c) centre-cracked tension specimen loaded in tension and shear (Otsuka and Tohgo, 1987); (d) double compact tension specimen with inclined loads (Chambers et al., 1991); (e) four-point bending specimen with penny-shaped pre-crack on side surface (Tohgo et al., 1990); (f) three-point bending specimen with through-thickness pre-crack on side surface (Pook, 1985a); and (g) solid round specimen with circumferential crack under torsion (Pook, 1985b)

All the specimens of Figure 1.2 are meant for mixed mode I – II fracture testing, and so are specimens (a) to (d) of Figure 1.3 Specimen (e) of Figure 1.3 is aimed at studying crack growth behaviour under mixed mode II – III loading, while specimens (f) and (g) are designed to produce mixed mode I – III loading

1.1.2.2 Fracture Tests under Mixed Mode I – II Loading

Considerable attention has been given to the study of crack growth under mixed mode I–II loading Most of the experiments on crack growth in metal specimens, under mixed mode I–II loading, have been conducted on a plate specimen with an inclined central crack, which is subjected to tension (vide specimen S1 of Figure 1.2) Most of the proposed criteria for crack growth direction provided satisfactory predictions However, significant discrepancies occurred when mode II loading was predominant (Williams and Ewing, 1972) This discrepancies can be contributed to the specimen configuration, wherein the compactness reduces significantly when mode II loading are predominant

Figure 1.2 Mixed mode fracture test specimens proposed by Richard (1989)

12Figure 1.3 Additional specimen configurations used in mixed mode

fracture tests

Beam specimens have been widely used in experimental studies on cement-based materials Jenq and Shah (1988) conducted mixed mode experiments on

a series of three-point bending beam specimens made of cement paste, mortar and concrete A similar three-point bending configuration, but with two notches, was tested

by Swartz et al (1988a) On the other hand, Arrea and Ingraffea (1981), Swartz et al (1988b), and Schlangen and van Mier (1990) proposed the four-point shearing of single-notched beam specimens, while a double-notched beam specimen was designed

by Iosipescu (1967) and later adopted by Bazant and Pfeiffer (1985) In addition, an extensive list of research dealing with the measurement of fracture properties of concrete and rock, subjected to mixed mode I–II loading, has been provided by Carpinteri and Swartz (1991)

1.1.2.3 Fracture Tests under Mixed Mode I–III Loading

Cylindrical specimens have often been used to study crack growth behaviour involving mode III loading Mixed mode I - III tests were conducted on circumferentially-notched steel and aluminium alloy specimens by Yoda (1979), Pook (1985b), Suresh and Tshegg (1987) and Tschegg et al (1992), while mixed mode fracture tests have been conducted on similar specimens of brittle polymethyl methacrylate (Davenport and Smith, 1993) The mode III fracture toughness and fracture energy of concrete were also measured using similar specimens, by Xu and Reinhardt (1989) and Yacoub-Tokatly and Barr (1989), respectively

Hollow concrete cylinders were used by Keuser and Walraven (1989) to study fracture under torsional loading, as well as combined torsional and tensile loading A

tubular cylindrical specimen for direct torsional testing was proposed by Luong (1989) for the determination of the mode III fracture strength of concrete and rock materials Sarkar (1989) reported the results of an investigation on hollow, concrete specimens tested under the combined action of compression and torsion to failure, and found that none of the existing failure theories adequately explained the results

In other experimental work by Pook (1993) and Kamat et al (1998), the traditional compact tension test specimen was used, except that its geometry was modified to include an angled slot crack, so that the crack plane would not have to be perpendicular to the uniaxial force The angle was varied to produce different states of mixed mode I−III loading at the crack tip As an alternative, Kumar et al (1994) designed a triple “pantleg” configuration with two crack fronts, which produce near pure mode III loading

Ueda et al (1983) conducted brittle fracture tests on plate specimens of polymethyl methacrylate, with a three-dimensionally inclined centre notch, under uni- and bi-axial loading, while Farshad and Flueler (1998) proposed a method of anti-clastic plate bending (or plate twist) to apply a relatively pure shear stress field, and suggested that it might have the potential to measure the mode III fracture toughness Pook (1985a) adopted a three-point bending beam specimen, with a through-thickness pre-crack on its side surface, to study crack growth behaviour under mixed mode I and III loading A four-point bending and shear beam specimen, with a penny-shaped surface crack, was used by Tohgo et al (1990) to produce mixed mode

II and III loading

Since there is no standardized test method, it is difficult to evaluate the experimental results of the various methods Indeed, conflicting experimental results may be found in the literature Although the general consensus would appear to be that

a mode III loading component would reduce the resistance to a mode I brittle fracture (Shah, 1974; Chiang, 1978; Ueda et al., 1983), some researchers have found that transverse shear has no apparent effect on the mode I fracture toughness (Yoda, 1979; Rosenfield and Duckworth, 1987; Keuser and Walraven, 1989) On the other hand, it has been reported that different materials tend to exhibit different degrees of such dependence (Suresh and Tschegg, 1987; Kumar et al., 1994)

1.1.2.4 Fracture Tests under Mixed Mode I – II – III Loading

As shown in Figure 1.4, Richard and Kuna (1990) developed a loading device

in order to produce superimposed loading composed of all three modes The device consists of two force induction elements (1 and 2) which are diametrically connected

to one another by a centre-pre-cracked specimen (3), forming an octant of a sphere

The loading device is connected to a testing machine, which applies the tensile, F, by

way of threaded holes drilled into the force induction elements, into which the

transition pieces, 8 and 9, pass The direction of F may be varied according to which

pair of diametrically-opposite holes is chosen, and the specimen correspondingly subjected to pure tensile, in-plane or anti-plane shear, or any combination of the three, loading The stresses between the force induction elements and the ends of the specimens are transmitted through bolts 4-7 Fracture data of plexiglas and aluminium specimens under mixed mode I–II–III, as well as the respective pure mode, loading were thus obtained The advantage of the set-up is that it may be carried out on a

16Figure 1.4 Device for superimposing states of loading on fracture

specimen (Richard and Kuna, 1990)

tensile-testing machine, and various loading conditions may be applied by the device

I–II–III, loading at the fronts of the corresponding main notches By using four failure planes in different regions of the sample, the influence of inhomogeneity of the concrete, for instance due to poor compaction, may be minimized Special care has to

be exhibited in the loading arrangement, however, to ensure that uniformly distributed loading would be applied to the pair of L-shaped loading blocks at both ends of the specimen It was found that the test set-up was capable of investigating the behaviour

of plain and fibre-reinforced concrete, under mixed mode loading

Hyde and Aksogan (1994) adopted an axisymmetric bar-type specimen containing conical “crack-like” external flaws to obtain fracture data from mixed mode I–II–III loading, as shown in Figure 1.6 Modes I and II crack tip loading conditions were created in the specimen by subjecting it to an axial load, while mode III crack tip conditions were produced by applying a torsional load Although the specimen does not cover every KI : KII : KIII ratio, it was reported that any KI : KIII ratio could be covered, and, together with the results from the “compact mixed mode specimen” of Hyde and Chambers (1988), which covers any range of KI : KII ratios, a wide range of mixed mode I, II and III fracture loading could be obtained

Figure 1.5 Cube specimen proposed by Arslan et al (1991)

Figure 1.6 Axisymmetric bar-type specimen proposed by Hyde and

Aksogan (1994)

T

T F

F

conical flaw

specimen

Apart from the preceding reports, laboratory tests aimed at mixed mode I–II–III fracture are rarely found in the literature, and this may be due to the perceived complexities of specimen geometry and loading configuration that would be involved

in such an undertaking

1.1.3 Fracture in Cement Mortar

It is generally known that the fracture path in cement paste is, in contrast to plain concrete, planar, rather than tortuous (Li and Maalej, 1996) In addition, hardened cement paste exhibits almost perfectly linear elastic, brittle behaviour under compression as well as tension (Mindess, 1983) Thus, it has been found (Mindess et al., 1977; Birchall et al., 1981) that Griffith’s theory is applicable to cement paste

When fine aggregates are added to the cement paste matrix, a mortar is obtained According to Jenq and Shah (1985a), when a 46% by volume fraction of fine aggregates was added to a cement paste matrix, the fracture toughness of material increased by 55% A similar increase in fracture toughness was reported by Gustafsson (1985)

1.1.4 The Unified Model

In view of the anomalies of conventional fracture mechanics described in

§1.1.1.2, the unified model, which is essentially a natural extension as well as

generalization of Irwin’s approach, was developed by Lo et al (1996a) In contrast with the approach, which only caters to pure mode fracture along the θ0C direction, due

to pure mode loading on the θ0 plane, consideration is given to either pure or mixed mode crack extension along the general θC plane (where -π < θC < π), due to either pure or mixed mode applied loading In doing so, it was found that there is a distinction between the traditional stress intensity factors, KI and KII, and the proposed unified stress intensity factors, KIθ and KIIθ, respectively The factors, KI and KII, are noteworthy, only in that they may be related directly to corresponding pure modes of applied loading in the far field, as referred to the θ0 plane The unified factors, KIθ and

development of fracture in the generalized θC direction

The unified model may be represented by a fracture surface defined in terms

of the fracture toughness ratio K CR = KIC / KIIC and normalized, traditional modes I and

II stress intensity factors, K IR = KI0 / KIC and K IIR= KII0 / KIIC, respectively By means

of the model, the foregoing anomalies may be rationalized and hence resolved (Lo et al., 1996a) In addition, the model has predicted the extension of true shear fracture with reasonable accuracy, and it has been shown that laboratory test results obtained for a variety of engineering materials conform with the proposed fracture envelope The envelope is based on the conversion of the pure mode loading energy to mixed mode loading energy, in proportion to their corresponding fracture energies (Lo et al., 1996a)

1.2 Scope and Objective of Research

In view of the foregoing literature review, the present research study has two objectives, namely: (i) to develop a generalized criterion for mixed mode I–II–III fracture; and (ii) to verify the proposed fracture criterion by laboratory tests on cement mortar specimens

The original unified model is capable of predicting fracture behaviour of

planar fracture problem, that is pure mode I, pure mode II, and mixed mode I–II fracture In order to extend the model to handle a non-planar fracture problem, that is including mode III deformation, a generalized mixed mode I–II–III fracture criterion would be required This will be undertaken in Chapter 2, on the basis of a closure analysis of the generalized θ plane, under mixed mode I–II–III deformation, and then adopting the conversion of pure to mixed mode loading energy, in the proportion of

their corresponding fracture energies, as in the case of the unified model The proposed

criterion may be represented by a three-dimensional fracture surface, defined in terms

of the normalized, unified modes I, II and III stress intensity factors, KIθ/KIC, KIIθ/KIIC

and KIIIθ/KIIIC, respectively (Lo et al., 2000)

The application of the finite element (FE) method to LEFM, and numerical methods of calculating the stress intensity factors from two- and three-dimensional modelling, will then be dealt with in Chapter 3

As a means of verifying the proposed criterion, laboratory test methods based

on cement mortar specimens were designed To start with, a beam specimen was

adopted to evaluate the pure modes I and II fracture toughness Next, mixed mode I–III fracture, as well as the respective pure mode fracture tests, was conducted on compact tension specimens Finally, mixed mode I–II–III fracture was performed on beam specimens, including a three-dimensionally inclined throat segment Two- or three-dimensional finite element analyses of the corresponding LEFM problems were conducted, as a result of which, the stress intensity factors due to the respective modes

of loading were evaluated numerically The results of the numerical analyses and test measurements made in the laboratory are presented in Chapter 4 and shown to have reasonably good agreement with the proposed fracture criterion (Lo et al., 2001, 2002; Zhong et al., 2001)

Finally, in Chapter 5, various conclusions will be drawn on the research findings and recommendations made on future directions to be taken to develop them further