Issue 20

Anno VI Numero 20 Aprile 2012

Rivista Internazionale Ufficiale del Gruppo Italiano Frattura Fondata nel 2007

Editor-in-chief:

Francesco Iacoviello

ISSN 1971-8993

Associate Editors:

Luca Susmel John Yates

Editorial Advisory Board:

Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Robert O. Ritchie Cetin Morris Sonsino Ramesh Talreja David Taylor

Frattura ed integrità strutturale The International Journal of the Italian Group of Fracture

www.gruppofrattura.it

Frattura ed Integrità Strutturale, 20 (2012); Rivista Ufficiale del Gruppo Italiano Frattura

R. Brighenti, A. Carpinteri, A. Spagnoli, D. Scorza Crack path dependence on inhomogeneities of material microstructure .................................................. 6 P. Rezakhani Current state of existing project risk modeling and analysis methods with focus on fuzzy risk assessment – Literature Review ……………..………………............................................................................… 17 A. Borruto, G. Narducci, P. Pietrosanti Analysis of the causes of failure in 5Cr-1Mo pipes mounted in a preheating furnace ………………..… 22 H. Jasarevic, S. Gagula Case studies in numerical simulation of crack Trajectories in brittle materials ………….....…………... 32

Segreteria rivista presso: Francesco Iacoviello Università di Cassino – Di.M.S.A.T. Via G. Di Biasio 43, 03043 Cassino (FR) Italia http://www.gruppofrattura.it iacoviello@unicas.it

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Frattura ed Integrità Strutturale, 20 (2012); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello

(Università di Cassino, Italy)

Associate Editors Luca Susmel

(University of Sheffield, UK) (University of Manchester, UK)

John Yates

Advisory Editorial Board Alberto Carpinteri

(Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy)

Andrea Carpinteri

Donato Firrao M. Neil James Gary Marquis

(University of Plymouth, United Kingdom) (Helsinki University of Technology, Finland)

Robert O. Ritchie Cetin Morris Sonsino

(University of California, USA) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

Ramesh Talreja David Taylor

Journal Review Board Stefano Beretta

(Politecnico di Milano, Italy) (Università di Cassino, Italy) (Università di Trieste, Italy) (EADS, Munich, Germany) (IMWS, Wien, Austria) (Politecnico di Torino, Italy) (Politecnico di Milano, Italy) (University of Porto, Portugal) (Politecnico di Torino, Italy) (Università di Parma, Italy)

Nicola Bonora Lajos Borbás Francesca Cosmi

(Budapest University Technology and Economics, Hungary)

Claudio Dalle Donne Josef Eberhardsteiner Giuseppe Ferro Tommaso Ghidini Mario Guagliano

(European Space Agency - ESA-ESRIN)

Lucas Filipe Martins da Silva

Marco Paggi

Alessandro Pirondi Ivatury S. Raju Roberto Roberti Marco Savoia

(NASA Langley Research Center, USA)

(Università di Brescia, Italy) (Università di Bologna, Italy)

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

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Frattura ed Integrità Strutturale, 20 (2012); Rivista Ufficiale del Gruppo Italiano Frattura

Descrizione e scopi Frattura ed Integrità Strutturale è la rivista ufficiale del Gruppo Italiano Frattura . E’ una rivista open-access pubblicata on-line con periodicità trimestrale (luglio, ottobre, gennaio, aprile). Frattura ed Integrità Strutturale riguarda l’ampio settore dell’integrità strutturale, basato sulla meccanica della fatica e della frattura, per la valutazione dell’affidabilità e dell’efficacia di componenti strutturali. Scopo della rivista è la promozione di lavori e ricerche sui fenomeni di frattura, nonché lo sviluppo di nuovi materiali e di nuovi standard per la valutazione dell’integrità strutturale. La rivista ha un carattere interdisciplinare e accetta contributi da ingegneri, metallurgisti, scienziati dei materiali, fisici, chimici e matematici. Contributi Frattura ed Integrità Strutturale si prefigge la rapida disseminazione di contributi originali di natura analitica, numerica e/o sperimentale riguardanti la meccanica della frattura e l’integrità strutturale. Si accettano lavori di ricerca che contribuiscano a migliorare la conoscenza del comportamento a frattura di materiali convenzionali ed innovativi. Note tecniche, lettere brevi e recensioni possono essere anche accettati in base alla loro qualità. L’ Editorial Advisory Board sollecita anche la pubblicazione di numeri speciali contenenti articoli estesi presentati in occasione di conferenze e simposia tematici. Istruzioni per l’invio dei manoscritti I manoscritti devono essere scritti in formato word senza necessità di utilizzare un particolare stile e devono essere inviati all'indirizzo iacoviello@unicas.it. Il lavoro proposto può essere in lingua Italiana (con riassunto in inglese di almeno 1000 parole e didascalie bilingue) o Inglese. La conferma della ricezione avverrà entro 48 ore. Il processo di referaggio e pubblicazione on-line si concluderà entro tre mesi dal primo invio. Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. The paper may be written in English or Italian (with an English 1000 words abstract). A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission.

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Frattura ed Integrità Strutturale, 20 (2012); ISSN 1971-9883

T

rent’anni fa, all’ordine del giorno d’una delle primissime riunioni del Gruppo Italiano Frattura, figurava il punto Marchio IGF. Alcuni tra i fondatori del nostro Gruppo – i più entusiasti – si presentarono alla riunione proponendo ciascuno un proprio bozzetto. Dopo un disattento esame degli elaborati, senza procedere a una valutazione comparativa autoreferenziale e pertanto invalidabile, fu facile per il Presidente pro tempore disattendere le aspirazioni degli improvvisati designer senza ferire il loro amor proprio. Un luogo comune mise tutti d’accordo: La guerra è una cosa troppo seria per farla combattere dai generali: un bravo professore, ahimè, non necessariamente è un eccellente comunicatore, un esperto nel campo della meccanica della frattura non necessariamente è un maestro della comunicazione visiva. Un’accoglienza unanime trovò la mia proposta di interessare al nostro problema Franco Grignani. Nato a Pieve di Porto Morone, classe 1908, milanese d’adozione, grafico e fotografo, architetto e pittore, Franco Grignani era noto per i suoi contributi sperimentali nel campo della psicologia della forma – basti pensare alla Pura lana vergine, il più famoso dei suoi logotipi (Fig. 1).

Figura 1 : Franco Grignani, Il marchio Pura lana vergine. Figura 2 : Franco Grignani, Psicoplastica nel campo , acrilico, 1968. Carpire a questo personaggio la promessa di creare il marchio IGF non mi fu difficile. L’avevo incontrato sette anni prima a Milano, l’8 gennaio 1975, alla vernice della sua Mostra “Una metodologia della visione”, alla Rotonda di via Besana. Quell’incontro fu per me un colpo di fulmine. Ancora oggi in me è vivissimo il ricordo. Al cospetto dei suoi Psicoplastici (v. ad esempio la Fig. 2) mi colpisce la violenza del messaggio visivo. Ma non è tutto. Qualche minuto di riflessione e ravviso una sconcertante analogia formale tra la percezione dinamica delle sue strutture modulari risonanti e il processo, a me familiare, di misura spettroscopica delle strutture quantistiche appartenenti al nanoscopico mondo degli atomi. Traumatizzato, vinco la mia timidezza di allora, e mi presento a lui. “Maestro – gli chiedo – come mai le misure di spettroscopia quantistica che noi fisici facciamo in laboratorio, sulle quali poggiano le nostre certezze scientifiche e si fondano gli sviluppi concreti delle nuove tecnologie, rassomigliano in modo così evidente alle illusioni ottiche che provo all’atto della percezione di queste sue strutture indecidibili, ambigue, assurde, inconcretizzabili?” Mi risponde: “Queste opere non sono fatte per essere osservate da persone facilmente impressionabili. E poi. Perché inconcretizzabili? Le tocchi pure, professore!”. Quell’incontro mi turbò profondamente. Le rassicuranti certezze della vita di laboratorio si struggevano come neve al sole. Seguirono notti da incubo: C’era soltanto un modo per uscirne: studiare il problema con l’aiuto di colui che, inconsapevolmente, l’aveva sollevato. Con opere stimolate dai nostri incontri, protrattisi fino alla sua scomparsa nel 1999, Franco Grignani è riuscito a rendere per_cepibile visivamente la rappresentazione fedele di strutture precluse ai nostri sensi: di strutture e di concetti che, come scriveva negli anni Venti del secolo scorso Werner Heisenberg, uno dei padri della meccanica quantistica, non si prestano ad essere facilmente com_presi utilizzando il linguaggio ordinario i cui concetti derivano dall’esperienza della vita di ogni giorno. Custodisco gelosamente le rappresentazioni dinamiche dei sistemi quantistici bistabili sviluppate da Grignani. Mi riferisco ad esempio alla rappresentazione (Fig. 3) del quantum bit – il qu-bit ovvero l’unità elementare del quanto di

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Frattura ed Integrità Strutturale, 20 (2012); Rivista Ufficiale del Gruppo Italiano Frattura

informazione – e al tempo stesso della molecola di idrogeno mentre assorbe un fotone durante una misura spettroscopica caratterizzata da un trasferimento risonante della carica elettronica. A mio avviso questa immagine, carica di ambiguità, propone alla nostra percezione non un’illusione ottica, bensì una rappresentazione fedele del comportamento dinamico del sistema soggetto – sistema-quantistico durante il processo di misurazione.

Figura 4 : Franco Grignani, il marchio IGFI, 1982.

Figura 3 : Franco Grignani, Rappresentazione dinamica del qu-bit e della molecola d’idrogeno

Torniamo al marchio IGF (Fig. 4). L’occhio è chiamato prepotentemente a percorrere e a chiudere il perimetro della figura disegnandolo mentalmente, cosicché l’immagine non tarda a presentarcisi come una EFFE maiuscola adagiata sul piano. Ma il processo percettivo non si ferma qui. I moduli strutturali di questa composizione s’insinuano nella mente, chiamata a interiorizzarli e a correlarli ulteriormente. Dopo un prolungato lavorio nascosto, le sinapsi quiescenti di chi, incuriosito, intensifica lo sforzo di at-tenzione, si attivano bruscamente: all’improvviso – è lo stadio critico della “catastrofe percettiva” – l’immagine della EFFE, dianzi compattata mentalmente, sembra sgretolarsi per lasciare il posto a una sorta di griglia che sguscia via dalla superficie piana dell’acrilico e invade la terza dimensione. E’ una griglia che, nella versione originale del marchio (è doveroso puntualizzarlo), è tutta rigorosamente nera. Si sa, i marchi evolvono. Pensiamo al marchio FIAT. Fagocitato dagli anni, l’acronimo dell’antica Fabbrica-Italiana-Automobili-Torino nella versione odierna è tornato a riveder le stelle. Il marchio autentico IGF stimola così il nostro pensiero visivo a evolversi in guisa di metafora del processo di frattura: una instabilità dinamica, questa, che consiste appunto in un prolungato lavorìo durante il quale si intensifica nel provino lo sforzo tensile, fino a pervenire ad uno stadio critico – la catastrofe – che si risolve nello sgretolamento. Franco Grignani donò questo marchio al Gruppo Italiano Frattura. Valga, questo scritto, a mantenere viva la sua memoria.

Giuseppe Caglioti Cofondatore e primo Presidente del Gruppo Italiano Frattura Professore emerito di fisica della materia al Politecnico di Milano giuseppe.caglioti@polimi.it www.piezomusicolor.it

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

Crack path dependence on inhomogeneities of material microstructure

Roberto Brighenti, Andrea Carpinteri, Andrea Spagnoli, Daniela Scorza Department of Civil and Environmental Engineering & Architecture, University of Parma, Viale Usberti, 181/A, 43124 Parma, Italy spagnoli@unipr.it

A BSTRACT . Crack trajectories under different loading conditions and material microstructural features play an important role when the conditions of crack initiation and crack growth under fatigue loading have to be evaluated. Unavoidable inhomogeneities in the material microstructure tend to affect the crack propagation pattern, especially in the short crack regime. Several crack extension criteria have been proposed in the past decades to describe crack paths under mixed mode loading conditions. In the present paper, both the Sih criterion (maximum principal stress criterion) and the R-criterion (minimum extension of the core plastic zone) are adopted in order to predict the crack path at the microscopic scale level by taking into account microstress fluctuations due to material inhomogeneities. Even in the simple case of an elastic behaviour under uniaxial remote stress, microstress field is multiaxial and highly non-uniform. It is herein shown a strong dependence of the crack path on the material microstructure in the short crack regime, while the microstructure of the material does not influence the crack trajectory for relatively long cracks. S OMMARIO . L’andamento dei percorsi di frattura sotto diverse condizioni di carico e caratteristiche microstrutturali del materiale ha un ruolo importante nella determinazione delle condizioni di nucleazione e propagazione a fatica della fessura. Inevitabili disomogeneità nella microstruttura del materiale tendono a influenzare il percorso di propagazione della fessura, particolarmente nel caso di fessure corte. In letteratura sono stati proposti numerosi criteri volti a descrivere i percorsi di frattura in condizioni di modo misto. Nel presente lavoro, sia il criterio di Sih (della massima tensione principale) che il criterio R (della minima estensione della zona plastica) sono stati adottati per predire i percorsi di frattura alla scala microscopica tenendo conto di fluttuazioni delle microtensioni dovute alle disomogeneità del materiale. Anche nel semplice caso di comportamento elastico del materiale in presenza di tensione monoassiale remota, il campo tensionale alla microscala risulta essere multiassiale e non uniforme. Viene evidenziata una significativa dipendenza del percorso di frattura nel caso di fessure corte dalla microstruttura del materiale, mentre nel caso di fessure lunghe tale dipendenza risulta trascurabile. K EYWORDS . Crack path; Kinked crack; Microstress fluctuation; Material microstructure.

I NTRODUCTION

he evaluation of fatigue crack growth at small scale is still an open problem. As a matter of fact, when the crack size is comparable with a characteristic length of the material (e.g. grain size in metallic materials), mechanical barriers to crack growth are produced by the material microstructure, and the plastic zone size at crack tip T

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

happens to be comparable with the crack size, leading to the violation of the small-scale yielding hypothesis. The effects of the material microstructure on the crack growth at small scale [1] can be modelled by taking into account the non- uniform stress field induced by embedded inohomogeneities. Even for a uniform remote stress applied to the structural component, an oscillating stress field might develop at the microscale. In the present paper, by using both the solution of a homogeneous elastic infinite plane with a circular elastic inclusion and the superposition principle, the stress field for a regular arrangement of inclusions is determined, and the corresponding mixed mode Stress Intensity Factors (SIFs) are computed. Crack paths are evaluated by applying both the maximum principal stress criterion (the Sih criterion [2, 3]) and the minimum plastic zone extension criterion (the R- criterion [4, 5]). The trajectory described by the crack tip is computed through an incremental method, where the Mode I and Mode II SIFs of the kinked crack are approximately evaluated as a function of the SIFs related to a projected straight crack. Finally, some examples related to metallic alloys are examined. It is shown that small-scale fluctuations of the stress field heavily affect the crack path for short cracks while, after reaching a transition point during the crack propagation process, such an influence disappears for sufficiently long cracks.

M ICROSTRESS FIELD INDUCED BY MATERIAL INHOMOGENEITIES

S

tructural materials always present heterogeneity features due to either the composite nature of the materials (e.g. composite materials characterised by a matrix and a reinforcing phase; concrete-like materials having a cement- based paste with dispersion of aggregates of different sizes) or unavoidable inhomogeneities (e.g. metallic alloys composed by a base material and secondary inclusions), see Fig. 1. Due to such inhomogeneity characteristics, the stress field in the material at microscopic level might be non-uniform and multiaxial even if a uniaxial uniform remote stress is applied. A local fluctuation of the microstress field can play a crucial role in the crack path assessment for cracks having length comparable with a characteristic material length.

(a) (b) Figure 1 : (a) Micrograph of pure iron with ferrite inclusions and crystals having a polygonal shape. (b) Typical concrete material with aggregates, cement paste and voids. The modelling rationale here adopted to describe the inhomogeneities contained in the material is based on a periodic distribution of spherical particles embedded in the base material. By considering, for the sake of simplicity, a single inclusion of radius embedded in an infinite plane under remote uniform stress (Fig. 2), the elastic stress field can be determined by applying the superposition principle together with the Kirsch solution [6]. The resulting stress field, , , x y xy    , is uniform within the inclusion, and can be expressed as a fraction of the remote applied stress 0 y  :



x k  

0 



k  

0 





,

,

0

(1)

x

y

xy

y

y

y

, , x y xy    under plane stress condition in the region around the inclusion (see point P

On the other hand, the stress field

in Fig. 2) can be expressed as follows [6]:

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

      

   

2

 

) k k R

(1

  

  

  

  

2

2 3 18 

2

2

x k R

R y

y

2

y

x

0    y





F G   





3

1

x

2

2

2

2

r

r

r

r

2

   

2

 

) k k R

(1

 

     

  

  

2

2 3 10 

2

2

x k R

R y

y

2

y

x

0      1 y

1   

F G   





1

(2)

y

2

2

2

2

r

r

r

r

2

   

   

2 k k R xy )

 

(1

  

2 4 R xy r

2 R y  6 8 12 3  2

2 2

k

2

R y

y

x



0    y

 



x

xy

4

2

4

r

r

r

2

2

where r x y   . The x-y coordinate system has origin in the inclusion centre (Fig. 2). Further, the coefficients , x y k k depend on the elastic constants of the base material ( 1 1 , E  ) and of the inclusion ( 2 2 , E  ) [6]:   (3 1) (1 3 ) c E E E      

2

2

1

1 2



k

x

2 2 (8 2 6 1 ) c c  

2        2 2 (1 )  E

1 2  

2  2 6 2 ) c c

  

E c

E E

(2

2

1

1

1

2

1 2

(3a)





2  2 E c    

1  (8 3 )

2 1 c E E 

(3 )



k

y

2 2 (8 2 6 1 ) c c  

2        2 2 (1 )  E

1 2  

2  2 6 2 ) c c

  

E c

E E

(2

2

1

1

1

2

1 2

2 1 1 c   

with

, and

2 8 (3 2 ) , y R y  2 2

2 4

R y

24





F

G

(3b)

4

6

r

r

Figure 2 : Circular elastic inclusion in an infinite elastic plane under remote uniform tensile stress 0 y  .

The elastic stress field in such heterogeneous materials can be computed by exploiting both Eqs 1-3 for a single inclusion and the superposition principle, provided that the inclusions are assumed to be non-interacting (as reasonably occurs for widely spaced inclusions). By considering point P belonging to the base material (Fig. 3), the resulting stress field is determined approximately by summing up the effects of the inclusions (such as particles 1, 2, 3, 4, etc. in Fig. 3), that is:

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

Figure 3 : Equally-spaced circular inclusions in an infinite domain arranged in a hexagonal cell pattern having characteristic size d , under remote uniform tensile stress 0 y  .



x 







( ) P

r

(

,

)

( ) i x i P i P ( ) ( )

i



 

 



0    y





0 



( ) P

r

(

,

)

(4)

y

i y

( ) i P i P ( )

y

( )

i











( ) P

r

(

,

)

xy

( ) i xy i P i P ( ) ( )

i

where the cartesian stress tensor components ( ) ( ) ( ) ( , ) i x i P i P r   , ( ) ( ) ( ) 0 ( ( , ) ) i y i P i P y r     , ( ) ( ) ( ) ( , ) i xy i P i P r  

indicate the stress

1, 2, 3, 4,..... i  ), see Eq. 2, under the

fluctuations evaluated in P in an elastic infinite plane containing a single inclusion i (

remote stress 0 y  . In the above expressions, the summation might be performed by taking into account all the inclusions that are within a significant influence region around the point P under consideration, since the inclusions located at a sufficiently large distance from P produce vanishing fluctuations of the stress components. In Fig. 4, sample spatial distributions of the fluctuating stress components along different lines normal to the remote loading axis are shown.

-0.008 0.008 dimensionless stresses,  x /  0y ,  xy /  0y

1.002

-0.004 0.004 dimensionless stresses,  x /  0y ,  xy /  0y (a) -0.002 0 0.002

dimensionless stresses,  y /  0y

1.004

dimensionless stresses,  y /  0y -0.004 0 0.004 (b)

1.001

1

1

0.999

 y /  0y  x /  0y  xy /  0y

 y /  0y  x /  0y  xy /  0y

0.996

0.998

0E+000 4E-004 8E-004 Position (m)

0E+000 4E-004 8E-004 Position (m)

Figure 4 : Stresses along a horizontal straight path (dashed line) located at (a) half distance and (b) one-third distance between two lines of inclusions, in an infinite plane under plane stress remote uniform tension stress 0 y  . Dots indicate the positions of inclusions in the material.

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

A PPROXIMATE SIF S FOR A NOMINALLY - MODE I KINKED CRACK

I

 , the SIF of a straight crack of semi-length l

n the case of an infinite cracked plane under a uniform remote stress 0 y

aligned with the X-axis is . Assume that such a straight crack is embedded in the stress field (given by Eqs 2-4) within the base material. Such a stress field can be decomposed in the remote uniform uniaxial tensile stress 0 y  and a fluctuating multiaxial stress field ( ) x T  here assumed to be a one-dimensional function of the x coordinate. Furthermore, by observing the courses (reported in Fig. 4) of the stress components due to the presence of inclusions, we can suppose that ( ) x T  is a self- balanced microstress field characterized by a material length d (related to the inclusion spacing), with two non-zero stress components ( / ) y a f x d         and ( / ) xy a f x d         . For the sake of simplicity, we assume     / cos 2 f x d x d   (this could be regarded as a first order approximation through Fourier series of a general periodic function), Fig. 5a. Under the self-balanced microstresses   and   , the SIFs (of the projected crack) are obtained using Buckner’s superposition principle:     cos 2 l l l f x d x d    0  I y K l    2 ( )

l



l

l

l

  

  

K 







a  

a  

  









dx

dx

dx

l J

2

2

2

I

a

0







d

2

2

2

2

2

2







0

0

0

l

x

l

x

l

x

(5)

 







f x d

x d

cos 2

l 





2

l

l

l

  

  

l

l

l

K 













a 

a 

a 











dx

dx

dx

l J

2

2

2

II

0







d

2

2

2

2

2

2







0

0

0

l

x

l

x

l

x

where J 0

is the zero-order Bessel function [7].

(a) (b) Figure 5 : (a) Self-balanced microstress field and periodically kinked crack. (b) Kinked crack in an infinite plane.

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

The total SIFs of the straight crack with semi-length l are the sums of the two contributions due to remote and microstress fields, that is:



( ) II K K K K K K        ( ) I I I II II

(6)

In the self-balanced microstress field, we assume that the crack might kink at each material microstructure semi-period, namely at each reversal in the microstress spatial courses. Obviously, kinking occurs since the microstress field is multiaxial. Because of the symmetry condition related to the Y-axis, the crack propagates symmetrically with respect of such an axis. Now, considering at first a singly-kinked crack (of projected crack length 2 l ), we have that the SIFs at the tips of the inclined part of the crack can be expressed through the SIFs I K and II K of a straight crack of length equal to the projected length of the kinked crack [8,9], that is:         11 12 21 22 , , , , I I II II I II k a b a K a b a K k a b a K a b a K         (7) where ij a are coefficients which depend on the slant angle  (positive counter-clockwise for tip coordinate x > 0) and the length ratio b a between the deflected leading segment and the horizontal trailing (preceding) segment (Fig. 5). If a geometry different from that of an infinite plate with a central crack were examined, the SIFs defined with respect to the projected crack would change but not the expressions in Eq. 7. The coefficients ij a for b a  (and, with good approximation, also for 0.3 b a  ) are [8]:

           

3 2

  



a a a a

cos

11

1 2

2sin cos 



12

(8)

1 2 sin cos cos 2 cos  



 

21



1 2



22

Note that the local SIFs in Eq. 8 are equal to those of an inclined straight crack of projected semi-length l forming an angle 2    with respect to the loading axis of 0 y  [8], Fig. 6.

Figure 6 : Infinite cracked plane with an inclined crack under remote tensile stress 0 y  .

Then, we assume that, as the crack propagates following the path in Fig. 5a, only the latter deflection of the crack path influences the stress field near the crack tips (e.g. along the straight segment 2-3 in Fig. 5a, the deflection point 2 has an

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

effect, while the deflection point 1 does not have). The local SIFs at the crack tip are assumed to be given by Eqs 7 and 8 for deflected (Mode I+II) segments (the segments 1-2 and 1 2   in Fig. 5b). The approximate calculation (based on the assumption that the near-tip stress field depends on the local crack direction at the crack tip) of the local SIFs for the kinked crack is examined for the case of an edge cracked plane under uniform tensile stress, Fig. 7a [10]. In particular, a two-equal-segment kinked crack with 1 45    and 2  ranging from 0° to 60° is considered. In Fig. 7b, we can observe that the SIFs computed by means of a FE model agree quite satisfactory with the analytical values corresponding to an equivalent slant straight crack (see thin line in Fig. 7a) having both the direction of the leading segment of the kinked crack and a projected length (normal to the loading axis) equal that of the kinked crack.

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 Dimensionless SIFs, F I(II) = K I(II) /  0y (  a 2 ) 1/2

FEM Present Study

 1 =  da = a 1

F I F II

30

45

60

75

90

Angle,  2 (degrees)

(a) (b) Figure 7 : (a) Edge-cracked plane. (b) SIFs obtained from the simplified method (present study, dashed line) and a FE analysis (continuous line) for a two-segment crack.

M IXED - MODE CRACK PROPAGATION CRITERIA

T

he kinked pattern of a crack embedded in the microstress field above described (see Sections 2 and 3) can be analysed by adopting a mixed-mode crack propagation criterion. Several criteria for both stable and unstable crack propagation have been proposed during the last decades for different materials. According to the MTS-criterion (Maximum Tensile Stress) proposed by Erdogan and Sih [2, 3], the crack grows in the direction perpendicular to the maximum principal stress (   ) direction or, equivalently, parallel to the maximum tangential stress. Analytically, the criterion can be stated as follows: 2 (9) where the polar coordinate  is used to identify the position vector with respect to the crack tip direction. By means of the stress field expressions (2), Eq. 9 can be written as follows: 2 0, 0            

1 2 2 2 2     tan

2

/ k k  

 

tan

0

with

(10)

I

II

This classical criterion, used to describe the mixed-mode crack propagation under the local SIFs I k and II kinking angle  , defined with respect to the general inclined axis of the crack, given by:

k , provides a

   

   

2

I II   k   k

k

1 4

1 4







8   

(11)

2 arctan

I

k

II

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

Several others criteria have been proposed, for instance the zero shear stress criterion by Maiti et al. [11], the M-criterion proposed by Kong et al. [12] based on the maximum value of the stress triaxiality ratio / H eq M    ( H  is the hydrostatic stress, whereas eq  is an equivalent stress which can be assumed to be equal to the Von Mises stress), the maximum dilatational strain energy density criterion ( T-criterion ) proposed by Theocaris et al. [13-15]. From experimental tests, it has been observed that the crack propagation direction usually tends to follow the local or global minimum extension of the plastic core region. From a physical point of view, this phenomenon could be explained by considering that the plastic core region is a highly-strained area, and the crack tends to reach the elastic region of the material outside the plastic zone, propagating through the plastic region which develops around the crack tip. Therefore, it is reasonable to assume that the crack follows the “easiest” path to reach the elastic region. Such a path can be assumed to coincide with the shortest path from the crack tip to the elastic material outside the plastic zone, as is stated by the R- criterion proposed by Shafique et al. [4, 5] (Fig. 8). The R-criterion can mathematically be written as follows : 2 p R is the function which defines the radial distance from the crack tip to a generic point of the plastic zone boundary 1 2 ( , ) 0 F I J  , with 1 I = first stress tensor invariant and 2 J = second deviatoric stress tensor invariant. When the conditions stated in Eq. 12 are fulfilled, the direction of minimum radial distance is determined, and the crack propagation direction vector t is assumed to be coincident with such a direction (Fig. 8). 2  0, 0 p p R R        (12) where

y

R ( ) p 

plastic region

crack



x

t

elastic region

F(I ,J )=0 1 2 Figure 8 : Graphical representation of the R-criterion.

 is proportional to the square root of

The above criterion can also be justified by considering that the fracture stress f

f w , which is the fracture energy per unit surface area. Such an energy for a quasi-brittle elastic-plastic material is equal to the summation of the surface energy s  and the plastic work p  consumed to create a unit surface area, that is,

s   



. For structural materials, where typically p 

  , the fracture stress

f  appears to be primarily dependent

w

f

p

s

on p  only. The shortest distance from the crack tip to the elastic-plastic boundary corresponds to the minimum plastic work which is needed to create a new portion of crack area, that is, such a shortest distance corresponds to the minimum values of fracture energy and fracture stress.

A PPLICATIONS TO SHORT - CRACK PROPAGATION REGIME

he above described model for the assessment of crack propagation at the microscale is herein applied to a carbon steel D6ac whose composition and mechanical parameters are presented in Tab. 1, where only the main secondary elements are listed. By performing a weighted average of the physical and mechanical parameters of the secondary constituents, a single equivalent inclusion with the features reported in Tab. 2 can be defined. T

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

volume fraction

Young modulus E [Gpa]

Poisson’s ratio

Thermal expansion coeff.

Element

Mass density



 [kg/m 3 ]

 [K -1 ]

[%]



Iron

Fe

~ 98.00 ~ 1.05 ~ 1.05

7870 10220 7190

200 330 248

0.29 0.38 0.30

1.20E-05 5.35E-06

Molibden Cromium

Mb

Cr 6.20E-06 Table 1 : Physical and mechanical parameters of the main elements in a carbon steel D6ac.

volume fraction

Young modulus E [GPa]

Poisson’s ratio

Thermal expansion coeff.

Element

Mass density



 [kg/m 3 ]

 [K -1 ]

[%]



Base material

Fe

~ 98.00

7870

200

0.29

1.20E-05

Equivalent inclusion

--

~ 2.10

8705

289

0.34

5.78E-06

Table 2 : Mean physical and mechanical parameters of the base material and the equivalent inclusion in a carbon steel D6ac. Now consider an infinite plane under remote uniform tensile stress 0 y  , containing an initial straight crack normal to the applied stress. By adopting the equivalent inclusion volume fraction (Tab. 2) and considering an average inclusion diameter equal to about 20 m  (e.g. see Ref. [16]), an inclusion spacing d equal to about 234 m  can be computed for a regular hexagonal distribution of inclusions (Fig. 3). The static crack extension is determined by applying the above described criteria (the Erdogan-Sih criterion and the R-criterion) on the crack growth direction. The mixed mode SIFs are computed by taking into account only the remote stress 0 y  (the local fluctuation of the stress component y  is negligible, as is shown in Fig. 4) and the micro shear stress fluctuations   . In Fig. 9, the crack path predicted for an initially straight crack developing at half distance between two horizontal lines of inclusions (see Fig. 4a, with 0 / 0.0026 a y     ) is represented. The crack path evaluated by the Erdogan-Sih criterion is similar to that determined by the R-criterion (Fig. 9). Nevertheless, it can be observed that the R-criterion produces a slight crack path deviation since the plastic zone shape is influenced in a complex way by the Mode I and Mode II SIFs which continuously change during the whole process of crack propagation.

(a) (b) Figure 9 : (a) Path of an initially straight crack developing at half distance between two lines of inclusions in an infinite plane under remote uniform tensile stress y 0  . (b) Detail of the crack path at the microscale where the distribution of inclusions is shown.

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

Fig. 10 shows the crack path determined for an initially straight crack developing at a vertical distance equal to one-third between two horizontal lines of inclusions (see Fig. 4b, with 0 / 0.0037 a y     ). As in the previous case, the crack path evaluated by the Erdogan-Sih criterion is rather similar to that determined by the R-criterion.

(a) (b) Figure 10 : (a) Path of an initially straight crack developing at one-third distance between two lines of inclusions in an infinite plane under remote uniform tensile stress 0 y  . (b) Detail of the crack path at the microscale where the distribution of inclusions is shown. By comparing Fig. 9 and Fig. 10, it can be observed that the kinking angle of the crack tends to increase with increasing the value of the 0 / a y    ratio.

C ONCLUSIONS

I

n the present paper, a simple analytical model to describe the trajectory of a plane crack propagating within an inhomogeneous material is proposed. With reference to metals, the inhomogeneities are treated by considering a two-phase material with an equivalent mean inclusion characterized by a regular spatial distribution. Even under remote uniaxial loading, the equivalent inclusions generate a multiaxial fluctuating stress field which is assumed to be responsible for mixed-mode crack propagation. By adopting different mixed-mode crack growth criteria, the crack path can be connected with the main features of the material microstructure, here accounted in terms of an appropriate microstress field. It is shown that both the maximum principal stress criterion and the R- criterion (based on the minimum extension of the core plastic zone) predict a zig-zag crack pattern, characterised by a length scale related to both the volume fraction of inclusions and their mean size. Moreover, it is shown a strong dependence of the crack path on the material microstructure in the short crack regime, while the microstructure of the material does not influence the crack trajectory for relatively long cracks.

A CKNOWLEDGEMENTS

T

he authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR).

R EFERENCES

[1] S. Suresh Metallurgical Trans, 14A (1985) 2375. [2] F. Erdogan, G. C. Sih, J Basic Engng, 85 (1963) 519.

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R. Brighenti et alii, Frattura ed Integrità Strutturale, 20 (2012) 6-16; DOI: 10.3221/IGF-ESIS.20.01

[3] G.C. Sih, Int. J. Fract, 10 (1974) 305. [4] M.A.K. Shafique, K. K. Marwan, Engng Fract Mech, 67 (2000) 397. [5] M.A.K. Shafique, K. K. Marwan, Int. J Plasticity, 20 (2004) 55. [6] Ye.Ye. Deryugin, G.V. Lasko, S. Schmauder, Field of stresses in an isotropic plane with circular inclusion uncere tensile stress (http://www.ndt.net/article/cdcm2006/papers/lasko.pdf) [7] A.Carpinteri, A. Spagnoli, S. Vantadori, In: The 13 th Int. Congress on Mesomechanics (Mesomechanics 2011), 6-8 July 2011, Vicenza (Italy). [8] H. Kitagawa, R. Yuuki, T. Ohira, Engng Fract Mechs, 7 (1975) 515. [9] YZ. Chen, Theoretical Applied Fract Mech, 31 (1999) 223. [10] A. Carpinteri, R. Brighenti, S. Vantadori, D. Viappiani, Special Issue Engng Fract Mech, 75 (3-4) (2007) 510. [11] S. K.Maiti, R. A. Smith, Int J Fract, 23 (1983) 281. [12] X. M. Kong, M. Schulter, W. Dahl, Engng Fract. Mech. , 52 (1995) 379. [13] P. S. Theocaris, NP. Adrianopoulos, Engng Fract Mech, 16 (1982) 425. [14] P. S.Theocaris, G. A. Kardomateas, NP. Adrianopoulos, Engng Fract. Mech, 17 (1982) 439. [15] P. S. Theocaris, NP. Adrianopoulos, Int. J. Fract, 20 (1982) R125. [16] Y. Murakami, Metal Fatigue: Effects of Small Defects and Nonmetallilc Inclusions. Elsevier, Amsterdam, (2002).

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P. Rezakhani, Frattura ed Integrità Strutturale, 20 (2012) 17-21; DOI: 10.3221/IGF-ESIS.20.02

Current state of existing project risk modeling and analysis methods with focus on fuzzy risk assessment – Literature Review

Pejman Rezakhani Kyungpook National University, School of Civil and Architectural Engineering, 1370 Sankyuk-dong, Buk-gu, Daegu 702-701, Korea rezakhani@knu.ac.kr

A BSTRACT . Risk modeling and analysis is one of the most important stages in project success. There are many approaches for risk assessment and an investigation of existing methods helps in developing new models . This paper is an extensive literature survey in risk modeling and analysis methods with main focus on fuzzy risk assessment. K EYWORDS . Risk modelling; Fuzzy risk assessment; Project risk. risk is defined as the potential for complications and problems with respect to the completion of a project and the achievement of a project goal [1] and as an uncertain future event or condition with the occurrence rate of greater than 0% but less than 100% that has an effect on at least one of project objectives (i.e., scope, schedule, cost, or quality, etc). In addition, the impact or consequences of this future event must be unexpected or unplanned [2]. It is well accepted that risk can be effectively managed to mitigate its’ adverse impacts on project objectives, even if it is inevitable in all project undertakings. The source of risk includes inherent uncertainties and issues relative to company’s fluctuating profit margin, competitive bidding process, weather change, job- site productivity, the political situations, inflation, contractual rights, and market competition, etc [3]. It is important for the construction companies to face these uncertain risks by assessing their effects on the project objectives because a risk quantitative method allows deciding which of the project is more risky, planning for the potential sources of risk in each project, and managing each source during construction [4]. It is noteworthy that risk is distinguished from uncertainty. The one is measurable uncertainty; the other is immeasurable risk [3, 5, 6]. Therefore, managing risks is involved in identifying, assessing and prioritizing risks by monitoring, controlling, and applying managerial resources with a coordinated and economical effort so as to minimize the probability and/or impact of unfortunate events and so as to maximize the realization of project objectives [7]. Project risk management, which has been practiced since the mid-1980s, is one of the nine main knowledge areas of the project management institute’s project management body of knowledge [8]. Effective risk management may lead the project manager to several benefits such as identification of favorable alternative course of action, increased confidence in achieving project objective, improved chances of success, reduced surprises, more precise estimates (through reduced uncertainty), reduced duplication of effort (through team awareness of risk control actions), etc [9]. Systemic project risk management has an effect on the project success. It is found that there is a strong relationship between the amount of risk management efforts undertaken in a project and the level of the project success [10]. Several project risk management approaches are proposed as follows; i.e., PRAM [11], RAMP [12], PMBOK [13], RMS [14], etc [15]. Existing approaches may be summarized into a four phase process for effective project risk management, i.e., Identifying risks, assessing risks, responding risks, and monitoring and/or reviewing risks. Identifying risks is the first step which determines which risk components may adversely affect which project objectives and documents their characteristics [3]. Construction risks are classified in many ways by risk types (i.e., natures, and magnitudes, etc), the sources and/or origins, or project phase [16-19]. Some of the existing researchers propose a hierarchical structure of risks which classifies the risks according to their origin and the location which the risk impacts to the project [20, 21]. A

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P. Rezakhani, Frattura ed Integrità Strutturale, 20 (2012) 17-21; DOI: 10.3221/IGF-ESIS.20.02

Assessing risks is the step which prioritizes the risks for further analysis by quantifying their occurrence rates. Risk assessment method is an essential component for this step. The existing methods are classified into (1) simple classical methods, and (2) advanced mathematical models [3]. The existing risk assessment methods are either qualitative or quantitative which require different information and the level of detail [22]. The simple classical methods integrate deterministic risk modeling and analysis into CPM scheduling. The deterministic methods include sensitivity analysis [23], critical path method [24], fault tree analysis [25], event tree analysis [26], failure mode, and effects and criticality analysis, etc [27]. Other advanced approaches were proposed as follows; a Monte Carlo Simulation [23] for stochastic quantitative modeling and analysis; scenario analysis [28], and fuzzy set theory for qualitative judgment [28]. There are many factors which should be considered when a project risk manager selects a risk assessment method as follows; i.e., the cost of employing the technique, the level of external party`s approval, organizational structure, agreement, adoptability, complexity, completeness, level of risk, organizational size, organizational security philosophy, consistency, usability, feasibility, validity, and credibility and automation [29]. It is essential for the risk manager to have high quality data in order to effectively apply the quantitative methods, even if it is not easy to obtain such high quality data relative to risk items in the construction industry. The difficulty is attributed to address the uncertainties and subjectivities associated with construction activities [30]. Beside the lack of collectability, the uniqueness, and non-repetitive nature of construction projects impedes using probabilistic risk quantification approaches [31]. Responding risks is involved in developing options and/or actions to enhance opportunities to achieve the project objectives. Finally, monitoring and reviewing risks is to implement a risk response plan, to keep tracking of the risks identified, to monitor residual risks, to identify new risks, and to evaluates the effectiveness of the project risk management process [15]. For this step, each engineering expertise should use specialized risk management tool as shown in Table 1 for risk analysis depending on project phase.

Planning/ Programming

Preliminary Engineering

Final Design

Construction

Discipline

Planning Environmental Funding Approval Project Management Engineering Civil, Structural, Systems Cost Estimating Scheduling Budgeting Controls Real Estate/Right of Way Construction Management/Oversight Constructability/Contractor Other Technical (e.g. Legal, Permitting, Procurement) Risk Facilitation Table 1 : Key expertise for risk analysis by project phase (Adapted from [32]).

Highly desirable; Desirable but optional

depending upon circumstances.

F UZZY R ISK A SSESSMENT fter Zadeh [33] introduced the concept of Fuzzy sets, and Fuzzy set theory, several researchers such as Kangari [34], Kangari and Riggs [35], Peak et al. [36], Tah and McCaffer [20], Wirba et al. [21], , Cho et al. [38], Choi et al. [39], Lyons and Skitmore [40], Baker and Zeng [41], Dikmen et al. [42], Zeng et al. [30], Wang and Elang [43], Karimiazar et al. [3], and Nieto et al. [15] -introduced fuzzy set theory(FST)-based risk modeling and analysis methods that deal with ill-defined, vague, imprecise, and complex risk analysis problems. For example, Kangari [34] proposes the application of fuzzy theory in risk analysis method using linguistic terms. The fuzzy theory-based risk analysis method was implemented as a part of construction project risk management system which consists of five steps (i.e., risk identification, policy definition, risk sharing and allocation, risk analysis, and risk minimization and response planning, etc). The fourth component, risk analysis, consists of three steps as follows; natural language computation, fuzzy set risk evaluation, and linguistic approximation. A

18