Issue 34

Frattura ed Integrità Strutturale, 34 (2015); International Journal of the Italian Group of Fracture

Table of Contents

Z. Marciniak, D. Rozumek Models of initiation fatigue crack paths proposed by Macha ……………………...……………. 1 F. Berto Local approaches for the fracture assessment of notched components: the research work developed by Professor Paolo Lazzarin ………………………….……………………………………... 11 N. R. Gates, A. Fatemi Crack paths in smooth and precracked specimens subjected to multiaxial cyclic stressing …………... 27 Y. Sumi Fracture morphology and its evolution. A review on crack path stability and brittle fracture along butt- weld …………………………………...………………………………………………. 42 R. Brighenti, A. Carpinteri, D. Scorza Effect of fibre arrangement on the multiaxial fatigue of fibrous composites: a micromechanical computational model ……………………………………………………….…………….. 59 D. Scorza, A. Carpinteri, G. Fortese, S. Vantadori, D. Ferretti, R. Brighenti Investigation of Mode I fracture toughness of red Verona marble after thermal treatment …………. 69 C. Ronchei, A. Carpinteri, G. Fortese, A. Spagnoli, S. Vantadori, M. Kurek, T. Łagoda Life estimation by varying the critical plane orientation in the modified Carpinteri-Spagnoli criterion ... 74 R. Brighenti, A. Carpinteri, N. Corbari A unified approach for static and dynamic fracture failure in solids and granular materials by a particle method ……………………………………………..………...………………………… 80 I. Ivanova, J. Assih Static and dynamic experimental study of strengthened reinforced short concrete corbel by using carbon fabrics, crack path in shear zone …………………………………………………………... 90 C. Fischer, W. Fricke Influence of local stress concentrations on the crack propagation in complex welded components ……... 99 G. Meneghetti, B. Atzori, A. Campagnolo, F. Berto A link between the peak stresses and the averaged strain energy density for cracks under mixed-mode (I+II) loading …………………………………………………………………………... 109 L. Náhlík, K. Štegnerová, P. Hutař Estimation of stepwise crack propagation in ceramic laminates with strong interfaces ……………... 116

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Fracture and Structural Integrity, 34 (2015); ISSN 1971-9883

R. D. Caligiuri Critical crack path assessments in failure investigations ………………………………….……. 125 Y. Hos, M. Vormwaldl Measurement and simulation of crack growth rate and direction under non-proportional loadings ….... 133 J. Pokluda, T. Voijtek, A. Hohenwarter, R. Pippan Effects of microstructure and crystallography on crack path and intrinsic resistance to shear-mode fatigue crack growth …………………………………………………………………..……….... 142 L. P. Pook Crack paths and the linear elastic analysis of cracked bodies …………………………………... 150 B. Žužek, M. Sedlaček, B. Podgornik Effect of segregations on mechanical properties and crack propagation in spring steel ……………… 160 F. Berto Crack initiation at V-notch tip subjected to in-plane mixed mode loading: An application of the fictitious notch rounding concept …………………………………………………………... 169 P. Gallo, F. Berto High temperature fatigue tests and crack growth in 40CrMoV13.9 notched components ………….. 180 A. Campagnolo, F. Berto, L. P. Pook Three-dimensional effects on cracked discs and plates under nominal Mode III loading ……………. 190 A. Shanyavskiy Crack path for run-out specimens in fatigue tests: is it belonging to high- or very-high-cycle fatigue regime? 199 P.O. Judt, A. Ricoeur Crack path predictions and experiments in plane structures considering anisotropic properties and material interfaces ……………………………………………………………………….. 208 M. Ševčík, P. Hutař, A. P. Vassilopoulos, M. Shahverdi Analytical model of asymmetrical Mixed-Mode Bending test of adhesively bonded GFRP joint …… 216 L. E. Kosteski, F. S. Soares, I. Iturrioz Applications of lattice method in the simulation of crack path in heterogeneous materials ………….. 226 T.-T.-G. Vo, P. Martinuzzi, V.-X. Tran, N. McLachlan, A. Steer Modelling 3D crack propagation in ageing graphite bricks of Advanced Gas-cooled Reactor power plant 237 Y. Nakai, D. Shiozawa, S. Kikuchi, K. Sato, T. Obama, T. Makino, Y. Neishi In situ observation of rolling contact fatigue cracks by laminography using ultrabright synchrotron radiation ……………………………………………………………………………….. 246 Yu.G. Matvienko, M.M. Semenova The concept of the average stress in the fracture process zone for the search of the crack path ……...…… 255 S. Kikuchi, T. Imai, H. Kubozono, Y. Nakai, A. Ueno, K. Ameyama Evaluation of near-threshold fatigue crack propagation in Ti-6Al-4V Alloy with harmonic structure created by Mechanical Milling and Spark Plasma Sintering …………………………………... 261

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Frattura ed Integrità Strutturale, 34 (2015); International Journal of the Italian Group of Fracture

K. Nambu, N. Egami Influence of vacuum carburizing treatment on fatigue crack growth characteristic in DSG2 ………... 271 B. Schramm, H.A. Richard Crack propagation in fracture mechanical graded structures ………………………………..…... 280 G. Lesiuk, M. Szata Kinetics of fatigue crack growth and crack paths in the old puddled steel after 100-years operating time 290 J. Saliba, A. Loukili, J.P. Regoin, D. Grégoire, L. Verdon, G. Pijaudier-Cabot Experimental analysis of crack evolution in concrete by the acoustic emission technique …………….... 300 K. Tanaka, K. Oharada, D. Yamada, K. Shimizu Fatigue crack propagation in short-fiber reinforced plastics ……………………..……………… 309 M. Kikuchi, Y. Wada, Y. Li Crack growth simulation in heterogeneous material by S-FEM and comparison with experiments ….. 318 S. Henschel, L. Krüger Effect of inhomogeneous distribution of non-metallic inclusions on crack path deflection in G42CrMo4 steel at different loading rates …………………………………………………………….... 326 T. Makino, Y. Neishi, D. Shiozawa, S. Kikuchi, S. Okada, K. Kajiwara, Y. Nakai Effect of defect length on rolling contact fatigue crack propagation in high strength steel ……………. 334 P. Hess Graphene as a model system for 2D fracture behavior of perfect and defective solids ………………. 341 A. Tajiri, Y. Uematsu, T. Kakiuchi, Y. Suzuki Fatigue crack paths and properties in A356-T6 aluminum alloy microstructurally modified by friction stir processing under different conditions …………………………………………………….. 347 S. Takaya, Y. Uematsu, T. Kakiuchi EBSD-assisted fractographic analysis of crack paths in magnesium alloy ……………………….. 355 O. Ševeček, M. Kotoul, D. Leguillon, E. Martin, R. Bermejo Understanding the edge crack phenomenon in ceramic laminates ……………………………….. 362 S. Keck, M. Fulland Investigation of crack paths in natural fibre-reinforced composites ………………………………. 371 E. Marcisz, D. Rozumek, Z. Marciniak Influence of control parameters on the crack paths in the aluminum alloy 2024 under bending ……... 379 L. Marsavina, E. Linul, T. Voiconi, D. M. Constantinescu, D. A. Apostol On the crack path under mixed mode loadings on PUR foams …………………………............. 387 A. Satoh, M. Satoh, K. Yamada Improvement of adhesion performance of mortar-repair interface with inducing crack path into repair ... 397 F. Iacoviello, V. Di Cocco Degenerated graphite nodules influence on fatigue crack paths in a ferritic ductile cast iron ……......... 406

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Fracture and Structural Integrity, 34 (2015); ISSN 1971-9883

V. Di Cocco, F. Iacoviello, S. Natali, A. Brotzu Fatigue crack micromechanisms in a Cu-Zn-Al shape memory alloy with pseudo-elastic behavior …... 415 V. Oborin, M. Bannikov, O. Naimark, M. Sokovikov, D. Bilalov Multiscale study of fracture in aluminum-magnesium alloy under fatigue and dynamic loading ……... 422 M. Goto, K. Morita, J. Kitamura, T. Yamamoto, M. Baba, S.-z. Han, S. Kim Growth behavior of fatigue cracks in ultrafine grained Cu smooth specimens with a small hole ……... 427 A. Riemer, H. A. Richard, J.-P. Brüggemann, J.-N. Wesendahl Fatigue crack growth in additive manufactured products ……………………………………… 437 F. Curà, A. Mura, C. Rosso Effect of centrifugal load on crack path in thin-rimmed and webbed gears ……………………….. 447 J. Bär, A. Vshivkov, O. Plekhov Combined lock-in thermography and heat flow measurements for analysing heat dissipation during fatigue crack propagation …………………………………………………………………. 456 S. Henkel, E. Liebelt, H. Biermann, S. Ackermann, L. Zybell Crack growth behavior of aluminum alloy 6061 T651 under uniaxial and biaxial planar testing condition ……………………………………………………………………………….. 466 K.L. Yuan, Y.Sumi Modelling of ultrasonic impact treatment (UIT) of welded joints and its effect on fatigue strength …… 476 T. Itoh, M. Sakane, T. Morishita, H.Nakamura, M. Takanashi Crack mode and life of Ti-6Al-4V under multiaxial low cycle fatigue …………………………. 487 G. M. Domínguez Almaraz, E. Correa Gómez, J.C. Verduzco Juárez, J.L. Avila Ambriz Crack initiation and propagation on the polymeric material ABS (Acrylonitrile Butadiene Styrene), under ultrasonic fatigue testing …………………………………………….........................… 498 K. Nowak Paths of interactive cracks in creep conditions .……………………………………………..… 507 R. Citarella, V. Giannella, M. Lepore DBEM crack propagation for nonlinear fracture problems .…………………………………… 514 S. Ahmad, J. M. Tulliani, G. A. Ferro, R. A. Khushnood, L. Restuccia, P. Jagdale Crack path and fracture surface modifications in cement composites …………………………….. 524 R. A. Khushnood, S, Ahmad, G, A, Ferro, L, Restuccia, J. M. Tulliani, P. Jagdale Modified fracture properties of cement composites with nano/micro carbonized bagasse fibers ………. 534 Q. Like, D. Jun, Y. Liqun Meso-mechanics simulation analysis of microwave-assisted mineral liberation …………………..… 543 R. Citarella, M. Perrella Robust design of a polygonal shaft-hub coupling ………………………………………..…..… 554 J. He, P. Zhang, Q. Yin, K., H. Liu Study of drilling muds on the anti-erosion property of a fluidic amplifier in directional drilling …….... 564

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Frattura ed Integrità Strutturale, 34 (2015); International Journal of the Italian Group of Fracture

X. Zhengbing, Z. Kefeng, D. Yanfeng, G. Fuwen A comparative study on dynamic mechanical performance of concrete and rock …………………… 574 S. Ackermann, T. Lippmann, D. Kulawinski, S. Henkel, H. Biermann Biaxial fatigue behavior of a powder metallurgical TRIP steel .……………………………....… 580 Z. Jijun, M. Xiangqing Defect detection of wire rope for oil well based on adaptive angle .……………………………..… 590 Y. Li, K. Zhang, B. Liu, Z. Pan On the decay of strength in Guilin red clay with cracks .…….………………………………… 599 C. Baron Saiz, T. Ingrassia, V. Nigrelli, V. Ricotta Thermal stress analysis of different full and ventilated disc brakes .…………………………...… 608 M. Scafidi, D. Cerniglia, T. Ingrassia 2D size, position and shape definition of defects by B-scan image analysis .……………………… 622

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Fracture and Structural Integrity, 34 (2015); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello

(Università di Cassino e del Lazio Meridionale, Italy)

Associate Editors Alfredo Navarro

(Escuela Superior de Ingenieros, Universidad de Sevilla, Spain) (Ecole Nationale Supérieure d'Arts et Métiers, Paris, France)

Thierry Palin-Luc

Luca Susmel John Yates

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

Guest Editors A. Carpinteri

(University of Parma)

Les P. Pook

(21 Woodside Road, Sevenoaks TN13 3HF, UK)

L. Susmel R. Tovo

(University of Sheffield, UK)

(University of Ferrara)

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (University of Plymouth, UK)

Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Ashok Saxena Darrell F. Socie Shouwen Yu Ramesh Talreja David Taylor Robert O. Ritchie Cetin Morris Sonsino Elisabeth Bowman Roberto Citarella Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Eugenio Giner Tommaso Ghidini Daniele Dini Editorial Board Stefano Beretta Nicola Bonora

(Helsinki University of Technology, Finland)

(University of California, USA)

(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)

(University of Illinois at Urbana-Champaign, USA)

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

(Politecnico di Milano, Italy)

(Università di Cassino e del Lazio Meridionale, Italy)

(University of Sheffield) (Università di Salerno, Italy) (EADS, Munich, Germany) (EDAM MIT, Portugal)

(Università di Cassino e del Lazio Meridionale, Italy)

(Imperial College, UK)

(Politecnico di Torino, Italy)

(Universitat Politecnica de Valencia, Spain) (European Space Agency - ESA-ESRIN)

Paolo Leonetti Carmine Maletta Liviu Marsavina

(Università della Calabria, Italy) (Università della Calabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)

Lucas Filipe Martins da Silva

Hisao Matsunaga Mahmoud Mostafavi

(Kyushu University, Japan) (University of Sheffield, UK)

Marco Paggi

(IMT Institute for Advanced Studies Lucca, Italy)

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Frattura ed Integrità Strutturale, 34 (2015); International Journal of the Italian Group of Fracture

Oleg Plekhov

(Russian Academy of Sciences, Ural Section, Moscow Russian Federation)

Alessandro Pirondi

(Università di Parma, Italy)

Luis Reis

(Instituto Superior Técnico, Portugal)

Giacomo Risitano Roberto Roberti

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

Marco Savoia

Andrea Spagnoli Charles V. White

(Kettering University, Michigan,USA)

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Fracture and Structural Integrity, 34 (2015); ISSN 1971-9883

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. Papers should be written in English. 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. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

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

Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)

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Frattura ed Integrità Strutturale, 34 (2015); International Journal of the Italian Group of Fracture

…. ECF21 is near!!

D

ear friends, just a few words about the most important event that IGF is organizing: the 21 st European Conference on Fracture, under the auspices of the European Structural Integrity Society. This event will be held in Catania, Italy (June 20-24, 2015) and, according to the ECFs events tradition, will allows to hundred of researchers to meet in the wonderful land of Sicily. I don’t want to bother you describing the sea, the culture, the food, the people of Sicily … join us and you will see by your own!!! In these lines I wish only to underline some details: - Abstract summission deadline: 30.11.2015; - Proceedings will be published in Procedia Structural Integrity (Elsevier); - Special issues will be published in Engineering Fracture Mechanics, Engineering Failure Analysis and International Journal of Fatigue; - A Summer School titled “Understanding and modelling fracture and fatigue of materials and structures” will be organized before the ECF event (June 18-19, 2015) You can find all the details in the ECF21 website www.ecf21.eu …. Looking forward to meeting you in Catania!!!

Francesco Iacoviello F&IS Chief Editor

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

Focussed on Crack Paths

Models of initiation fatigue crack paths proposed by Macha

Z. Marciniak Opole University of Technology, Poland z.marciniak@po.opole.pl D. Rozumek Opole University of Technology, Poland d.rozumek@po.opole.pl

A BSTRACT . Professor E. Macha devoted his academic life to solving the problems connected with random multiaxial fatigue in components of machines and structures. In his studies he formulated stress, strain and energy criteria related to critical plane concept. He also proposed several methods to determine critical plane position. In particular, he formulated and verified weight functions applied in order to determine critical plane position. The variance method constituted another significant contribution to the development of methods for determining critical plane position. Apart from these criteria, Macha was exploring energy approach in fatigue of materials and the development of fatigue cracks. He has also observed that strain characteristics multiplied by stress amplitude determined at specimen half-life are applied to estimate fatigue life using energy criteria. However, for cyclically instable materials, stress amplitude value may differ a lot; therefore he proposed the method to determine energy fatigue characteristics directly from experimental research. K EYWORDS . Multiaxial fatigue criteria; Energy; Crack growth; Variance method; Weight function. redicting service life of different objects is a very important issue for modern engineering. Wrong service life estimation may result in accidents and disasters. Therefore, studies aimed to understand and control this phenomenon, started already in the 19 th century, are continued today. The multitude of problems connected with it suggests that scientists still have plenty of work ahead. Initially, the scope of studies was limited to uniaxial, constant- amplitude issues only. With increasing knowledge on the phenomenon, the interest in multiaxial fatigue (most frequent in engineering practice) was growing. At the same time, many stress assessment criteria were proposed. Another step in the development involved attempts to assess life for random loads. This problem was explored by Professor Macha as well [1- 4]. He started his work from proposing mathematical models to assess fatigue life for materials in the conditions of random complex stress state, where besides stress criteria he demonstrated the method for determining critical plane position using weight functions. Further studies were connected with strain and energy criteria, and methods used to P I NTRODUCTION

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

determine critical plane position. Many scientists became interested in these studies, which resulted in numerous contacts and team work, e.g. with Carpinteri [5, 6], Sakane [7], Sonsino [8], Dragon [9], Petit [9], and others. The purpose of this study is to present academic achievements of Professor Macha.

M ATHEMATICAL MODELS AND THEIR EXPERIMENTAL VERIFICATION

O

ne of the first criteria proposed by E. Macha for multiaxial random loads [1, 2] has the following form:   max ( ) ( ) ns n t B t K t F     (1) n t  are: shear stress and normal stress in fracture plane, respectively; and B, K, F – constants for the selection of a given criterion version. Initially, in this criterion fracture plane was regarded as the critical plane. However subsequent analyses make it possible to observe that this plane changes especially for elastic-plastic materials. Detailed criterion guidelines are: (i) fatigue crack is generated (caused) by the activity of normal stresses σ n (t) and shear stresses τ ns (t) in the direction s  in plane with normal n  , (ii) direction s  is concurrent with average direction of shear stresses. In the criteria related to critical plane it is very important to determine critical plane position. In order to determine its position, it was proposed to apply the weight function method. The weight function method involves finding averaged positions of main axes directions through properly selected weight functions W k . where ( ) ns t  and ( )

1 ˆ cos L

1 ˆ cos L m  

1 ˆ cos L n  

,

,

,











l

k k W

k k W

k k W

(2)

1

1 1

2

2 1

3

3 1

W

W

W







k

k

k

where:

L

k   - sum of weights,

1 k W W 

L – number of averages,  1 , β 2 ,  3

– angles between main stresses and axes in the Cartesian coordinates, (  1 Then, critical plane position is being determined relative to these averaged directions. 6 weight functions are demonstrated in the study [1]: - Weight I – W k

, x), (  2

, y), (  3

, z), respectively

= 1 – it is assumed that each position of the main axes has the same effect on the critical plane position,

1 





for k = 1, 2,…,N – this weight reduces the impact of maximum main stress  1



W

(t) value

- Weight II –

k

1min

k







1max

1min

on the critical plane position,

1 

a  



for

0  

k

af

- Weight III –

for k = 1, 2,…,N – according to this weight , only those

a  

W

0

1

  

k

a  

1 



for

1

k

af

positions of main axes are averaged, for which maximum stress value is  1

(t)  a·  f

, where  f

is fatigue limit,

1 

  

for

R

0     1 

k

e

- Weight IV –

W

for k = 1, 2,…,N – only those positions of main axes are averaged,

k

  

1 

for

R

k

e

for which maximum stress value  1

(t) is higher than product of Poisson’s ratio and yield point,

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

1 

f    a

k

1 



a  

for



k

af



1min    

1 max k

- Weight V –

– this weight was developed as a result of combining weights

W

k

  

1 



a  

for

0

k

af

II and III,

      

m

   

1 

1 



a  

for

k

k

af

f 

a

- Weight VI –

– in this proposal only those positions of main axes are taken for

   

W

k

1 



a  

for

0



k

af

averaging, in which  1 (t) is higher than fatigue limit fraction, while their share is in exponential function dependent on Wöhler curve inclination. However, the selection of proper angles for averaging creates problems, and there are no physical guidelines, which angles should be averaged. The issue of averaging proper angles is discussed in the study [10], where direction cosines were made dependent on Euler angles. Matrix of direction cosines defined in this way is expressed in the following form

cos cos cos sin sin cos cos sin sin cos cos sin sin cos cos cos sin sin cos sin cos cos sin sin sin cos sin sin cos                                               .

(3)

Nevertheless, some transformations are required in order to obtain values of Euler angles. The first step involves calculation of the quantity:

3 2sin m n 

2 2sin n l  

2 2sin l 

m

1 arccos





1    , m n

,

,

.

 





l

u

u

u

(4)

2

1 3

1

1

2

3

1

3







2

Then, Euler - Rodriguez parameters are used:









,

,

,

 

 

 

 

u

u

u

(5)

sin

sin

sin

cos

1

2

3

2

2

2

2

to determine values of angles

     

       

 

  

     

       

sin m

,

,

.





  

arcsin  





  



arctg

arctg

arctg

arctg

(6)

3



Euler angles calculated in this way are averaged using the following relations:

1 1 ˆ L k

1 1 ˆ L k

1 1 ˆ L k

    k W k

    k W k

    k W k

,

,

.



W   





W   





W   



(7)

Then, Macha and Będkowski [11] developed variance method to determine critical plane position. In this method, the critical plane is considered to be the plane, for which the variance of equivalent stress reduced by selected criterion reaches maximum. The study [12] contains comparison of lives of steel specimens using variance method with damage accumulation method for criterion of maximum shear stress in the critical plane. According to this criterion, the equivalent stress  eq (t) takes the following form

3

Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

,



x   

( ) sin(2 ) 2 ( ) cos(2 ) t t     

( ) t

(8)

eq

xy

where:  x

(t) - normal stress along the specimen axis,  xy (t) - shear stress in the specimen cross section,

 - angle determining the critical plane position. From Eq. (8) it appears that the equivalent stress  eq

(t) is linearly dependent on the stress state components  x

(t) and  xy

(t), so it can be expressed as

n



,





1 1 a x a x a x   2 2 j j

(9)

eq



j

1

where: a 1 . From theory of probability [13] it results that the variance of random variable being a linear function of some random variables is expressed by the following formula = sin (2  ), a 2 = 2cos (2  ), x 1 =  x , x 2 =  xy

n

2   2 j xj  a 

,

2    2 a a x  x  1 2









a a

a a

(10)

2



eq

j k xjk

x x

1 2

1

2

1 2



j k 

j

1

where:   eq

- variance of equivalent stress  eq , - variance of normal stress  x , - variance of shear stress  xy ,

 x1  x2

 x1x2 stresses. Under biaxial random stationary and ergodic stress state, the variances  x1 ,  x2 - covariance of normal  x and shear stress  xy

and the covariance  x1x2

in Eq. (10) are

constant. In the method of variance for determination of the critical plane position the maximum function of Eq. (10) is searched in relation of the angle  occurring in coefficients a 1 and a 2 . After reduction, the variance of equivalent stress   eq versus the angle  can be written as       1 2 1 2 2 2 sin 2 4 cos 2 2sin 4 eq x x x x            . (11) An exemplary assessment of the critical plane position for loading combination K01 [12] obtained using the variance and damage accumulation methods is shown in Fig. 1.

a) b) Figure 1 : Dependence of the normalized value of: a) variance, b) damage accumulation on the angle  of critical plane position for loading combination K01 (λ  = 0.189) [14].

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

Fig. 2 compares calculated life values and experimental values for variance and damage accumulation [12].

a) b) Figure 2 : Comparison of calculated and experimental fatigue lives with critical planes determined according to: a) variance and b) damage accumulation methods [14]. Strain fatigue criterion [3] expressed as   max ( ) ( ) ns n t b t k t q     (12) (t) are shear and normal strain in critical plane, respectively; and a, b, k, q – constants for the selection of a given criterion version. In 1991, Macha, Grzelak and Łagoda [14] attempted to apply spectral method to determine fatigue life. Studies on these issues were continued further in cooperation with Niesłony [15]. In these studies, assuming linear effort criteria, a generalised spectral method was formulated for determining fatigue life of materials put to multiaxial loading, using the function of power spectral density in the field of frequency. Multiaxial state of stress is reduced to uniaxial state, and accumulation of damage is carried out using standard material characteristics. The study proves that the results for lives assessed using spectral method in the field of frequency and cycle counting method in the field of time are much the same. Whereas, determination of expected critical plane position using variance method for time histories gives results equivalent to the function of power spectral density. Then, Professor Macha focused his attention on stress distribution in notch root. Like in Neuber [16] and Molski-Glinka [17] criteria, Łagoda-Macha [18] proposed an energy equation for determining the state of stresses in notch bottom as is another proposal to formulate multiaxial random fatigue in the field of strains, where ε ns (t) and ε n

1

 

2 max 1 2 1 



n

  



n

,

 



W

(13)

max

 

LM

max





E n 

K

where: n  - exponent of cyclic strain curve, K  - coefficient of cyclic strain curve. Experimental verification proved that the values obtained through this relation are between the results obtained using Neuber and Molski-Glinka relations. Tab. 1 contains sample calculation results for the above three models [19].

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

Model

K t

Neuber

Molski-Glinka  max (MPa)

Macha-Łagoda  max (MPa)

 max

(MPa)

9.61 4.30 3.23 1.85

421 329 294 233

386 302 270 215

405 316 282 224

Table 1 : The presentation  max

depending on K t

and strain energy density models.

Professor Macha was interested most in energy criteria of multiaxial random fatigue. In this field, in cooperation with Łagoda he proposed a generalised criterion of energy density parameter for normal and shear strains in critical plane, shown as [18, 20]   max ( ) ( ) ns n t W t W t Q     or max ( ) eq t W t Q      (14) where  ,  , Q – constants for the selection of a given criterion version. Guidelines of the proposed criterion are as follows [21]: “a) this portion of strain energy density is responsible for fatigue crack, which matches the work of normal stress  n (t) in normal strain  n (t), that is W n (t) and work of shear stress  ns (t) in a shear strain  ns (t) in the direction s in plane with normal n, that is W ns (t), b) direction s in the critical plane matches average direction, in which density of shear strain energy is maximal, c) in boundary state, material effort is determined by the maximum value of linear combination of energy parameters W n (t) and W ns (t).” For uniaxial stress state, strain energy density parameter is expressed as For multiaxial stress state, the course of equivalent strain energy density parameter is calculated in the critical plane with normal n and shear direction s as     ( ) ( ) ( ) 0.5 ( ) ( )sgn ( ), ( ) 0.5 ( ) ( )sgn ( ), ( ) eq ns n n n n n ns ns ns ns W t W t W t t t t t t t t t               . (16) The proposed energy criterion in the critical plane is applicable for cyclic and random loads for small and large number of cycles. Depending on the coefficients chosen, different criteria are obtained and thus, for: -  = 0,  = 1 we have the criterion of maximum energy density for normal strain in the critical plane, -  = 1,  = 0 we have the criterion of maximum energy density for shear strain in the critical plane, -  = 1,  = 1 we have the criterion of maximum energy density for normal and shear strain in the critical plane. When applying energy fatigue criteria to assess life, energy characteristics are used, developed as a result of the Coffin- Manson-Basquin characteristic multiplication [22-24] by stress amplitude determined for specimen half-life. However, this characteristic not fully illustrates the behaviour of cyclically unstable materials. Being aware of these differences, Professor Macha and Słowik proposed a new model to determine energy fatigue characteristics directly from experimental research. This model is described as [25]       0.5 pl i W t t t       , (17)       sgn ( ) sgn ( ) t t    ( ) 0.5 ( ) ( )sgn ( ), ( ) 0.5 ( ) ( ) W t t t t t t t         2 . (15)

where  i pl =  (t i

) for  (t i

) = 0 and i = 1, 2, 3,.... are successive numbers of the hysteresis loop (σ-ε).

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

In Eq. (17), W(t), σ(t), ε(t) are continuous functions of time t, and ε i pl and ε i+1 pl are constant values in time t in the hysteresis loop with the number i, while  i pl is the plastic strain registered in the moment t i , when the stress  (t i ) is equal to zero, and remains constant to the moment t i+1 when the stress reaches zero again, i.e.  (t i+1 ) = 0. Then the new registered value of plastic strain  i+1 pl replaces the previous one  i pl . This procedure is repeated for each cycle of loading. Fig. 3 shows sample hysteresis loops and energy parameter course calculated on the basis of variable-amplitude history of stresses and strains. Energy parameter course calculation procedure for variable-amplitude loads: Step 1. In point 0, individual values of stresses, strains and energy parameter are: σ(t 0 ) = 0, ε(t 0 ) = 0, ε 0 pl = 0, W(t 0 ) = 0. Step 2. In point A, individual parameters have the following values: σ(t A ) = σ A , ε(t A ) = ε A , ε A pl = ε 0 pl =0, W(t A ) = 0.5·σ A ׀ ε A - ε 0 pl ׀ = 0.5·σ A ·ε A . Step 3. Then, going to point B we obtain: σ(t B ) = σ B =0, ε(t B ) = ε B = 0, ε B pl = ε B pl , W(t B ) = 0.5·σ B · ׀ ε B - ε B pl ׀ = 0. Step 4. Point C: σ(t C ) = σ C , ε(t C ) = ε C , ε C pl = ε B pl , W(t C ) = 0.5·σ C · ׀ ε C - ε B pl ׀ . Step 5. Point D: σ(t D ) = σ D =0, ε(t D ) = ε D =0, ε D pl = ε D pl , W(t D ) = 0.5·σ D · ׀ ε D - ε D pl ׀ = 0. Step 6. Point E: σ(t E ) = σ E , ε(t E ) = ε E , ε E pl = ε D pl , W(t E ) = 0.5·σ E · ׀ ε E - ε D ׀ pl , etc. Fig. 3d presents energy parameter course calculated based on above procedure .

Figure 3 : Sample hysteresis loops a) , stress courses b) , strain courses c) , energy parameter courses d) .

Fig. 4 shows energy fatigue characteristic for steel C45 according to the formula (17). The models and methods proposed above were used to assess fatigue life until crack initiation. Whereas, as regards development of fatigue cracks, Rozumek and Macha proposed an energy criterion based on parameter J for three crack modes [26]. This criterion was verified for mode I and mode III [27].

2

2

2

I         Ic    J J

      

  





J

J

,



(18)

1

II

III

J

J

IIc

IIIc

where J Ic are critical values for modes I, II and III. The criterion (18) was successfully verified while tests of aluminium alloy and steels. Different bending (cracking mode I) to torsion (cracking mode III) ratio in steel 18G2A is shown in Fig. 5 [28]. It provides grounds to observe shift of experimental points towards increasing the value of parameter  J I , except of the , J IIc , J IIIc

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

angle  = 60  , where decrease in these values was confirmed. Considerable increase of parameter  J I and  J III observed with rising fatigue crack growth rate (Fig. 5, curves 1 to 2). Diagrams 1 and 2 shown in Fig. 5 concern fatigue crack growth rates: da/dN = 1.68  10 -8 m/cycle and da/dN = 4.23  10 -8 m/cycle, respectively. values was

Figure 4 : Energy fatigue characteristic for steel C45.

Figure 5 : Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq. (18) for 18G2A steel [27]. Different bending (cracking mode I) to torsion (cracking mode III) ratio in AlCuMg1 alloy is shown in Fig. 6. Fig. 6 [28] provides grounds to observe shift of experimental points towards increasing the values of parameter  J I - these increment values were lower than for steel 18G2A. Experimental results of interdependences between cracking mode I and III, for constant da/dN ratio value were defined by Eq. (18). Diagrams 1 and 2 shown in Fig. 6 concern fatigue crack growth rates: da/dN = 7.64  10 -8 m/cycle and da/dN = 1.41  10 -7 m/cycle, respectively.

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Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01

Figure 6 : Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq. (18) for AlCuMg1 alloy.

S UMMARY

T

his brief description of professor Macha activity irrefutably proves his wide interests and great influence on progress regarding the issues of fatigue life assessment for components of machines and structures. During his academic career, Macha with colleagues proposed many fatigue criteria concerning the parameters of stress, strain and strain energy density both in the field of time and frequency. Macha’s interests covered initiation range and propagation of fatigue cracks. Many times these criteria were verified in various load conditions for different materials, and were presented during various scientific conferences.

R EFERENCES

[1] Macha, E., Mathematical models of the life to fracture for materials subject to random complex stress systems, Monographs no 13, Wrocław University of Technology, Wrocław, (1979) (in Polish). [2] Macha, E., Generalization of fatigue fracture criteria for multiaxial sinusoidal loadings in the range of random loadings. Biaxial and Multiaxial Fatigue, EGF 3, Eds M.W. Brown and K.J. Miller, Mechanical Engineering Publications, London, (1989) 425-436. [3] Macha, E., Generalization of strain criteria of multiaxial cyclic fatigue to random loadings, Studies and Monographs, no. 23, Opole University of Technology, Opole, (1988) (in Polish). [4] Macha, E., Simulations investigations of the position of fatigue plane in materials with biaxial loads, Mat.-wiss. U. Werkstofftech., 20 (1989) 132-136. [5] Carpinteri, A., Macha, E., Brighenti, R., Spagnoli, A., Expected principal stress directions for multiaxial random loading - Part I, Theoretical aspects of the weight function method, Int. J. Fatigue, 21 (1999) 83-88. [6] Carpinteri, A, Macha, E, Brighenti, R, Spagnoli, A., Expected principal stress directions under multiaxial random loading. Part II: numerical simulation and experimentally assessment through the weight function method. Int. J. of Fatigue, 21 (1999) 89-96.

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[7] Będkowski, W., Macha, E., Ohnami M., Sakane M., Fracture plane of cruciform specimen in biaxial low cycle fatigue – estimate by variance method and experimental verification, Journal of Engineering Materials and Technology, 117 (1995) 183-190. [8] Macha, E., Sonsino, C.M., Energy criteria of multiaxial fatigue failure, Fatigue Fract. Engng. Mater. Struci., 22 (1999) 1053-1070. [9] Lagoda, T, Macha, E, Dragon, A, Petit, J., Influence of correlations between stresses on calculated fatigue life of machine elements, Int. J. Fatigue, 18 (1996) 547–555. [10] Carpinteri, A, Karolczuk, A., Macha, E., Vantadori, S., Expected position of the fatigue fracture plane by using the weighted mean principal Euler angels, Int. J. of Fracture, 115 (2002) 87-99. [11] Będkowski, W., Macha, E., Fatigue fracture plane under multiaxial random loadings – prediction by variance of equivalent stress based on the maximum shear and normal stresses, Mat.-wiss. U. Werkstofftech . , 23 (1992) 82-94. [12] Marciniak, Z., Rozumek, D., Macha, E., Verification of fatigue critical plane position according to variance and damage accumulation methods under multiaxial loading, Int. J. of Fatigue, 58 (2014) 84-93. [13] Korn, GA, Korn TM., Mathematical Handbook, Sec. Ed., Mc Graw-Hill Book Company, New York, (1968). [14] Grzelak, J., Lagoda, T., Macha, E., Spectral-analysis of the criteria for multiaxial random fatigue, Mat.-wiss. U. Werkstofftech., 22 (1991) 85-98. [15] Niesłony, A., Macha, E., Spectral Method in Multiaxial Random Fatigue, Springer-Verlag Berlin Heidelberg, (2007) 147. [16] Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law, ASME J. Applied Mech. 28 (1961) 544-550. [17] Molski K., Glinka G. A method of elastic-plastic stress and strain calculation at a notch root, Mat. Sci. and Engng. 50 (1981) 93-100. [18] Łagoda, T., Macha, E., Multiaxial random fatigue of machine elements and structures, Cyclic energy based multiaxial fatigue criteria to random loading, Part III, Studies and Monographs 104, Opole University of Technology, Opole, (1998) (in Polish). [19] Rozumek, D., Marciniak, Z., Fatigue properties of notched specimens made of FeP04 steel, Materials Science, 47 (2012) 462-469. [20] Łagoda, T., Macha, E., Będkowski, W., A critical plane approach based on energy concepts: application to biaxial random tension–compression high-cycle fatigue regime, Int. J. Fatigue, 21 (1999) 431–443. [21] Karolczuk, A., Macha, E., Critical planes in multiaxial fatigue of materials, monograph. Fortschritt-Berichte VDI, Mechanik/Bruchmechanik, reihe 18, nr. 298. Düsseldorf: VDI Verlag, (2005) 204. [22] Manson, S.S., Behaviour of materials under conditions of thermal stress, NACA TN-2933, (1953). [23] Coffin, L.F., A study of the effects of cyclic thermal stresses on a ductile metal, Trans. ASME 76 (1954) 931-950. [24] Basquin, O.H., The experimental law of endaurance test, ASTM, 10 (1910) 625-630. [25] Macha, E., Słowik, J., Pawliczek, R., Energy based characterization of fatigue behavior of cyclically unstable materials, Solid State Phenomena, 147-149 (2009) 512-517. [26] Rozumek, D., Macha, E., A survey of failure criteria and parameters in mixed-mode fatigue crack growth, Materials Science, 45 (2009) 190-210. [27] Rozumek, D., Macha, E., J-integral in the description of fatigue crack growth rate induced by different ratios of torsion to bending loading in AlCu4Mg1, Mat.-wiss. U. Werkstofftech . , 40 (2009) 743-749. [28] Rozumek, D., Mixed mode fatigue cracks of constructional materials, Studies and Monographs, no. 241, Opole University of Technology, Opole, (2009) (in Polish).

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F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 11-26; DOI: 10.3221/IGF-ESIS.34.02

Focussed on Crack Paths

Local approaches for the fracture assessment of notched components: the research work developed by Professor Paolo Lazzarin

F. Berto University of Padova, Italy berto@gest.unipd.it

A BSTRACT . Brittle failure of components weakened by cracks or sharp and blunt V-notches is a topic of active and continuous research. It is attractive for all researchers who face the problem of fracture of materials under different loading conditions and deals with a large number of applications in different engineering fields, not only with the mechanical one. This topic is significant in all the cases where intrinsic defects of the material or geometrical discontinuities give rise to localized stress concentration which, in brittle materials, may generate a crack leading to catastrophic failure or to a shortening of the assessed structural life. Whereas cracks are viewed as unpleasant entities in most engineering materials, U- and V-notches of different acuities are sometimes deliberately introduced in design and manufacturing of structural components. Dealing with brittle failure of notched components and summarising some recent experimental results reported in the literature, the main aim of the present contribution is to present a review of the research work developed by Professor Paolo Lazzarin. The approach based on the volume strain energy density (SED), which has been recently applied to assess the brittle failure of a large number of materials. The main features of the SED approach are outlined in the paper and its peculiarities and advantages accurately underlined. Some examples of applications are reported, as well. The present contribution is based on the author’s experience over about 15 years and the contents of his personal library. This work is in honor and memory of Prof. Paolo Lazzarin who suddenly passed away in September 2014. K EYWORDS . Notch stress intensity factors; Strain Energy Density; Fracture assessment; Fatigue strength. ealing with fracture assessment of cracked and notched components a clear distinction should be done between large and small bodies [1-6]. The design rules applied to large bodies are based on the idea that local inhomogeneities, where material damage starts, can be averaged being large the volume to surface ratio. In small bodies the high ratio between surface and volume makes not negligible the local discontinuities present in the material and the adoption of a multi-scaling and segmentation scheme is the only way to capture what happens at pico, nano and micro levels [4-6]. In this scheme the crack tip has no dimension or mass to speak; it is the sink and source that absorbs and dissipates energy while the stress singularity representation at every level is the most powerful tool to quantify the energy packed by an equivalent crack reflecting both material effect and boundary conditions. This new revolutionary D I NTRODUCTION

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F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 11-26; DOI: 10.3221/IGF-ESIS.34.02

representation implies also a new definition of mass [4, 6]. The distinction between large and small bodies should ever be considered by avoiding to transfer directly the design rules valid for large components to small ones under the hypothesis that all material inhomogeneities can be averaged [1-6]. Keeping in mind the observations above and limiting our considerations to large bodies (i.e. large volume to surface ratio), for which an averaging process is still valid, the paper is addressed to review a volume-based Strain Energy Density Approach applied to Static and Fatigue Strength Assessments of notched and welded structures [7-14]. The concept of “elementary” volume and “structural support length” was introduced many years ago [15-17] and it states that not the theoretical maximum notch stress is the static or fatigue strength-effective parameter in the case of pointed or sharp notches, but rather the notch stress averaged over a short distance normal to the notch edge. In high cycle fatigue regime, the integration path should coincide with the early fatigue crack propagation path. A further idea was to determine the fatigue-effective notch stress directly (i.e. without notch stress averaging) by performing the notch stress analysis with a fictitiously enlarged notch radius,  f , corresponding to the relevant support [15-17]. Fundamentals of Critical Distance Mechanics applied to static failure, state that crack propagation occurs when the normal strain [18] or circumferential stress    [19] at some critical distance from the crack tip reaches a given critical value. This “Point Criterion” becomes a “Line criterion” in Refs. [20, 21] who dealt with components weakened by sharp V-shaped notches. A stress criterion of brittle failure was proposed based on the assumption that crack initiation or propagation occurs when the mean value of decohesive stress over a specified damage segment d 0 reaches a critical value. The length d 0 is 2-5 times the grain size and then ranges for most metals from 0.03 mm to 0.50 mm. The segment d 0 was called “elementary increment of the crack length”. Dealing with this topic a previous paper, Ref. [22], was quoted in Refs [20, 21]. Afterwards, this critical distance-based criterion was extended also to structural elements under multi-axial loading [23, 24] by introducing a non-local failure function combining normal and shear stress components, both normalised with respect to the relevant fracture stresses of the material. Dealing with notched components the idea that a quantity averaged over a finite size volume controls the stress state in the volume by means of a single parameter, the average value of the circumferential   stress [25]. For many years the Strain Energy Density (SED) has been used to formulate failure criteria for materials exhibiting both ductile and brittle behaviour. Since Beltrami [26] to nowadays the SED has been found being a powerful tool to assess the static and fatigue behaviour of notched and un-notched components in structural engineering. Different SED-based approaches were formulated by many researchers. Dealing here with the strain energy density concept, it is worthwhile contemplating some fundamental contributions [27- 35]. The concept of “core region” surrounding the crack tip was proposed in Ref. [27]. The main idea is that the continuum mechanics stops short at a distance from the crack tip, providing the concept of the radius of the core region. The strain energy density factor S was defined as the product of the strain energy density by a critical distance from the point of singularity [28]. Failure was thought of as controlled by a critical value S c , whereas the direction of crack propagation was determined by imposing a minimum condition on S . The theory was extended to employ the total strain energy density near the notch tip [29], and the point of reference was chosen to be the location on the surface of the notch where the maximum tangential stress occurs. The strain energy density fracture criterion was refined and extensively summarised in Ref. [30]. The material element is always kept at a finite distance from the crack or the notch tip outside the “core region” where the in-homogeneity of the material due to micro-cracks, dislocations and grain boundaries precludes an accurate analytical solution. The theory can account for yielding and fracture and is applicable also to ductile materials. Depending on the local stress state, the radius of the core region may or may not coincide with the critical ligament r c that corresponds to the onset of unstable crack extension [30]. The ligament r c depends on the fracture toughness K IC , the yield stress σ y , the Poisson’s ratio  and, finally, on the ratio between dilatational and distortional components of the strain energy density. The direction of  max determines maximum distortion while  min relates to dilatation. Distortion is associated with yielding, dilatation tends to be associated to the creation of free surfaces or fracture and occurs along the line of expected crack extension [30, 31]. A critical value of strain energy density function (d W /d V ) c has been extensively used since 1965 [32-35], when first the ratio (d W /d V ) c was determined experimentally for various engineering materials by using plain and notched specimens. The deformation energy required for crack initiation in a unit volume of material is called Absorbed Specific Fracture Energy (ASFE) and its links with the critical value of J c and the critical factor S c were widely discussed. This topic was deeply considered in Refs. [28-30] where it was showed that (d W /d V ) c is equivalent to S c / r being S c the critical strain energy density factor and the radius vector r the location of failure. Since distributions of the absorbed specific energy W in notched specimens are not uniform, it was assumed that the specimen cracks as soon as a precise energy amount has been absorbed by the small plastic zone at the root of the notch. If the notch is sufficiently sharp, specific energy due to the elastic deformation is small enough to be neglected as an initial approximation [34]. While measurements of the energy

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