Advanced in Strength of Materials

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Advances in

Strength of Materials



Advances in

Strength of Materials

Selected, peer reviewed papers from theStrength of Materials Laboratory at 85 years ,

21 – 22 November 2008,Timisoara, Romania

Edited by

Liviu MARSAVINA

TRANS TECH PUBLICATIONS LTD

Switzerland • UK • USA



Copyright 2009 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this book may be reproduced ortransmitted in any form or by any means without the written permission of the publisher.

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Volume 399 of Key Engineering Materials

ISSN 1013-9826

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Strength of Materials Laboratory at 85 years

21 – 22 November 2008, Timisoara, Romania

Local Organizing Committee

Prof. Tiberiu D. BABEUProf. Liviu BERETEUProf. Ion DUMITRU

Prof. Nicolae FAURDr. Mihai HLUŞCU

Prof. Liviu MARSAVINA

Eng. Radu NEGRUProf. Nicolae NEGUTProf. Pavel TRIPA

International Scientific Committee

Prof. Holm ALTENBACH - Halle, GermanyProf. Costică ATANASIU – Bucureşti, Romania

Dr. Vyacheslav BURLAYENKO – Harkov, UkrainaProf. Ionel CHIRICA – Galati, Romania

Prof. Dan CONSTANTINESCU – Bucureşti, Romania

Prof. Eduard Marius CRACIUN – Constanta, RomaniaProf. Ioan CURTU – Brasov, RomaniaProf. Geert DeSCHUTTER – Gent, Belgium

Prof. Lluis GIL – Barcelona, SpainProf. Mihail HARDAU – Cluj, Romania

Prof. Nicolae ILIESCU – Bucureşti, Romania

Prof. Ivelin IVANOV – Ruse, Bulgaria;Prof. Vasile NĂSTĂSESCU – Bacău, RomaniaProf. Guy PLUVINAGE – Metz, France

Dr. Marko RAKIN – Belgrad, SerbiaProf. Mircea RAŢIU – Oakland, USA

Prof. Marin SANDU - Bucureşti, RomaniaProf. Tomasz SADOWSKI – Lublin, Polonia

Prof. Stojan SEDMAK – Belgrad, Serbia

Prof. Vadim SILBERSCHMIDT – Loughborough, UK



Preface

The Strength of Materials Laboratory of POLITEHNICA University in Timisoara

celebrated in November 2008 its 85th

anniversary. The Laboratory was officially openedin 1923, in the presence of the Governing King FERDINAND and Prime Minister I. C.

BRATIANU. The laboratory was equipped with the support of the Rector of Politehnica

School in Timisoara, professor Traian LALESCU, and Strength of Materials professor,C. C. TEODORESCU. The first tests attended by students were held in 1923, and the

first student “ Laboratory notebook ” was published in 1924. In 1931, the laboratory

hosted the visit of King CAROL the Second and Prime Minister N. IORGA.

Over the years, the Laboratory has developed considerably and now represents one of themain facilities for Strength of Materials research and teaching in the Western side of

Romania. Special mention should be made of the important contributions in Strength ofMaterials and Fatigue - theoretical, experimental and teaching aspects - of Acad. Stefan

NADASAN, Prof. Lazar BOLEANTU, Prof. Iosif HAJDU, Dr. Mircea RATIU, Prof.Tiberiu BABEU, Prof. Ionel DOBRE, Prof. Constantin CRISTUINEA Prof. Nicolae

NEGUT, Prof. Ion DUMITRU.

In order to celebrate the 85th

anniversary of the lab, POLITEHNICA University of

Timisoara organized an international conference - „Strength of Materials Laboratory at

85 years”. The conference that brought together scientists from 12 countries representeda forum for the latest analytical, experimental and numerical developments in the field of

Strength of Materials, Fracture Mechanics and Fatigue. The conference program

consisted of five plenary sessions with key note lectures provided by the attendees andone poster session.

This volume is a collection of the papers presented at the „Strength of Materials

Laboratory at 85 years” conference, held in Timisoara, Romania, during 21 - 22 November, 2008. The proceedings of the conference were structured in four parts:

Metallic Materials, Composite Materials, Construction and Building Materials and Bio –

Materials. The proceedings were published with the financial support of NationalAuthority for Scientific Research (ANCS) of the Romanian Ministry of Education,

Research and Youth through the Research for Excellence grant 202/20.07.2006.

The Editor acknowledges the careful work and support of the members of the LocalOrganizing Committee and International Scientific Committee that were essential to the

success of the conference. As well the participation of invited lecturers: Holm

ALTENBACH (Germany), Geert DeSCHUTTER (Belgium), Lluis GIL (Spain), IvelinIVANOV (Bulgaria), Guy PLUVINAGE (France), Marko RAKIN (Serbia), Tomasz

SADOWSKI (Poland), Stojan SEDMAK (Serbia) and Vadim SILBERSCHMIDT (UK)

is also acknowledged.

Editor

Liviu MARSAVINA



Table of Contents

Committees

Preface

I. Metallic Materials

Constraint Parameter for a Longitudinal Surface Notch in a Pipe Submitted to InternalPressureM. Hadj Meliani, M. Benarous, Z. Azari and G. Pluvinage 3

Application of Complete Gurson Model for Prediction of Ductile Fracture in Welded SteelJointsB. Međo, M. Rakin, N. Gubeljak and A. Sedmak 13

Fatigue Crack Growth under Variable Amplitude Loadings: A Theoretical StudyI. Dumitru and A. Cernescu 21

Fracture Mechanics and Non-Destructive Testing for Structural Integrity AssessmentS. Sedmak and A. Sedmak 27

Bulk Amorphous Soft Magnetic Iron Based Alloy with Mechanical Strength and CorrosionResistanceV.A. Şerban, C. Codrean and I.D. Uţu 37

Fractal Analysis of Fracture Surfaces of Steel Charpy SpecimensC. Secrieru and I. Dumitru 43

The Assessment of Remaining Life of Chemical Reactor Exposed to Creep and FatigueH. Mateiu, T. Fleşer and A.C. Murariu 51

II. Composite Materials

On the Time-Dependent Behavior of FGM PlatesH. Altenbach and V.A. Eremeyev 63

Impact Fatigue of Adhesive JointsV.V. Silberschmidt, J.P. Casas-Rodriguez and I.A. Ashcroft 71

Multiscale Modelling of Damage Processes in Polycrystalline Ceramic Porous CompositesT. Sadowski and L. Marsavina 79

The Effect of Geometry and Material Properties on the Load Capacity of Single-StrappedAdhesive Bonded JointsM. Sandu, A. Sandu, D.M. Constantinescu and Ş. Sorohan 89

Behaviour Analysis of Adhesive Joints Used in Ship StructuresI. Chirica, E.F. Beznea and A. Chirica 97

Evaluation of Interlaminar Damage and Crack Propagation through Digital ImageCorrelation MethodD.M. Constantinescu, M. Sandu, L. Marsavina, R. Negru, M.C. Miron and D.A. Apostol 105

Experimental and Numerical Analysis of Buckling Behaviour of the Ship Plates Made of Composite MaterialsE.F. Beznea, I. Chirica and R. Chirica 113

Polyurethane Foams Behaviour. Experiments versus ModelingL. Marsavina, T. Sadowski, D.M. Constantinescu and R. Negru 123

Investigations Regarding the Thermoplastic Resistance Evaluation with SimulatedImperfectionsA.C. Murariu, V.I. Safta and T. Fleşer 131

III. Construction and Building Materials

A Pull-Shear Test for Debonding of FRP- Laminates for Concrete StructuresL. Gil, J.J. Cruz and M.A. Pérez 141



b Advances in Strength of Materials

Influence of Cracks on the Service Life of Concrete Structures in a Marine EnvironmentK. Audenaert, L. Marsavina and G. De Schutter 153

Experimental and Numerical Analysis of the Behaviour of Ceramic Tiles under ImpactD.M. Constantinescu, M. Sandu, E. Volceanov, M. Gavan and Ş. Sorohan 161

Micromechanical Material Model of Wooden Veneers for Numerical Simulations of Plywood Progressive Failure

I.V. Ivanov and T. Sadowski 169Mathematical Modelling of the Crack Propagation in Wood MaterialsE.M. Craciun and T. Sadowski 177

IV. Bio-Materials

Experimental Assessment by Finite Elements Method of the Residual Stress State and of theHeat Flow from the Laser Weldings of the Alloys of CoCrMo Used in RPD (RemovablePartial Dentures) TechnologyC. Bortun, N. Faur, A. Cernescu and L. Sandu 185

Investigation of Implant Bone Interface with Non-Invasive Methods: Numerical Simulation,Strain Gauges and Optical Coherence TomographyS. Antonie, C. Sinescu, M. Negrutiu, C. Sticlaru, R. Negru, P.L. Laissue, M. Rominu and A.G.Podoleanu 193

Biomechanical 3D Analysis of Stress Induced by Orthodontic ImplantsC. Szuhanek, N. Faur and A. Cernescu 199

In Vitro Experimental Testing of a Cervical Implanted UnitD.I. Stoia, N. Faur, M. Toth-Taşcău and L. Culea 205

Stress Analysis of the Human Skull due to the Insertion of Rapid Palatal Expander withFinite Element Analysis (FEA)L. Culea and C. Bratu 211



I. Metallic Materials I. Metallic Materials



Constraint Parameter for a Longitudinal Surface Notch

in a Pipe Submitted to Internal Pressure

M. Hadj Meliani1,a

, M. Benarous2,b

, Z. Azari3,c

G. Pluvinage3,d

1

J0201/03/01/05, Département de Mécanique. BP 151, Hay Salem, Université de Chlef, Algérie.2 Laboratoire de Physique théorique et Physique des Matériaux. Université de Chlef, Algérie.

3 Laboratoire de Fiabilité Mécanique, LFM-ENIM, île de saulcy 57045, Université de Metz, France

[email protected], [email protected], [email protected], [email protected]

Keywords: Constraint, T-stress, Finite Element Analysis, Castem2000.

Abstract. The use of two parameters fracture mechanics criterion as a tool for structural design and

analysis has increased significantly in recent years. First, we discuse the elastic solution for the

stress distribution at crack tip for two dimensional geometries and particularly constraint as T-stressunder various loading conditions. Secondly, using notch fracture mechanics and particularly the

Volumetric Method approach, we study the stress distribution at the tip of a notch in pipes

submitted to internal pressure. The Notch Stress Intensity Factor K ρ and the effective T-stress are

combined into a two-parameter fracture criterion ( K I ρ-T ef ). This approach is then used to quantify

the constraint of notch-tip fields for various pipe geometry and loading conditions.

Introduction

Recent numerical and experimental studies have attempted to describe fracture in terms of two

parameters [1] characterising the crack tip stress distribution. One of candidate parameters is theelastic T stress [2]. The T-stress is not singular as r→0 but it can alter the elastic crack-tip stress

distribution described by the stress intensity factor K I. T is a function of geometry, loading

conditions and is proportional to the applied gross stress [3-4]. For brittle mode I fracture in mainly

elastic regime, current fracture assessment methodology uses plane strain fracture toughness K IC,

which is assumed to be a material property. Recent studies [5-10] have shown that fracture

toughness can be strongly affected by specimen size, crack depth and loading configuration. This

dependency is often referred to the effect of crack-tip constraint.

At first, we use the elastic solution for two dimensional stress analysis of different cracked

specimen‟s geometries, including stress intensity factor and constraint parameter under various

loading conditions. This work has started by the development of an FEA data base containing the

values of the second higher order stress term coefficients. This paper exploited the K-T crackapproach which was derived from a rigorous asymptotic solution and has been developed for a two-

parameters fracture criterion. For the particular problem of a longitudinal crack in a pipe submitted

to internal pressure, it has been seen that the T stress is closed to those of a central crack in a panel

in tension. In this paper, the notch tip stress distribution is described using Notch Fracture

Mechanics and particularly the Volumetric Method approach in the aim to study constraint in pipes

submitted to internal pressure and exhibiting gouges as external defects. With the notch stress

intensity factor K as the driving force and an effective T stress Tef as constraint parameter, this

approach has been successfully used to quantify the notch-tip fields for various proposed geometry

and loading configurations. .

The K-T approach used for crack tip distribution

In some works [5-8] the T-stress is a stress acting parallel to the crack with its magnitude

proportional to the gross stress applies to the crack.

Constraint Parameter for a Longitudinal Surface Notch

in a Pipe Submitted to Internal Pressure

M. Hadj Meliani1,a

, M. Benarous2,b

, Z. Azari3,c

G. Pluvinage3,d

1

J0201/03/01/05, Département de Mécanique. BP 151, Hay Salem, Université de Chlef, Algérie.2 Laboratoire de Physique théorique et Physique des Matériaux. Université de Chlef, Algérie.

3 Laboratoire de Fiabilité Mécanique, LFM-ENIM, île de saulcy 57045, Université de Metz, France

[email protected], [email protected], [email protected], [email protected]

Keywords: Constraint, T-stress, Finite Element Analysis, Castem2000.

Abstract. The use of two parameters fracture mechanics criterion as a tool for structural design and

analysis has increased significantly in recent years. First, we discuse the elastic solution for the

stress distribution at crack tip for two dimensional geometries and particularly constraint as T-stressunder various loading conditions. Secondly, using notch fracture mechanics and particularly the

Volumetric Method approach, we study the stress distribution at the tip of a notch in pipes

submitted to internal pressure. The Notch Stress Intensity Factor K ρ and the effective T-stress are

combined into a two-parameter fracture criterion ( K I ρ-T ef ). This approach is then used to quantify

the constraint of notch-tip fields for various pipe geometry and loading conditions.

Introduction

Recent numerical and experimental studies have attempted to describe fracture in terms of two

parameters [1] characterising the crack tip stress distribution. One of candidate parameters is theelastic T stress [2]. The T-stress is not singular as r→0 but it can alter the elastic crack-tip stress

distribution described by the stress intensity factor K I. T is a function of geometry, loading

conditions and is proportional to the applied gross stress [3-4]. For brittle mode I fracture in mainly

elastic regime, current fracture assessment methodology uses plane strain fracture toughness K IC,

which is assumed to be a material property. Recent studies [5-10] have shown that fracture

toughness can be strongly affected by specimen size, crack depth and loading configuration. This

dependency is often referred to the effect of crack-tip constraint.

At first, we use the elastic solution for two dimensional stress analysis of different cracked

specimen‟s geometries, including stress intensity factor and constraint parameter under various

loading conditions. This work has started by the development of an FEA data base containing the

values of the second higher order stress term coefficients. This paper exploited the K-T crackapproach which was derived from a rigorous asymptotic solution and has been developed for a two-

parameters fracture criterion. For the particular problem of a longitudinal crack in a pipe submitted

to internal pressure, it has been seen that the T stress is closed to those of a central crack in a panel

in tension. In this paper, the notch tip stress distribution is described using Notch Fracture

Mechanics and particularly the Volumetric Method approach in the aim to study constraint in pipes

submitted to internal pressure and exhibiting gouges as external defects. With the notch stress

intensity factor K as the driving force and an effective T stress Tef as constraint parameter, this

approach has been successfully used to quantify the notch-tip fields for various proposed geometry

and loading configurations. .

The K-T approach used for crack tip distribution

In some works [5-8] the T-stress is a stress acting parallel to the crack with its magnitude

proportional to the gross stress applies to the crack.

Key Engineering Materials Vol. 399 (2009) pp 3-11© (2009) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.399.3



T stress is non-singular term represents a tension or compression stress. Positive T-stress

strengthens the level of crack tip stress triaxiality and leads to high crack-tip constraint while

negative T-stress produces the inverse effect. It was noted that T-stress characterizes the local crack

tip stress field for linear elastic material. Various studies have shown that T-stress has significant

influence on crack growth direction, crack growth stability, crack tip constraint and fracture

toughness. Near the tip of the crack, where the higher order terms of the series expansion arenegligible, stresses for mode I are written as :

T 2

3 sin

2 sin1

2cos

r 2

K I xx

2

3 sin

2 sin1

2cos

r 2

K I yy

T E 2

cos2r 2

K zz

I zz

2

3cos

2cos

2 sin

r 2

K I xy

(1)

In this paper, the T -stress is evaluated by the Difference Method proposed by Yang et al [9] and

coupling with finite element (FE) analysis.

0 ,0r yy xxT

- as r → ∞, (2)

Then the Stress Intensity Factor takes the following form :

j1i1ijij r 2T )( f .r 2 K as r → ∞, (3)

Levers and Radon [10] state that T must be proportional to the gross stress, g and therefore can be

normalized by the stress intensity factor to give the stress biaxiality ratio

I K

aT (4)

a is the crack length for edge cracks. The parameter is an elastic stress parameter depending of

geometry and has been tabulated by [10] for several specimen geometries.

Volumetric method for a two parameters notch fracture concept.

Notch Fracture Mechanics (NFM) principles are here applied to describe stress distribution at notch

tip of a a longitudinal gouge in a pipe submitted to internal pressure. Volumetric Method, presented

by Pluvinage [11] is a meso-mechanical approach belonging to this NFM. It is assumed, according

to the mesofracture principle that the fracture process requires a physical volume. This assumption

is supported by the fact that fracture resistance is affected by loading mode, structural geometry,

and scale effect. By using the value of the “hot spot stress” i.e. the maximum stress value, it is not

possible to explain the influence of theses parameters on fracture resistance. It is necessary to take

into account the stress value and the stress gradient in all neighbouring points within the fracture process volume, this volume is assumed to be quasi-cylindrical by with a notch plastic zone of

similar shape. The diameter of this cylinder is called the “effective distance “. By computing the

average stress value within this zone, the fracture stress can be estimated. This leads to a local

fracture stress criterion based on two parameters, the effective distance X ef and the effective stress

σ ef.,A graphical representation of this local fracture stress criterion is given in Figure 1.a, where the

stress normal to the notch plane is plotted against the distance ahead of notch. For X ef determination,

a graphical procedure is used. It has been observed that the effective distance is related to the

minimum value of the relative stress gradient χ and corresponds to the beginning of the pseudo

stress singularity as indicated in Figure 1.b. In a bi-logarithmic graph, we plot the opening stress

r yy , the relative stress gradient χ and the T-stress versus distance to notch tip. The effective

distance X ef is easily determined using the graphical procedure associated with the minimum of χ.

r

r

r

1r

yy

yy

(5)

T stress is non-singular term represents a tension or compression stress. Positive T-stress

strengthens the level of crack tip stress triaxiality and leads to high crack-tip constraint while

negative T-stress produces the inverse effect. It was noted that T-stress characterizes the local crack

tip stress field for linear elastic material. Various studies have shown that T-stress has significant

influence on crack growth direction, crack growth stability, crack tip constraint and fracture

toughness. Near the tip of the crack, where the higher order terms of the series expansion arenegligible, stresses for mode I are written as :

T 2

3 sin

2 sin1

2cos

r 2

K I xx

2

3 sin

2 sin1

2cos

r 2

K I yy

T E 2

cos2r 2

K zz

I zz

2

3cos

2cos

2 sin

r 2

K I xy

(1)

In this paper, the T -stress is evaluated by the Difference Method proposed by Yang et al [9] and

coupling with finite element (FE) analysis.

0 ,0r yy xxT

- as r → ∞, (2)

Then the Stress Intensity Factor takes the following form :

j1i1ijij r 2T )( f .r 2 K as r → ∞, (3)

Levers and Radon [10] state that T must be proportional to the gross stress, g and therefore can be

normalized by the stress intensity factor to give the stress biaxiality ratio

I K

aT (4)

a is the crack length for edge cracks. The parameter is an elastic stress parameter depending of

geometry and has been tabulated by [10] for several specimen geometries.

Volumetric method for a two parameters notch fracture concept.

Notch Fracture Mechanics (NFM) principles are here applied to describe stress distribution at notch

tip of a a longitudinal gouge in a pipe submitted to internal pressure. Volumetric Method, presented

by Pluvinage [11] is a meso-mechanical approach belonging to this NFM. It is assumed, according

to the mesofracture principle that the fracture process requires a physical volume. This assumption

is supported by the fact that fracture resistance is affected by loading mode, structural geometry,

and scale effect. By using the value of the “hot spot stress” i.e. the maximum stress value, it is not

possible to explain the influence of theses parameters on fracture resistance. It is necessary to take

into account the stress value and the stress gradient in all neighbouring points within the fracture process volume, this volume is assumed to be quasi-cylindrical by with a notch plastic zone of

similar shape. The diameter of this cylinder is called the “effective distance “. By computing the

average stress value within this zone, the fracture stress can be estimated. This leads to a local

fracture stress criterion based on two parameters, the effective distance X ef and the effective stress

σ ef.,A graphical representation of this local fracture stress criterion is given in Figure 1.a, where the

stress normal to the notch plane is plotted against the distance ahead of notch. For X ef determination,

a graphical procedure is used. It has been observed that the effective distance is related to the

minimum value of the relative stress gradient χ and corresponds to the beginning of the pseudo

stress singularity as indicated in Figure 1.b. In a bi-logarithmic graph, we plot the opening stress

r yy , the relative stress gradient χ and the T-stress versus distance to notch tip. The effective

distance X ef is easily determined using the graphical procedure associated with the minimum of χ.

r

r

r

1r

yy

yy

(5)

4 Advances in Strength of Materials



where r and r yy are the relative stress gradient and maximum principal stress or crack opening

stress, respectively. The relative stress gradient characterised the influence of loading mode and at a

less level, the influence of specimen geometry on fracture. The effective stress is defined as the

average of the weighted stress inside the fracture process zone:

dr r .r X

1 ef X

0

yyef

ef (6)

where ef , ef X , r yy and r are effective stress, effective distance, opening stress and weight

function, respectively. Therefore, the, noted, Notch Stress Intensity Factor, NSIF described an

important part of the notch tip stress distribution and is defined as a function of effective distance

and effective stress by the following relationship

ef ef I X 2 K (7)

Critical NSIF in mode I, c I K , , can be considered as a fracture toughness value with units m MPa if

notch opening angle are equal to zero. Like for the crack tip stress distribution, T -stress is function

of geometry, loading conditions and is proportional to the applied gross stress. T stress, as we can

see further, is only constant at some distance of the notch tip Based on these observations, we

propose a formulation of the effective elastic T -stress, noted T ef as the value of T at effective

distance of the notch-tip, noted X ef where is practically constant. T ef can be written as :

yy xx xxT for x = X ef . (8)

The effective Notch Stress Intensity Factor K Iρ, take the form of :

ef ef ef ef I I X X T K K 22* , (9)

and the stress biaxiality ratio at effective distance can by rewritten as :

I

ef ef

ef K

X 2T (10)

Fig. 1 (a) Schematic presentation of a local stress criterion for fracture emanating from notches and (b) Notch stressintensity factor and the effective T-stress at notch root together with the relative stress gradient versus distance from thenotch tip.

The Notch Stress Intensity Factor K Iρ, the effective T -stress, T ef , near the notch rot and the relative

distance are shown in Figure 1.b. Effective T-stress has been then used as constraint parameter. This

addition to the classical plastic notch tip parameter K I provides an effective two-parameters

where r and r yy are the relative stress gradient and maximum principal stress or crack opening

stress, respectively. The relative stress gradient characterised the influence of loading mode and at a

less level, the influence of specimen geometry on fracture. The effective stress is defined as the

average of the weighted stress inside the fracture process zone:

dr r .r X

1 ef X

0

yyef

ef (6)

where ef , ef X , r yy and r are effective stress, effective distance, opening stress and weight

function, respectively. Therefore, the, noted, Notch Stress Intensity Factor, NSIF described an

important part of the notch tip stress distribution and is defined as a function of effective distance

and effective stress by the following relationship

ef ef I X 2 K (7)

Critical NSIF in mode I, c I K , , can be considered as a fracture toughness value with units m MPa if

notch opening angle are equal to zero. Like for the crack tip stress distribution, T -stress is function

of geometry, loading conditions and is proportional to the applied gross stress. T stress, as we can

see further, is only constant at some distance of the notch tip Based on these observations, we

propose a formulation of the effective elastic T -stress, noted T ef as the value of T at effective

distance of the notch-tip, noted X ef where is practically constant. T ef can be written as :

yy xx xxT for x = X ef . (8)

The effective Notch Stress Intensity Factor K Iρ, take the form of :

ef ef ef ef I I X X T K K 22* , (9)

and the stress biaxiality ratio at effective distance can by rewritten as :

I

ef ef

ef K

X 2T (10)

Fig. 1 (a) Schematic presentation of a local stress criterion for fracture emanating from notches and (b) Notch stressintensity factor and the effective T-stress at notch root together with the relative stress gradient versus distance from thenotch tip.

The Notch Stress Intensity Factor K Iρ, the effective T -stress, T ef , near the notch rot and the relative

distance are shown in Figure 1.b. Effective T-stress has been then used as constraint parameter. This

addition to the classical plastic notch tip parameter K I provides an effective two-parameters

Key Engineering Materials Vol. 399 5



characterization of elastic notch-tip fields in a variety of notch configurations and loading

conditions.

Material properties and Finite Element Modelling

The material used in this study is an X52 steels meeting requirements of API 5L standard. API X52,was the most common gas pipeline material for transmission of oil and gas during 1950-1960.

Chemical composition of the studied steels is given in Table 1.

Table 1: Chemical composition of specimens material (weight %)

C Mn P Si Cr Ni Mo S Cu Ti Nb Al

0.22 1.220 - 0.240 0.16 0.14 0.06 0.036 0.19 0.04 <0.05 0.032

Ref[12 ] 0.15 1.250 0.025 0.350 - - - - - - - -

In Table 2, mechanical properties of API X52 are presented. E, y , u , A%, n, k and K are the

Young‟s modulus, yield stress, ultimate stress, relative elongation, hardening exponent and fracturetoughness, respectively.

Table 2: Mechanical properties of API X52E

(GPA) y

(MPa)

u

(MPa) A% n k K CharpyEnergy

(J/mm2)

210 410 528 32 0.164 876 116.6 -

Ref[12] in longitudinal sense 210 493 623 30 - - - ~ 75

Ref[12] in circumferential sense 210 410 638 19 - - - 22

The pipe geometry of this study is a cylinder with a V-shaped longitudinal surface notch subjectto different internal pressure P as shown in Table 2. The effects of the following three parameters:

ratio of inner radius of the cylinder to thickness ( Ri /t ), ratio of notch depth to cylinder thickness

(a/t), and pressure P on effective T-stress (T ef ) and NSIF ( K Iρ) are systematically considered.

Table 3: List of analysis cases for the FE analysis

P R i /t a/t P R i /t a/t P R i /t a/t P R i /t a/t

20 5 0.1 20 10 0.1 20 20 0.1 20 40 0.10.3 0.3 0.3 0.30.5 0.5 0.5 0.5

0.75 0.75 0.75 0.7530 5 0.1 30 10 0.1 30 20 0.1 30 40 0.1

0.3 0.3 0.3 0.30.5 0.5 0.5 0.50.75 0.75 0.75 0.75

40 5 0.1 40 10 0.1 40 20 0.1 40 40 0.1

0.3 0.3 0.3 0.30.5 0.5 0.5 0.50.75 0.75 0.75 0.75

50 5 0.1 50 10 0.1 50 20 0.1 50 40 0.10.3 0.3 0.3 0.3

0.5 0.5 0.5 0.50.75 0.75 0.75 0.75

To cover practical and interesting ranges of these three variables, four different values of R i/t i.e 5,

10, 20 and 40, were selected. In terms of notch depth, four different values of a/t were selected,

characterization of elastic notch-tip fields in a variety of notch configurations and loading

conditions.

Material properties and Finite Element Modelling

The material used in this study is an X52 steels meeting requirements of API 5L standard. API X52,was the most common gas pipeline material for transmission of oil and gas during 1950-1960.

Chemical composition of the studied steels is given in Table 1.

Table 1: Chemical composition of specimens material (weight %)

C Mn P Si Cr Ni Mo S Cu Ti Nb Al

0.22 1.220 - 0.240 0.16 0.14 0.06 0.036 0.19 0.04 <0.05 0.032

Ref[12 ] 0.15 1.250 0.025 0.350 - - - - - - - -

In Table 2, mechanical properties of API X52 are presented. E, y , u , A%, n, k and K are the

Young‟s modulus, yield stress, ultimate stress, relative elongation, hardening exponent and fracturetoughness, respectively.

Table 2: Mechanical properties of API X52E

(GPA) y

(MPa)

u

(MPa) A% n k K CharpyEnergy

(J/mm2)

210 410 528 32 0.164 876 116.6 -

Ref[12] in longitudinal sense 210 493 623 30 - - - ~ 75

Ref[12] in circumferential sense 210 410 638 19 - - - 22

The pipe geometry of this study is a cylinder with a V-shaped longitudinal surface notch subjectto different internal pressure P as shown in Table 2. The effects of the following three parameters:

ratio of inner radius of the cylinder to thickness ( Ri /t ), ratio of notch depth to cylinder thickness

(a/t), and pressure P on effective T-stress (T ef ) and NSIF ( K Iρ) are systematically considered.

Table 3: List of analysis cases for the FE analysis

P R i /t a/t P R i /t a/t P R i /t a/t P R i /t a/t

20 5 0.1 20 10 0.1 20 20 0.1 20 40 0.10.3 0.3 0.3 0.30.5 0.5 0.5 0.5

0.75 0.75 0.75 0.7530 5 0.1 30 10 0.1 30 20 0.1 30 40 0.1

0.3 0.3 0.3 0.30.5 0.5 0.5 0.50.75 0.75 0.75 0.75

40 5 0.1 40 10 0.1 40 20 0.1 40 40 0.1

0.3 0.3 0.3 0.30.5 0.5 0.5 0.50.75 0.75 0.75 0.75

50 5 0.1 50 10 0.1 50 20 0.1 50 40 0.10.3 0.3 0.3 0.3

0.5 0.5 0.5 0.50.75 0.75 0.75 0.75

To cover practical and interesting ranges of these three variables, four different values of R i/t i.e 5,

10, 20 and 40, were selected. In terms of notch depth, four different values of a/t were selected,




ranging from a/t = 0.1 to 0.75. In terms of pressure, four different values of P were selected, ranging

from pressure of 2 MPa to 5 MPa . Thus, a total of 84 different experimental setups are considered

in this investigation, the details of which are listed in table 3.

The finite element method was used to determine the stress field parameters T ef and K Iρ for the pipe

specimens. The structure were modelled by CASTEM 2000 code in two dimensions under plane

strain conditions, using free meshed isoperimetric triangular elements and only on half of thespecimen. The elastic analysis comprises 31485 elements and 63526 nodes.

A fan-like mesh focused at the notch-tip was employed, because this yields more accurate non-

singular terms estimates. Further, more a detailed mesh sensitivity study have show that refinement

mesh leads to only a small changes (<1%). Pipe specimen geometry is illustrated in Figure 2. The

wall thickness is 10 mm the pipe length is 40 mm, dimensions of the notch are 5 mm depth and 50

mm length, notch radius 0.15mm. Support and symmetric boundary conditions are used in this

model.

Fig. 2 (a) Geometry boundary conditions and loading configuration using in the half of the pipe. (b) Typical 2D finiteelement mesh used to model the cracked on the half pipeline for elastic analysis.

Results and discussion

A detailed stress analysis was carried out in the vicinity of the notch tip, in order to emphasize the

characteristics of the two dimensional stress field . Due the fact in Mode I, the crack propagation is

determined by the stress perpendicular to the notch namely yy .

T-stress evolution along of ligament

T stress can be determined along any direction where the singular term of xx vanishes or can be set

to zero by superposing with a fraction of yy .The Stress Difference Method (SDM) is used todetermine the T-stress versus distance behind the notch. This method was proposed by Yang et al

[9] using directly a single finite element (FE). The underlying idea is that errors in the numerically

obtained values of xx and yy near the notch tip evolve with distance and their difference must

eliminate the errors effectively. They calculated the T-stress using the difference of the normal

stresses along θ = 0, i.e.( xx - yy ).This corresponds to mode I positions around the notch tip.

0 ,0r yy xxT

- (11)

The results show the variation of the T-stress near the notch tip for different pipe diameters and (a/t)

ratios. An example of the effects of notch length on T-stress is depicted for small (a/t =0.1)

(Figure.3.a) and long notches (a/t =0.75) (Figure.3.b). Results are influenced by numerical errors

normally expected form FE results in highly stresses zones. The effects of higher terms in William‟s

series expansion are significant along the ligament. The T-stress values are normalized with respect

to gross stress σg (the definition of σg for a pipe subject to internal pressure is given by σg = PR i/t).

2RtP

a

ranging from a/t = 0.1 to 0.75. In terms of pressure, four different values of P were selected, ranging

from pressure of 2 MPa to 5 MPa . Thus, a total of 84 different experimental setups are considered

in this investigation, the details of which are listed in table 3.

The finite element method was used to determine the stress field parameters T ef and K Iρ for the pipe

specimens. The structure were modelled by CASTEM 2000 code in two dimensions under plane

strain conditions, using free meshed isoperimetric triangular elements and only on half of thespecimen. The elastic analysis comprises 31485 elements and 63526 nodes.

A fan-like mesh focused at the notch-tip was employed, because this yields more accurate non-

singular terms estimates. Further, more a detailed mesh sensitivity study have show that refinement

mesh leads to only a small changes (<1%). Pipe specimen geometry is illustrated in Figure 2. The

wall thickness is 10 mm the pipe length is 40 mm, dimensions of the notch are 5 mm depth and 50

mm length, notch radius 0.15mm. Support and symmetric boundary conditions are used in this

model.

Fig. 2 (a) Geometry boundary conditions and loading configuration using in the half of the pipe. (b) Typical 2D finiteelement mesh used to model the cracked on the half pipeline for elastic analysis.

Results and discussion

A detailed stress analysis was carried out in the vicinity of the notch tip, in order to emphasize the

characteristics of the two dimensional stress field . Due the fact in Mode I, the crack propagation is

determined by the stress perpendicular to the notch namely yy .

T-stress evolution along of ligament

T stress can be determined along any direction where the singular term of xx vanishes or can be set

to zero by superposing with a fraction of yy .The Stress Difference Method (SDM) is used todetermine the T-stress versus distance behind the notch. This method was proposed by Yang et al

[9] using directly a single finite element (FE). The underlying idea is that errors in the numerically

obtained values of xx and yy near the notch tip evolve with distance and their difference must

eliminate the errors effectively. They calculated the T-stress using the difference of the normal

stresses along θ = 0, i.e.( xx - yy ).This corresponds to mode I positions around the notch tip.

0 ,0r yy xxT

- (11)

The results show the variation of the T-stress near the notch tip for different pipe diameters and (a/t)

ratios. An example of the effects of notch length on T-stress is depicted for small (a/t =0.1)

(Figure.3.a) and long notches (a/t =0.75) (Figure.3.b). Results are influenced by numerical errors

normally expected form FE results in highly stresses zones. The effects of higher terms in William‟s

series expansion are significant along the ligament. The T-stress values are normalized with respect

to gross stress σg (the definition of σg for a pipe subject to internal pressure is given by σg = PR i/t).

2RtP

a




Fig. 3 Evolution of the T-stress values for a longitudinal surface notch under different internal pressure (P = 20, 30, 40and 50 bars) for (a) small and (b) length notches (R/t=20).

For any pressure values and pipe diameter, the T-stress values are negative along the ligament when

the crack length ratio is less than a/t < 0.5. The ligament is then submitted to compression. On the

other hand, when the ratios a/t exceeds 0.75; the T-stress values become positives (tension case). In

the first case and under predominant mode I loading, the T-stress is negative (compressive) and the

initiating crack will propagate along original notch direction which is the directional stable crack

trajectory. Thus, for a given notch length, a rapid transition from negative to positive zone (tension)

is its (see Figure 3). The initiating crack will rapidly deviate away from the original path and the

resultant crack path will be directionally unstable. Eventually, near the notch-tip, it is shown that the

effect of R/t, a/t and P on the T-stress distribution is significant. The change of the T-stress sign fornegative to positive values may be due to the fact that the magnitude of local moment closing the

notch increases with increasing in notch depth. Negative values indicate low crack front constraint

and extended plastic deformation around the notch front. The T-stress becomes more positive when

the free surface is approached due to the loss of notch – front constraint. The elastic T-stress seems to

be convenient for this constraint analysis.

Determination of effective T-stress and Notch Stress Intensity factor (NSIF)

T -stress is function of geometry, loading conditions and is proportional to the gross applied stress.

Based on the observation than T stress of the notch tip stress distribution becomes constant when a

given distance from the notch tip is reached, we propose an effective elastic T -stress, noted T ef determined at the effective distance from the notch-tip noted X ef . This effective distance is defined

as the minimum of the relative stress gradient as previously. T ef can be rewritten as:

0 , X r yy xxef ef

T

as r = X ef (12)

The distribution of the T-stress along of the ligament (r ) is shown in Figure 3.

The effective Notch Stress Intensity Factor K Iρ, is given by :

ef ef ef ef I I X X T K K 22* (13)

Figures 4.a-b show variations of the effective T-stress and Notch Stress Intensity Factor, K Iρ value

at the deepest points of an external surface notch in a pipe submitted to internal pressure and fordifferent relative notch depths, „a/t' (notch depth /cylinder thickness). It can be seen that T ef and K I ρ

values increase with an increase of the relative notch size, a/t. When a/t tends towards 0.1, K I ρ and

(a) (b)

Fig. 3 Evolution of the T-stress values for a longitudinal surface notch under different internal pressure (P = 20, 30, 40and 50 bars) for (a) small and (b) length notches (R/t=20).

For any pressure values and pipe diameter, the T-stress values are negative along the ligament when

the crack length ratio is less than a/t < 0.5. The ligament is then submitted to compression. On the

other hand, when the ratios a/t exceeds 0.75; the T-stress values become positives (tension case). In

the first case and under predominant mode I loading, the T-stress is negative (compressive) and the

initiating crack will propagate along original notch direction which is the directional stable crack

trajectory. Thus, for a given notch length, a rapid transition from negative to positive zone (tension)

is its (see Figure 3). The initiating crack will rapidly deviate away from the original path and the

resultant crack path will be directionally unstable. Eventually, near the notch-tip, it is shown that the

effect of R/t, a/t and P on the T-stress distribution is significant. The change of the T-stress sign fornegative to positive values may be due to the fact that the magnitude of local moment closing the

notch increases with increasing in notch depth. Negative values indicate low crack front constraint

and extended plastic deformation around the notch front. The T-stress becomes more positive when

the free surface is approached due to the loss of notch – front constraint. The elastic T-stress seems to

be convenient for this constraint analysis.

Determination of effective T-stress and Notch Stress Intensity factor (NSIF)

T -stress is function of geometry, loading conditions and is proportional to the gross applied stress.

Based on the observation than T stress of the notch tip stress distribution becomes constant when a

given distance from the notch tip is reached, we propose an effective elastic T -stress, noted T ef determined at the effective distance from the notch-tip noted X ef . This effective distance is defined

as the minimum of the relative stress gradient as previously. T ef can be rewritten as:

0 , X r yy xxef ef

T

as r = X ef (12)

The distribution of the T-stress along of the ligament (r ) is shown in Figure 3.

The effective Notch Stress Intensity Factor K Iρ, is given by :

ef ef ef ef I I X X T K K 22* (13)

Figures 4.a-b show variations of the effective T-stress and Notch Stress Intensity Factor, K Iρ value

at the deepest points of an external surface notch in a pipe submitted to internal pressure and fordifferent relative notch depths, „a/t' (notch depth /cylinder thickness). It can be seen that T ef and K I ρ

values increase with an increase of the relative notch size, a/t. When a/t tends towards 0.1, K I ρ and

(a) (b)




T ef values for both notch types become small and similar. When both K I ρ and T ef tend to reach value

of a/t =1, they become very sensitive to pipe geometry. We note that for all study cases,

From the Figure 5.a, it can be seen that the effective T-stress and the Notch Stress Intensity

Factor for the four pipe diameter are proportional to the internal pressure P. They can be very well

expressed by linear relationship versus internal pressure. Figure 5.b shows a comparison between

short and long notches according to pressure for a pipe diameter (R/t) which varies in the range of5-40. An increase of the pipe diameter to thickness ratio from 5 to 40 (for a ∕ t = 0.5) induces a

similar variation of the Notch Stress Intensity Factor and a variation of effective T-stress by a

factor of 8. For long notches, the NSIF and T ef increases more rapidly with internal pressure (Figure

5.b).

Fig. 4 Notch tip constraint represented by the effective T-stress for various situations of pressure and diameters of pipesand presence of K I ρ.

Fig. 5 (a) Evolution of the effective T-stress and the Notch Stress Intensity Factor with internal pressure P for a/t=0.5and (b) transition of Teff and NSIF for a/t=0.1 to 0.75 for the diameter pipe R/t =5 to 40.

Proposed fracture assessment procedure

It has been seen previously that there is good correlation between the effective T stress and the

Notch stress Intensity factor. Within these investigations for a wide range of specimens and with

different constraint, a master curve reference ( K ρc – T ef,c) can be determined in order to quantified in

terms of a two parameters fracture resistance criterion. The application of the master curve concept

enables testing on small specimens for determination of the material fracture toughness to be used

for assessment of components. However, small specimens could have a larger plastic deformation preceding failure and therefore possibly different constraint situation than larger specimens or

components, the use of the master curve allows a correction in the fracture toughness value. For a

wide range of specimens with different constraint, a master curve reference ( K ρc – T ef,c) is determined

(a)

(b)

T ef values for both notch types become small and similar. When both K I ρ and T ef tend to reach value

of a/t =1, they become very sensitive to pipe geometry. We note that for all study cases,

From the Figure 5.a, it can be seen that the effective T-stress and the Notch Stress Intensity

Factor for the four pipe diameter are proportional to the internal pressure P. They can be very well

expressed by linear relationship versus internal pressure. Figure 5.b shows a comparison between

short and long notches according to pressure for a pipe diameter (R/t) which varies in the range of5-40. An increase of the pipe diameter to thickness ratio from 5 to 40 (for a ∕ t = 0.5) induces a

similar variation of the Notch Stress Intensity Factor and a variation of effective T-stress by a

factor of 8. For long notches, the NSIF and T ef increases more rapidly with internal pressure (Figure

5.b).

Fig. 4 Notch tip constraint represented by the effective T-stress for various situations of pressure and diameters of pipesand presence of K I ρ.

Fig. 5 (a) Evolution of the effective T-stress and the Notch Stress Intensity Factor with internal pressure P for a/t=0.5and (b) transition of Teff and NSIF for a/t=0.1 to 0.75 for the diameter pipe R/t =5 to 40.

Proposed fracture assessment procedure

It has been seen previously that there is good correlation between the effective T stress and the

Notch stress Intensity factor. Within these investigations for a wide range of specimens and with

different constraint, a master curve reference ( K ρc – T ef,c) can be determined in order to quantified in

terms of a two parameters fracture resistance criterion. The application of the master curve concept

enables testing on small specimens for determination of the material fracture toughness to be used

for assessment of components. However, small specimens could have a larger plastic deformation preceding failure and therefore possibly different constraint situation than larger specimens or

components, the use of the master curve allows a correction in the fracture toughness value. For a

wide range of specimens with different constraint, a master curve reference ( K ρc – T ef,c) is determined

(a)

(b)




for any material. In this context, 2-dimensional finite element (FE) analyses on several standard and

non standard specimens (at least five) are performed to quantify relationship between variations of

constraint parameters and notch fracture toughness. The K I ρ-T ef space is fitted generally by a

parabolic line and the data points as shown in Figure 6. From the obtained relationship, prediction

the fracture of a specimen of another geometry or size, is then possible. In Figure 6 a, an example of

NSIF versus Tef relation ship is given (for a pipe with R/t = 20 and for an internal pressure, P =2-5MPa). Figure 6.b indicates that the window of possible toughness values increases with pressure.

Fig. 6a NSIF versus Tef relation ship ( R/t = 20 ; internal pressure, P =2-5MPa)

Fig. 6b Evolution of different NSIF-Teff windows with pressure.

Conclusions

We have extended the first non-vanishing terms from the series solutions of Williams‟ to notch tipstress distribution. The crack (K-T) methodology has been modifying to create the ( K – T eff ) two

parameters fracture resistance criterion. A parabolic relation ship has been found between these two

parameters and allows to build a window including data from a large range of pressure and

geometry. Application of this method has been made for pipe submitted too internal pressure with a

longitudinal notch but can be extended to any kind of components and defect types.

References

[1] Williams ML : On the stress distribution at the base of stationary crack. ASME J Appl Mech;

24 , 109-14 (1957).

[2] Rice JR. : Limitations to the-scale yielding approximation for crack-tip plasticity. J. Mech.

Solids, 22, 17-26 (1974).

[3] Larsson, S.G and Carlsson, A.J.: Influence of non-singular stress terms and specimen geometry

on small-scale yielding at crack tips in elastic-plastic materials. J. Mech. Phys. Solids 21, 263-278

(1973).

[4] Leevers PS, Radon JC : Inherent stress biaxiality in various fracture specimen geometries. Int.

J. Fract. 19, 311-25 (1982).

[5] Chao, Y.J. and Zhang, X.: Constraint effect in brittle fracture. 27th National Symposium onFatigue and fracture, ASTM STP 1296, R.S. Piascik, J.C. Newman, Jr. and D.E. Dowling, Eds.,

American Society for Testing and Materials, Philadelphia, pp. 41 – 60 (1997).

(a)

for any material. In this context, 2-dimensional finite element (FE) analyses on several standard and

non standard specimens (at least five) are performed to quantify relationship between variations of

constraint parameters and notch fracture toughness. The K I ρ-T ef space is fitted generally by a

parabolic line and the data points as shown in Figure 6. From the obtained relationship, prediction

the fracture of a specimen of another geometry or size, is then possible. In Figure 6 a, an example of

NSIF versus Tef relation ship is given (for a pipe with R/t = 20 and for an internal pressure, P =2-5MPa). Figure 6.b indicates that the window of possible toughness values increases with pressure.

Fig. 6a NSIF versus Tef relation ship ( R/t = 20 ; internal pressure, P =2-5MPa)

Fig. 6b Evolution of different NSIF-Teff windows with pressure.

Conclusions

We have extended the first non-vanishing terms from the series solutions of Williams‟ to notch tipstress distribution. The crack (K-T) methodology has been modifying to create the ( K – T eff ) two

parameters fracture resistance criterion. A parabolic relation ship has been found between these two

parameters and allows to build a window including data from a large range of pressure and

geometry. Application of this method has been made for pipe submitted too internal pressure with a

longitudinal notch but can be extended to any kind of components and defect types.

References

[1] Williams ML : On the stress distribution at the base of stationary crack. ASME J Appl Mech;

24 , 109-14 (1957).

[2] Rice JR. : Limitations to the-scale yielding approximation for crack-tip plasticity. J. Mech.

Solids, 22, 17-26 (1974).

[3] Larsson, S.G and Carlsson, A.J.: Influence of non-singular stress terms and specimen geometry

on small-scale yielding at crack tips in elastic-plastic materials. J. Mech. Phys. Solids 21, 263-278

(1973).


J. Fract. 19, 311-25 (1982).

[5] Chao, Y.J. and Zhang, X.: Constraint effect in brittle fracture. 27th National Symposium onFatigue and fracture, ASTM STP 1296, R.S. Piascik, J.C. Newman, Jr. and D.E. Dowling, Eds.,

American Society for Testing and Materials, Philadelphia, pp. 41 – 60 (1997).

(a)




[6] Chao, Y.J., Liu, S., and Broviak, B.J. : Variation of fracture toughness with constraint of

PMMA specimens. Proceedings of ASME-PVP conference 393, 113 – 120 (1999).

[7] Chao , Y.J., Liu, S., and Broviak, B.J.: Brittle fracture: variation of fracture toughness with

constraint and crack curving under mode I conditions. Experimental Mechanics 41(3), 232 – 241

(2001).[8] Du ZZ, Hancock JW : The effect of non-singular stresses on crack tip constraint. J Mech Phys

Solids. 39; 555-67 (1991).

[9] Yang, B. Ravi-Chandar, K.: Evaluation of elastic T-stress by the stress difference method.

Engng Fract Mech. 64; 589-605 (1999).


J. Fract. 19, 311-25 (1982).

[11] Pluvinage G. : Fracture and Fatigue Emanating from Stress Concentrators, Kluwer Publisher

(2003).

[12] Rousserie S : L‟amorçage de la fissuration des pipelines en milieu bicarboné à ph neutre .

Thèse présentée à l‟université de Bordeaux I (2000).

[6] Chao, Y.J., Liu, S., and Broviak, B.J. : Variation of fracture toughness with constraint of

PMMA specimens. Proceedings of ASME-PVP conference 393, 113 – 120 (1999).

[7] Chao , Y.J., Liu, S., and Broviak, B.J.: Brittle fracture: variation of fracture toughness with

constraint and crack curving under mode I conditions. Experimental Mechanics 41(3), 232 – 241

(2001).[8] Du ZZ, Hancock JW : The effect of non-singular stresses on crack tip constraint. J Mech Phys

Solids. 39; 555-67 (1991).

[9] Yang, B. Ravi-Chandar, K.: Evaluation of elastic T-stress by the stress difference method.

Engng Fract Mech. 64; 589-605 (1999).


J. Fract. 19, 311-25 (1982).

[11] Pluvinage G. : Fracture and Fatigue Emanating from Stress Concentrators, Kluwer Publisher

(2003).

[12] Rousserie S : L‟amorçage de la fissuration des pipelines en milieu bicarboné à ph neutre .

Thèse présentée à l‟université de Bordeaux I (2000).




Application of Complete Gurson Model for prediction of ductile fracturein welded steel joints

Bojan Međo1, Marko Rakin

1, a, Nenad Gubeljak

2, Aleksandar Sedmak

3

1

Faculty of Technology and Metallurgy, Belgrade University, Belgrade, Serbia2 Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia

3 Faculty of Mechanical Engineering, Belgrade University, Belgrade, Serbia

a [email protected]

Keywords: Ductile fracture, micromechanical model, Gurson flow criterion, FEM calculation

Abstract. Ductile fracture process includes three stages: void nucleation, their growth and

coalescence. The voids nucleate due to the fracture or separation of non-metallic inclusions and

secondary-phase particles from the material matrix. Micromechanical models based on the Gurson

plastic flow criterion are often used for analysis of ductile fracture. They consider the material as a

porous medium in which the effect of voids on the stress-strain state and plastic flow cannot be

neglected. Another important property of the Gurson criterion is that the hydrostatic stress

component influences the plastic flow of the material.

Two models that include the Gurson plastic flow criterion are frequently used: GTN (Gurson-

Tveergard-Needleman) and recently CGM (Complete Gurson Model). Their application includes a

combination of experimental and numerical procedure. The problem with the GTN model is

determining the critical void volume fraction at the beginning of void coalescence, because this

parameter depends on geometry and the initial state of the material. The CGM eliminates the

critical void volume fraction as a failure criterion, which is an important advantage of this model. Inthis paper, a detail insight into the GTN and the CGM is given, including the application of the

CGM in numerical simulation of ductile fracture of a pre-cracked specimen. Inhomogeneous

materials (welded joints) are analysed, considering the influence of initial parameters and the size

of the finite elements near the crack tip.

Introduction

The micromechanical approach describes the process of fracture in a way close to the actual

phenomena in the material. It is based on a large number of models of microscopic damages, as an

effort to explain and predict the macroscopic failure. The micromechanical models for prediction of

fracture initiation in steel and other metal alloys are constantly being improved. The main problems

in their application are the determination of numerous parameters present in these models and the

lack of physical significance of some of these parameters.

The process of ductile fracture of most metals and alloys includes void nucleation, growth and

coalescence. Void nucleation takes place around non-metallic inclusions and second-phase

particles, and void nucleates when the so-called critical stress within the inclusion or at the

inclusion – matrix interface is exceeded. In materials of distinct ductile behaviour, fracture occurs

after all three phases. During loading, these materials exhibit strain hardening, but they also exhibit

softening due to the presence of the voids.

According to the uncoupled modelling, void presence does not significantly alter the behaviour

of the material, hence the damage parameter is not incorporated into the constitutive equation. Inthat case, the von Mises criterion is the most frequently used as the yield criterion. Research efforts

have recently been directed toward the so-called coupled models of damage, with the damage

parameter “built into” the numerical procedure and estimated during the finite elements (FE)

Application of Complete Gurson Model for prediction of ductile fracturein welded steel joints

Bojan Međo1, Marko Rakin

1, a, Nenad Gubeljak

2, Aleksandar Sedmak

3

1

Faculty of Technology and Metallurgy, Belgrade University, Belgrade, Serbia2 Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia

3 Faculty of Mechanical Engineering, Belgrade University, Belgrade, Serbia

a [email protected]

Keywords: Ductile fracture, micromechanical model, Gurson flow criterion, FEM calculation

Abstract. Ductile fracture process includes three stages: void nucleation, their growth and

coalescence. The voids nucleate due to the fracture or separation of non-metallic inclusions and

secondary-phase particles from the material matrix. Micromechanical models based on the Gurson

plastic flow criterion are often used for analysis of ductile fracture. They consider the material as a

porous medium in which the effect of voids on the stress-strain state and plastic flow cannot be

neglected. Another important property of the Gurson criterion is that the hydrostatic stress

component influences the plastic flow of the material.

Two models that include the Gurson plastic flow criterion are frequently used: GTN (Gurson-

Tveergard-Needleman) and recently CGM (Complete Gurson Model). Their application includes a

combination of experimental and numerical procedure. The problem with the GTN model is

determining the critical void volume fraction at the beginning of void coalescence, because this

parameter depends on geometry and the initial state of the material. The CGM eliminates the

critical void volume fraction as a failure criterion, which is an important advantage of this model. Inthis paper, a detail insight into the GTN and the CGM is given, including the application of the

CGM in numerical simulation of ductile fracture of a pre-cracked specimen. Inhomogeneous

materials (welded joints) are analysed, considering the influence of initial parameters and the size

of the finite elements near the crack tip.

Introduction

The micromechanical approach describes the process of fracture in a way close to the actual

phenomena in the material. It is based on a large number of models of microscopic damages, as an

effort to explain and predict the macroscopic failure. The micromechanical models for prediction of

fracture initiation in steel and other metal alloys are constantly being improved. The main problems

in their application are the determination of numerous parameters present in these models and the

lack of physical significance of some of these parameters.

The process of ductile fracture of most metals and alloys includes void nucleation, growth and

coalescence. Void nucleation takes place around non-metallic inclusions and second-phase

particles, and void nucleates when the so-called critical stress within the inclusion or at the

inclusion – matrix interface is exceeded. In materials of distinct ductile behaviour, fracture occurs

after all three phases. During loading, these materials exhibit strain hardening, but they also exhibit

softening due to the presence of the voids.

According to the uncoupled modelling, void presence does not significantly alter the behaviour

of the material, hence the damage parameter is not incorporated into the constitutive equation. Inthat case, the von Mises criterion is the most frequently used as the yield criterion. Research efforts

have recently been directed toward the so-called coupled models of damage, with the damage

parameter “built into” the numerical procedure and estimated during the finite elements (FE)

Key Engineering Materials Vol. 399 (2009) pp 13-20© (2009) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.399.13



analysis. One of these models has been developed by Tvergaard and Needleman [1], based on the

constitutive equations suggested by Gurson [2]. The main variable in this model is the void volume

fraction, which is directly incorporated into the flow criterion. An important feature of

micromechanical models based on the Gurson flow criterion is that the hydrostatic stress

component influences the plastic flow of the material. This influence can be seen as deformation

softening during loading.

Micromechanical models

According to the uncoupled model of Rice and Tracey, improved by Beremin [3, 4], the models of

Huang [5] and Chaouadi et al. [6], void growth is strongly dependant on stress triaxiality. All these

models are uncoupled, hence damages are calculated by post processing routines, based on the

stress and strain fields determined experimentally and using FE analysis. Such models, that

consider the growth of an isolated void, cannot take into account the possible coalescence of

adjacent voids leading to local instability of the material.

The coupled approaches to material damage and ductile fracture initiation consider the material

as a porous medium, where the influence of nucleated voids on the stress-strain state and the plasticflow cannot be avoided. The GTN (Gurson-Tveergard-Needleman) model is based on the

hypothesis that void nucleation and growth can be macroscopically described by extending von

Mises plasticity theory to cover the effects of porosity occurring in the material. The void volume

fraction f (porosity) is introduced into the expression for plastic potential [1, 2, 7]:

2

* *21 12

3 32 cosh 1 0

2 2

ij ij m

eq eq

S S qq f q f

, (1)

where eq denotes the yield stress of the material matrix,m eq

is stress triaxiality, ijS is the

stress deviator, the parameters q1 and q2 were introduced by Tvergaard [7] to improve the ductile

fracture prediction of the Gurson model and f * is the damage function [1]:

c

c

for*

( ) forc c

f f f f

f K f f f f

(2)

where f c is the critical value of f , at the moment when the void coalescence begins. For f * = 0, the

plastic potential (2) is identical with that of von Mises. The parameter K defines the slope of a

sudden drop of force on the force – diameter reduction diagram, and is often referred to as an

accelerating factor. This factor defines the final stage of ductile fracture – void coalescence, which

leads to complete loss of the load carrying capacity of the material.

During the ductile fracture of steel, the voids nucleate due to separation or fracture of non-

metallic inclusions and secondary-phase particles from the material matrix. In the initial phase of

the ductile fracture, the voids nucleate mostly around non-metallic inclusions. To quantify this

micromechanism, the volume fraction of non-metallic inclusions, f V, that can be determined using

light microscopy, should be known [8].

Two phenomena contribute to the increase of the void volume fraction in FE analysis with the

embedded GTN yield criterion. One is the growth of the existing voids and the other is the

nucleation of new voids under external loading:

nucleation growth f f f

; p

nucleation eq f A

and (1 )

p growth ii f f

(3)

where pii is the plastic part of the strain rate tensor.

analysis. One of these models has been developed by Tvergaard and Needleman [1], based on the

constitutive equations suggested by Gurson [2]. The main variable in this model is the void volume

fraction, which is directly incorporated into the flow criterion. An important feature of

micromechanical models based on the Gurson flow criterion is that the hydrostatic stress

component influences the plastic flow of the material. This influence can be seen as deformation

softening during loading.

Micromechanical models

According to the uncoupled model of Rice and Tracey, improved by Beremin [3, 4], the models of

Huang [5] and Chaouadi et al. [6], void growth is strongly dependant on stress triaxiality. All these

models are uncoupled, hence damages are calculated by post processing routines, based on the

stress and strain fields determined experimentally and using FE analysis. Such models, that

consider the growth of an isolated void, cannot take into account the possible coalescence of

adjacent voids leading to local instability of the material.

The coupled approaches to material damage and ductile fracture initiation consider the material

as a porous medium, where the influence of nucleated voids on the stress-strain state and the plasticflow cannot be avoided. The GTN (Gurson-Tveergard-Needleman) model is based on the

hypothesis that void nucleation and growth can be macroscopically described by extending von

Mises plasticity theory to cover the effects of porosity occurring in the material. The void volume

fraction f (porosity) is introduced into the expression for plastic potential [1, 2, 7]:

2

* *21 12

3 32 cosh 1 0

2 2

ij ij m

eq eq

S S qq f q f

, (1)

where eq denotes the yield stress of the material matrix,m eq

is stress triaxiality, ijS is the

stress deviator, the parameters q1 and q2 were introduced by Tvergaard [7] to improve the ductile

fracture prediction of the Gurson model and f * is the damage function [1]:

c

c

for*

( ) forc c

f f f f

f K f f f f

(2)

where f c is the critical value of f , at the moment when the void coalescence begins. For f * = 0, the

plastic potential (2) is identical with that of von Mises. The parameter K defines the slope of a

sudden drop of force on the force – diameter reduction diagram, and is often referred to as an

accelerating factor. This factor defines the final stage of ductile fracture – void coalescence, which

leads to complete loss of the load carrying capacity of the material.

During the ductile fracture of steel, the voids nucleate due to separation or fracture of non-

metallic inclusions and secondary-phase particles from the material matrix. In the initial phase of

the ductile fracture, the voids nucleate mostly around non-metallic inclusions. To quantify this

micromechanism, the volume fraction of non-metallic inclusions, f V, that can be determined using

light microscopy, should be known [8].

Two phenomena contribute to the increase of the void volume fraction in FE analysis with the

embedded GTN yield criterion. One is the growth of the existing voids and the other is the

nucleation of new voids under external loading:

nucleation growth f f f

; p

nucleation eq f A

and (1 )

p growth ii f f

(3)

where pii is the plastic part of the strain rate tensor.




Numerical and experimental tests have shown that the GTN model can describe both damage

development at microscopic level and plastic strain as a global, macro-parameter of material

behaviour. Some recent investigations [9, 10] aim to reduce the weak points of this model, but in

spite of these imperfections, it can be used to predict the values of macroscopic ductile toughness

[11]. The GTN model has been successfully applied for various problems and has been incorporated

into the Finite Element software ABAQUS. Recently, the plastic limit load model proposed byThomason [12] for void coalescence has received attention as a possible improvement to the GTN

model. Basically, Thomason proposed that localized deformation modes should be considered for

ductile fracture prediction, in addition to homogenous modes.

The localized deformation state of void coalescence is very different from the homogenous

deformation state during void nucleation and growth. The coalescence of voids depends on the

competition between these two deformation modes. For small deformations, the voids are still

rather small and it is easier to follow the homogenous deformation mode, since the stress required

for this mode is less than the stress for the localized deformation mode. However, plastic

deformation grows with the increase of loading, with subsequent increase of the void volume

fraction. During this process, the stress needed for localized deformation decreases, and when it is

equal to the stress needed for homogenous deformation - the void coalescence starts.

Zhang et al [13] made a significant modification of the GTN model – they applied the

Thomason’s void coalescence criterion based on the plastic limit load model, thus introducing the

Complete Gurson Model - CGM. The criterion for determining the moment when void coalescence

takes place is:

21 11 1 r

r r

, (4)

where1

is the maximum principal stress, and are constants introduced by Thomason [12],

and r is the void space ratio [13], given by:

2 3

1 2 333

4 2

f er e

, (5)

1 ,

2 and

3 being principal strains. Therefore, the critical void volume fraction f c is not a material

constant in the CGM - but the material response to void coalescence. The value of f c depends on the

strain field, hence it is not necessarily the same in all finite elements and in all integration points

within one element.

Application of the Complete Gurson Model

In the remainder of the paper, the CGM is applied in analysis of overmatched welded joints. The

SENB specimens are used for estimation of fracture behaviour of welded joints, and the width of

the joint is varied (6, 12 and 18 mm – Fig. 1a). The specimens were previously fatigue precracked

in accordance with [14], with the same length of the fatigue precrack (a0/W = 0.32) for all

specimens. CTOD values are directly measured using a d5 clip gauge, developed by GKSS [15].

The base metal (BM) is high-strength low-alloyed (HSLA) steel NIOMOL 490, which is often used

for steel constructions. The flux-cored arc-welding (FCAW) process in shielding gas is applied and

the filler FILTUB 75 (designation according to producer “Elektrode Jesenice”) is chosen to ensure

overmatching. Details about the geometry of the specimens, welding process, preparation of the plates and experimental procedure are given in [16]. Welded joints are considered as bimaterial

joints, according to the recommendations given in the literature [14, 17, 18], since the crack is

located in the weld metal, along the axis of symmetry of the joint.

Numerical and experimental tests have shown that the GTN model can describe both damage

development at microscopic level and plastic strain as a global, macro-parameter of material

behaviour. Some recent investigations [9, 10] aim to reduce the weak points of this model, but in

spite of these imperfections, it can be used to predict the values of macroscopic ductile toughness

[11]. The GTN model has been successfully applied for various problems and has been incorporated

into the Finite Element software ABAQUS. Recently, the plastic limit load model proposed byThomason [12] for void coalescence has received attention as a possible improvement to the GTN

model. Basically, Thomason proposed that localized deformation modes should be considered for

ductile fracture prediction, in addition to homogenous modes.

The localized deformation state of void coalescence is very different from the homogenous

deformation state during void nucleation and growth. The coalescence of voids depends on the

competition between these two deformation modes. For small deformations, the voids are still

rather small and it is easier to follow the homogenous deformation mode, since the stress required

for this mode is less than the stress for the localized deformation mode. However, plastic

deformation grows with the increase of loading, with subsequent increase of the void volume

fraction. During this process, the stress needed for localized deformation decreases, and when it is

equal to the stress needed for homogenous deformation - the void coalescence starts.

Zhang et al [13] made a significant modification of the GTN model – they applied the

Thomason’s void coalescence criterion based on the plastic limit load model, thus introducing the

Complete Gurson Model - CGM. The criterion for determining the moment when void coalescence

takes place is:

21 11 1 r

r r

, (4)

where1

is the maximum principal stress, and are constants introduced by Thomason [12],

and r is the void space ratio [13], given by:

2 3

1 2 333

4 2

f er e

, (5)

1 ,

2 and

3 being principal strains. Therefore, the critical void volume fraction f c is not a material

constant in the CGM - but the material response to void coalescence. The value of f c depends on the

strain field, hence it is not necessarily the same in all finite elements and in all integration points

within one element.

Application of the Complete Gurson Model

In the remainder of the paper, the CGM is applied in analysis of overmatched welded joints. The

SENB specimens are used for estimation of fracture behaviour of welded joints, and the width of

the joint is varied (6, 12 and 18 mm – Fig. 1a). The specimens were previously fatigue precracked

in accordance with [14], with the same length of the fatigue precrack (a0/W = 0.32) for all

specimens. CTOD values are directly measured using a d5 clip gauge, developed by GKSS [15].

The base metal (BM) is high-strength low-alloyed (HSLA) steel NIOMOL 490, which is often used

for steel constructions. The flux-cored arc-welding (FCAW) process in shielding gas is applied and

the filler FILTUB 75 (designation according to producer “Elektrode Jesenice”) is chosen to ensure

overmatching. Details about the geometry of the specimens, welding process, preparation of the plates and experimental procedure are given in [16]. Welded joints are considered as bimaterial

joints, according to the recommendations given in the literature [14, 17, 18], since the crack is

located in the weld metal, along the axis of symmetry of the joint.




Materials: Basic mechanical properties of the materials (base metal – BM and weld metal – WM),

determined on round tensile (RT) specimens, are given in Table 1, while the true stress – true strain

diagrams of both materials can be found in [18].

Table 1: Properties of the materials

Mechanical propertiesWM BM

E [GPa] 183.8 202.9

R p0.2 [MPa] 648 545

R m [MPa] 744 648

M [R p0.2 WM/R p0.2 BM] 1.19 –

Microstructural parameters

f V 0.006342 0.012164

λ [µm] 157.4719 103.1336

Table 2: Chemical composition of the materialsMaterial C Si Mn P S Cr Mo Ni

WM 0.04 0.16 0.95 0.011 0.021 0.49 0.42 2.06

BM 0.123 0.33 0.56 0.003 0.002 0.57 0.34 0.13

Microstructural observation of the materials indicates presence of sulphides, oxides, silicates and

complex inclusions. A significant fraction of oxides is observed in the BM and the WM (for details

see [19]), and a micrograph with clusters of oxides in the BM is shown in Fig. 1b.

(a) (b)

Fig. 1: (a) Welded joint (2H = 6, 12 and 18 mm) (b) Optical micrograph of non-metallic inclusions in BM

Volume fractions of non-metallic inclusions, V V, are determined based on equality with surface

fractions, AA: V V = AA, [20]. The volume fraction f V (see Table 3) is determined as the mean valueof volume fractions of non-metallic inclusions for all measurement fields:

Ai AV V

A f V A

n

(6)

where n is the number of measurement fields. Microstructural observations on low-alloyed steel

with 0.22 wt% of carbon given in [8] have shown that the effect of secondary voids formed around

Fe3C particles is extremely low and present only during the final stage of ductile fracture. Since the

percentage of carbon in BM and WM is lower (Table 2) in comparison with this investigation,

nucleation of the secondary voids is neglected in this paper. Therefore, it is assumed that the initialvoid volume fraction, f 0, is equal to the volume fraction f V, according to [8, 21].

In order to determine the mean free path between non-metallic inclusions according to [20], in

each measurement field five horizontal measuring (scan) lines are drawn. Then the value of N L is

Materials: Basic mechanical properties of the materials (base metal – BM and weld metal – WM),

determined on round tensile (RT) specimens, are given in Table 1, while the true stress – true strain

diagrams of both materials can be found in [18].

Table 1: Properties of the materials

Mechanical propertiesWM BM

E [GPa] 183.8 202.9

R p0.2 [MPa] 648 545

R m [MPa] 744 648

M [R p0.2 WM/R p0.2 BM] 1.19 –

Microstructural parameters

f V 0.006342 0.012164

λ [µm] 157.4719 103.1336

Table 2: Chemical composition of the materialsMaterial C Si Mn P S Cr Mo Ni

WM 0.04 0.16 0.95 0.011 0.021 0.49 0.42 2.06

BM 0.123 0.33 0.56 0.003 0.002 0.57 0.34 0.13

Microstructural observation of the materials indicates presence of sulphides, oxides, silicates and

complex inclusions. A significant fraction of oxides is observed in the BM and the WM (for details

see [19]), and a micrograph with clusters of oxides in the BM is shown in Fig. 1b.

(a) (b)

Fig. 1: (a) Welded joint (2H = 6, 12 and 18 mm) (b) Optical micrograph of non-metallic inclusions in BM

Volume fractions of non-metallic inclusions, V V, are determined based on equality with surface

fractions, AA: V V = AA, [20]. The volume fraction f V (see Table 3) is determined as the mean valueof volume fractions of non-metallic inclusions for all measurement fields:

Ai AV V

A f V A

n

(6)

where n is the number of measurement fields. Microstructural observations on low-alloyed steel

with 0.22 wt% of carbon given in [8] have shown that the effect of secondary voids formed around

Fe3C particles is extremely low and present only during the final stage of ductile fracture. Since the

percentage of carbon in BM and WM is lower (Table 2) in comparison with this investigation,

nucleation of the secondary voids is neglected in this paper. Therefore, it is assumed that the initialvoid volume fraction, f 0, is equal to the volume fraction f V, according to [8, 21].

In order to determine the mean free path between non-metallic inclusions according to [20], in

each measurement field five horizontal measuring (scan) lines are drawn. Then the value of N L is




determined, representing the number of interceptions of oxides or sulphides per measurement line

unit (in mm). The mean free path λ, as the mean edge-to-edge distance between inclusions, is

determined as follows:

1 A

L

A

N

. (7)

The average value of the mean free path λ is determined based on calculated values of λ in all

measurement fields. Using this procedure, values for both materials from Table 1 are determined.

Numerical analysis: Finite Element software ABAQUS (www.simulia.com) is used for numerical

analysis and the CGM is applied through user material subroutine (by Z.L. Zhang). The specimen is

analysed under plane strain conditions, and 8-noded isoparametric reduced integration elements are

used. FE mesh is shown in Fig. 2, with the magnification of the mesh near the crack tip. The crack

tip is modelled using refined FE mesh without singular elements.

Fig. 2: FE mesh and detail around the crack tip

Influence of FE size at the crack tip and initial void volume fraction: A number of studies have

shown that the size of the element at the crack tip should be similar to the mean free path between

non-metallic inclusions in steel [10, 21]. However, disregarding the applied method for

determination of FE size at the crack tip, it is a common opinion that FE size strongly effects

anticipation of the onset of crack growth. In addition to FE size 0.15x0.15 mm (corresponding to

the mean free path between the inclusions and second phase particles in WM), two other FE sizes

are considered: 0.3x0.3 mm and 0.075x0.075 mm.

Results: By application of the CGM, the onset of crack growth in analysed welded joints is

successfully predicted. Distribution of void volume fraction, f , near the crack tip, at the crack

growth initiation (for a joint 6 mm wide), is shown in Fig. 3. Concentration of large values very

close to the crack tip is obvious. One can also see a large variation of f in the elements near thecrack tip.

The effect of FE size at the crack tip on CTOD i is shown in Fig. 4 (width of the joint is 6 mm). It

can be seen that all three curves obtained numerically do not differ significantly from the

experimental one (Fig 4a). The moment of failure at the Gauss point nearest to the crack tip is

considered in Fig. 4b, and significant differences are present if this failure is taken into account.

Calculation with FE size of 0.3x0.3 mm gives higher material resistance than the real one, while the

opposite is obtained with FE size of 0.075x0.075 mm. It is obvious that FE size that approximately

corresponds to the mean free path between the inclusions is the most suitable. The results for

CTODi, obtained using these three element sizes, are given in Table 3.

determined, representing the number of interceptions of oxides or sulphides per measurement line

unit (in mm). The mean free path λ, as the mean edge-to-edge distance between inclusions, is

determined as follows:

1 A

L

A

N

. (7)

The average value of the mean free path λ is determined based on calculated values of λ in all

measurement fields. Using this procedure, values for both materials from Table 1 are determined.

Numerical analysis: Finite Element software ABAQUS (www.simulia.com) is used for numerical

analysis and the CGM is applied through user material subroutine (by Z.L. Zhang). The specimen is

analysed under plane strain conditions, and 8-noded isoparametric reduced integration elements are

used. FE mesh is shown in Fig. 2, with the magnification of the mesh near the crack tip. The crack

tip is modelled using refined FE mesh without singular elements.

Fig. 2: FE mesh and detail around the crack tip

Influence of FE size at the crack tip and initial void volume fraction: A number of studies have

shown that the size of the element at the crack tip should be similar to the mean free path between

non-metallic inclusions in steel [10, 21]. However, disregarding the applied method for

determination of FE size at the crack tip, it is a common opinion that FE size strongly effects

anticipation of the onset of crack growth. In addition to FE size 0.15x0.15 mm (corresponding to

the mean free path between the inclusions and second phase particles in WM), two other FE sizes

are considered: 0.3x0.3 mm and 0.075x0.075 mm.

Results: By application of the CGM, the onset of crack growth in analysed welded joints is

successfully predicted. Distribution of void volume fraction, f , near the crack tip, at the crack

growth initiation (for a joint 6 mm wide), is shown in Fig. 3. Concentration of large values very

close to the crack tip is obvious. One can also see a large variation of f in the elements near thecrack tip.

The effect of FE size at the crack tip on CTOD i is shown in Fig. 4 (width of the joint is 6 mm). It

can be seen that all three curves obtained numerically do not differ significantly from the

experimental one (Fig 4a). The moment of failure at the Gauss point nearest to the crack tip is

considered in Fig. 4b, and significant differences are present if this failure is taken into account.

Calculation with FE size of 0.3x0.3 mm gives higher material resistance than the real one, while the

opposite is obtained with FE size of 0.075x0.075 mm. It is obvious that FE size that approximately

corresponds to the mean free path between the inclusions is the most suitable. The results for

CTODi, obtained using these three element sizes, are given in Table 3.




Fig. 3: Distribution of void volume fraction, f , near the crack tip, at the crack growth initiation

(a) (b)

Fig. 4: (a) Load F vs. Crack tip opening displacement CTOD (b) part of the same diagram with points

corresponding to crack growth initiation

Table 3: Dependence of CTODi on FE size at the crack tip

Result CTODi [mm]

WM6 WM12 WM18

Experimental 0.084 0.092 0.065

FE = 0.075 x 0.075 mm at the crack tip 0.04 0.039 0.039



In order to analyse the influence of the initial void volume fraction, computations are conducted

with values of f 0 obtained by quantitative microstructural analysis, but also with the values

recommended in literature [10]. In Table 4 it is shown that the initial void volume fraction

influences the results for CTODi significantly, and that the value of f 0 obtained by microstructural

analysis leads to better results. Results in Table 4 are obtained with the size of the elements near the

crack tip of 0.15x0.15 mm.

Fig. 3: Distribution of void volume fraction, f , near the crack tip, at the crack growth initiation

(a) (b)

Fig. 4: (a) Load F vs. Crack tip opening displacement CTOD (b) part of the same diagram with points

corresponding to crack growth initiation

Table 3: Dependence of CTODi on FE size at the crack tip

Result CTODi [mm]

WM6 WM12 WM18

Experimental 0.084 0.092 0.065




In order to analyse the influence of the initial void volume fraction, computations are conducted

with values of f 0 obtained by quantitative microstructural analysis, but also with the values

recommended in literature [10]. In Table 4 it is shown that the initial void volume fraction

influences the results for CTODi significantly, and that the value of f 0 obtained by microstructural

analysis leads to better results. Results in Table 4 are obtained with the size of the elements near the

crack tip of 0.15x0.15 mm.


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