Crack Paths 2012

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University of Cassino

Universify-of Parrna

Universify of Sheffield

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Materials Researcha n dTesfing Metallurgya n dMaferials

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G r o u pof Fraciure

G a e f a(Ifaly), 19 — 21 Sepfernber,2012

The 4Th Infernafional C o n f e r e n coen

CRACK PATHS (CP 2012)

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ISBN: 97888959 441

ISSN: 2281-1060

Edifors

A n d r e aCarpinferi, Parrna (Ifaly)

Francescolacoviello, Cassino (Ifaly)

Les P. Pook, Sevenoaks(UK)

LucaSusmel, Sheffield (UK)

Proceedings of the 4th International Conference on

C R A C PKA T H S(CP 2012)

Gaeta (Italy), 19 – 21 September, 2012

ISBN :9788895940441

ISSN :2281-1060

E D I T O R S

Andrea Carpinteri, Parma (Italy)

Francesco Iacoviello, Cassino (Italy)

Les P. Pook, Sevenoaks (UK)

Luca Susmel, Sheffield (UK)

Scientific Committee

José Alexander Araujo, Brasilia (Brazil)

Zenon Mroz, Warsaw(Poland)

Yukitaka Murakami, Fukuoka (Japan)

Harm Askes, Sheffield (UK)

Bruno Atzori, Padua (Italy)

Alfredo Navarro, Seville (Spain)

Thierry Palin-Luc, Talence (France)

Stefano Beretta, Milan (Italy)

Feargal Brennan, Cranfield (UK)

Jean Petit, Futuroscope (France)

G.M. DomínguezAlmaraz, Morelia (Mexico)

Jaroslav Pokluda, Brno (Czech Republic)

Véronique Doquet, Palaiseau (France)

Hans Albert Richard, Paderborn (Germany)

Andrey Shanyavskiy, Moscow(Russia)

Fernand Ellyin, Vancouver (Canada)

Grzegorz Glinka, Waterloo (Canada)

Malgorzata Skorupa, Cracow (Poland)

M. Neil James, Plymouth (UK)

C. Morris Sonsino, Darmstadt (Germany)

Paolo Lazzarin, Padua (Italy)

Andrea Spagnoli, Parma (Italy)

Yoichi Sumi, Yokohama(Japan)

Ewald Macha, Opole (Poland)

Gary Marquis, Helsinki (Finland)

Keisuke Tanaka, Nagoya (Japan)

Yuri Matvienko, Moscow(Russia)

Silvio Valente, Turin (Italy)

Michael Roger Mitchell, Flagstaff (USA)

John R. Yates, Manchester (UK)

Local OrganisingCommittee

andrea.carpinteri@unipr.it iacoviello@unicas.it brigh@unipr. t

Andrea Carpinteri, Parma

Francesco Iacoviello, Cassino

Roberto Brighenti, Parma

Vittorio Di Cocco, Cassino

v.dicocco@unicas.it sabrina.vantadori@unipr.it

Sabrina Vantadori, Parma

P R E F A C E

Crack growth can take place under both static and fatigue loading. The complete solution

of a crack growth problem includes the determination of the path taken by the crack. The

crack path in a critical component or structure in aerospace, automotive, offshore and

other industries can determine whether failure is benign or catastrophic. Knowledge of

potential crack paths is also needed for the selection of appropriate non-destructive

testing procedures. Agreement between theoretically predicted and experimentally

determined crack paths is sometimes poor.

experimental

The scope of this Conference has been to focus on the state-of-the-art

techniques, and numerical and analytical models for the determination of crack paths in

solids made of a broad range of materials, together with the application of crack path data

to the design of structures subjected to both static and fatigue loading.

This Conference follows the Conferences in Parma (Italy) in 2003 and 2006, and in

Vicenza (Italy) in 2009. Special issues of international journals have been devoted to

work presented at those Conferences (Fatigue and Fracture of Engineering Materials and

Structures, Vol.28, No.1-2, 2005; Engineering Fracture Mechanics, Vol.75, No.3-4, 2008;

and Engineering Fracture Mechanics, Vol.77, No.11, 2010).

Two Special Issues of International Journals (Engineering Fracture Mechanics and

International Journal of Fatigue) with extended versions of selected papers presented at

C P2012 will be published after the Conference:

Membersof different industrial laboratories and scientists from all over the world have

contributed with presentations on any of the following topics (in the case of both static

and fatigue loading):

Experimental Determination of CP

Theoretical Prediction of CP

Integrity Assessments based on CP EvaluationMicroscopic Aspects of CP

CPof Surface Cracks

CPof Short Cracks

Effect of Large Scale Yielding on CP

Effect of Material Inhomogeneities on CP

Effect of Non-Proportional Cyclic Loading on CP

Effect of Environmental Conditions on CP

CPin Advanced Materials

Laboratory Methods of Controlling CP

In-Service Inspection of CP

Application of CP Concepts and Data in Design

Industrial Application of CP Concepts and Data

List of Sponsors

University of Cassino

University of Parma

University of Sheffield

ESIS (European Structural Integrity Society)

GermanAssociation for Materials

Research and Testing

French Society for Metallurgy

and Materials

A S T MInternational

Standards Worldwide

IGF

Italian Group of Fracture

Proceedings of the 4 th International Conference on

C R A CPKA T H S(CP2012)

Gaeta (Italy), 19 – 21 September, 2012

ISBN: 9788895940441

P L E N A RLYE C T U R E S

Review of fatigue crack growth under non-proportional loading

1

Michael Vormwald, Patrick Zerres

Propagation of small cracks under RCF: a challenge to multiaxial fatigue criteria

15

Stefano Beretta, Stefano Foletti

Near-threshold behavior of modeII and III fatigue cracks in ferritic and austenitic steel

29

Jaroslav Pokluda

Small crack propagation in multiaxial notch fatigue

31

Keisuke Tanaka

Under cyclic loading, plastic dissipation in heat at the crack tip modifies the stress intensity factor

47

Nicolas Ranc, Thierry Palin-Luc, Paul C. Paris, Nicolas Saintier

Crazing, crack paths and plastic shielding in fatigue of polycarbonate

61

M. Neil James, Y. Lu, C.J. Christopher, Eann A. Patterson

INVITEDP A P E R S

A three-dimensional analysis of fatigue crack paths in thin metallic sheets

75

J.B. Esnault, V. Doquet, P. Massin

Crack path prediction in layered ceramics designed with residual stresses

87

O. Ševeþek, R. Bermejo, M. Kotoul

Crack initiation and path prediction of float glass with various constrain conditions

under thermal loading

99

Q.S. Wang, Y. Wang, J.H. Sun, L.H. He

i

Observation of fatigue crack propagation modetransition under cyclic torsion using

micro-CT imaging with ultra-bright synchrotron radiation

111

Y. Nakai, D. Shiozawa, T. Murakami, H. Nosho

Effect of pre-stressing on the growth direction of surface cracks in ultrafine grained copper

123

K. Kamil, M. Goto, S.Z. Han, K. Euh, T. Yakushiji, Y. Tatsukawa

On coupled fracture modes and three-dimensional fracture mechanics

135

A. Kotousov, P. Lazzarin, F. Berto, L.P. Pook

Structural health monitoring of wind towers: residual life estimation

147

M. Benedetti, V. Fontanari

Role of pearlitic and spheroiditic microstructure in fatigue crack paths

159

J. Toribio, B. González, J.C. Matos

Small crack growth path and rate under combined stresses

171

N. Shamsaei, A. Fatemi

Someproblems of fatigue crack propagation in marine structures under seaway loading

183

Y. Sumi

Fatigue critical plane according to variance and cumulative damage methods under multiaxial loading

195

Z. Marciniak, D. Rozumek, E. Macha

Fatigue crack initiation and early crack propagation in ultrafine-grained copper in

high-cycle fatigue region

207

L. Kunz, P. Lukáš, L. Navrátilová

Experimental investigations on mixed-mode-loaded cracks

219

H.A. Richard, N.-H. Schirmeisen, A. Eberlein

Microstructural investigations of crack path development in nodular cast iron under

variable amplitude loading

235

L. Zybell, H. Chaves, M. Kuna, T. Mottitschka, S. Henkel, G. Pusch, H. Biermann

Crack initiation direction for in-phase biaxial fatigue loading

247

V. Chaves, C. Madrigal, C. Vallellano, A. Navarro

Effect of initial geometric misalignments of the circular notched bar specimen on the

fatigue crack growth behavior

255

I. Kim, Y.J. Zhao, B.-H. Choi, J.M. Lee

An understanding of crack growth in V H C Ffrom an internal inclusion in high strength steel

267

C. Wang, A. Nikitin, A. Shanyavskiy, C. Bathias

ii

LIST O F P A P E R S

Fatigue fractal crack propagating in a self-balanced microstress field

279

Andrea Carpinteri, L. Montanari, A. Spagnoli

High-cycle fatigue in a hydraulic turbine runner

287

Andrea Carpinteri, C. Bagni, D. Scorza, S. Vantadori

Short crack and long crack propagation in metals based on damage mechanics concepts

295

R. Brighenti, Andrea Carpinteri, N. Corbari

Crack path in unidirectional fibre-reinforced brittle-matrix materials through two

computational models

303

R. Brighenti, Andrea Carpinteri, A. Spagnoli, D. Scorza

Local/global eyes into the crack path behavior in metal/hydrogen interactive system

311

Y. Katz

Microstrucural influences on crack initiation and growth in an equiatomic NiTi PE alloy

319

F. Iacoviello, V. Di Cocco, S. Natali, C. Maletta

Sn and Ti influence in bending cracks path in hot dip galvanizing coatings

327

V. Di Cocco, F. Iacoviello, S. Natali, L. Zortea

Analysis of fatigue damaging micromechanisms in a ferritic ductile iron

335

V. Di Cocco, F. Iacoviello, A. Rossi

Crack path for in-service subsurface fatigued turbine blades

343

A.A. Shanyavskiy

X F E Mfor fretting fatigue: straight VSmixed modecrack propagation

351

R. Hojjati-Talemi, M.A. Wahab

Crack paths in a borosilicate glass under triaxial loading

359

V. Doquet, N. Ben Ali, A. Constantinescu

Fatigue cracking in bifurcation area of titanium alloy at 20 kHz

367

A. Nikitin, A. Shanyavskiy, T. Palin-Luc, C. Bathias

A novel approach for estimation of crack paths in Fibre Metal Laminates

375

M. Gupta, R.C. Alderliesten, R. Benedictus

Threshold fatigue crack growth and crack paths in heat-treated nodular cast iron

385

R. Koneþná, G. Nicoletto, S. Fintová, L. Bubenko

Micro cracking of ceramic and carbon and fibres acoustic emission in channel-die

compressed Mg-Li and Mg-Al alloys matrix composites

393

A. Paweáek, S. Kúdela, Z. Ranachowski, A. Piatkowski, S. Kúdela Jr., P. Ranachowski, Z. Jasieski

iii

Fatigue crack paths in the VHCF-regime of 100Cr6

401

P. Grad, E. Kerscher

Fatigue endurance and crack propagation on polymeric material under ultrasonic fatigue testing

409

G.M. Domínguez Almaraz

Fibre optic sensor grids in structural testing

417

K.-H. Haase

Brittle failure of inclined key-hole notches in isostatic graphite under in-plane mixed modeloading

425

P. Lazzarin, F. Berto, M.R. Ayatollahi

Crack paths propagation under ModeII fracture in concrete composites containing fly-ash additive

433

G.L. Golewski, P. Golewski, T. Sadowski

Strength degradation analysis of an aging R Cgirder bridge due to existing cracks

441

J. Wang, Z. Shi, M. Nakano

449

Fatigue crack nucleation at a stress concentration point

D. Leguillon, S. Murer

A statistical evaluation of micro-crack initiation and growth in thermally cut structural elements

457

M. Šori, S. Glodež, U. Fevžer

LEFM-based simulation of fatigue crack growth under non-proportional mixed-mode loading

465

Y. Yang, M. Vormwald

Direction of the maximumvariance of the resolved shear stress and orientation of Stage I crack paths

473

L. Susmel, R. Tovo

A procedure for evaluating the crack propagation taking into account the material plastic behaviour

481

A. Rossetti, P. Zerres, M. Vormwald

The effect of residual stresses on crack shape in polymer pipes

489

P. Huta, M. Ševþík, M. Zouhar, L. Náhlík, J. Kuþera

Correlation between road public usage and experimental fatigue curves on brazed heat exchanger

497

A.-G. Villemiane, J. Paturaud, D. Delaux, A. Buteri

505

Relationship between Charpy impact energy and notch tip position in functionally graded steels

H. Samareh Salavati Pour, Y. Alizadeh, F. Berto, M. Abolghasemzadeh

Modelling of crack path evolution in round bars under cyclic tension and bending

513

J. Toribio, J.C. Matos, B. González, J. Escuadra

Analysis of failure paths in steel bolted connections

521

J. Toribio, B. González, J.C. Matos, F.J. Ayaso

iv

Newbox-counting method as interpretation of crack paths and mechanical properties

of concrete with interface layer

529

A. Satoh, K. Yamada, T. Homma,S. Ishiyama

Experimental and numerical sub-interface crack paths

537

L. Marsavina, T. Sadowski, M. Knec

On the calculation of crack paths in 3-dimensional anisotropic solids

545

M. Steigemann, B. Schramm, M. Specovius-Neugebauer, H.A. Richard

Crack propagation calculations in aircraft engines by coupled F E M - D B EaMpproach

553

R. Citarella, M. Lepore, C. Caliani, M. Perrella

Validation of the numerical stress intensity factor calculation of surface cracks using

crack propagation experiments

563

J. Lebahn, H. Heyer, M. Sander

Fatigue-fractured surfaces and crack paths of textured polycrystalline magnesium alloys

571

S. Morita, K. Matsushita, T. Mayama, T. Hirai, T. Enjoji, N. Hattori

Evaluation of rolling contact fatigue crack path of high strength steel with artificial defects

579

T. Makino, Y. Neishi, D. Shiozawa, Y. Fukuda, Y. Nakai

Strength analysis of attachment lugs under cyclic loading

587

Slobodanka Boljanoviü, S. Maksimoviü

Cracking of nanofilm-based devices for electrochemical energetics

595

B. Bozzini, M. Boniardi, A. Gianoncelli, B. Kaulich, C. Mele, M. Prasciolu,

G. Scarselli, M. Kiskinova

Crack repairing in AA2099by Cu electrodeposition (ECD)

603

B. Bozzini, P. Cavaliere, C. Mele, I. Sgura

Experimental investigation of fatigue crack propagation in 2xxx aluminum alloy with local

yield strength gradient at the crack path

611

A. Tzamtzis, A.T. Kermanidis

Changes of the chemical composition of continuously cast steel slab and their relation to breakout

619

F. Kaviþka, J. Dobrovská, K. Stránský, B. Sekanina, J. Stetina, T. Mauder, M. Masarik

Mixed mode I/II fracture path simulation in a typical jointed rock slope

627

M.R.M. Aliha, M. Mousavi, M.R. Ayatollahi

Influence of hydrogen environment on crack growth rate

635

L. Vergani, C. Colombo, A. Sciuccati

Variable T-stress and its implication for crack path

643

V.N. Shlyannikov

v

651

Prediction of post-cracking behaviour in SFRCelements under in-plane stresses

P. Bernardi, R. Cerioni, E. Michelini

Mode-II and Mode-III effects of cyclic crack propagation in specimens

661

G. Dhondt, M. Schrade

Mixed modecohesive crack growth at the bi-material interface between a damand the

foundation rock

669

A. Alberto, S. Valente

Fatigue crack growth path of 2324-T39 Aluminium alloy under constant-amplitude

and spectrum tension loading

677

R. Bao, T. Zhang, B. Fei

Study of fatigue crack initiation mechanism on an Armcoiron by dissipation assessments

and microstructural observations

685

C. Wang, A. Blanche, D. Wagner, A. Chrysochoos, C. Bathias

Infrared study of heat dissipation under fatigue crack propagation

693

O. Plekhov, M. Bannikov, A. Terekhina, A. Fedorova

Fatigue properties and analysis of fracture surface and crack path of ultrafine-grained

structures produced by severe plastic deformation

701

A. Tomasella, V. Kaune, V. Landersheim, H. Kaufmann, H. Hanselka, E. Bruder, C. Müller

Stress analysis around a through crack in a thin copper film using molecular dynamics

711

D. Johansson, P. Hansson, S. Melin

Crack paths in steel-titanium explosive cladding

719

D. Rozumek, Z. Marciniak, E. Macha

Paths of shear-mode cracks in ferritic and austenitic steel

727

T. Vojtek, J. Pokluda

Lynx: new tool to model ModeI fatigue crack propagation

735

R. Branco, F.V. Antunes, J.D.M. Costa

Mixed modecrack growth for titanium alloy in specimen various geometries

743

V.N. Shlyannikov, A.V. Tumanov, S.Yu. Kislova

751

Second-order deformation of the front of a ModeI crack propagating in a heterogeneous material

J.B. Leblond, L. Ponson, M. Vasoya

Determination of the cast structure parameter on the basis of micro-segregation analysis

759

J. Dobrovská, F. Kaviþka, K. Stránský, V. Dobrovská

Crack paths near the interface between anisotropic solids

767

M. Specovius-Neugebauer, M. Steigemann, S.A. Nazarov, H.A. Richard

vi

Enriched fracture mechanics from discrete elements method

775

E. Morice, S. Pommier, A. Delaplace

In-situ observation of initiation and propagation of (short) microstructural crack growth

using a rotating bending machine

783

P. Huter, M. Wohlfahrt, C. Oberwinkler

Fatigue crack path evaluation on two different micro-structures H Cand B C Cunder

791

multiaxial loading

L. Reis, V. Anes, B. Li, M. Freitas

Somenumerical assessments on intergranular crack propagation in polycrystals.

Application to J-TiAl

799

D. Geoffroy, J. Crépin, E. Héripré, A. Roos

807

Recent developments in textural fractography of fatigue fractures

H. Lauschmann, Z. Sekerešová, F. Šiška

Effect of notch depth and notch root radius on the J-integral in the plates made of

functionally graded steel

815

H. Monajjem, H. Samareh Salavati Pour, Y. Alizadeh

Experimental and numerical study on mixed mode I/II fatigue crack growth in planar specimens

823

I. Varfolomeev, M. Burdack, S. Moroz, D. Siegele, K. Kadau

Modelling crack propagation in A G Rgraphite bricks in Code_Aster using the eXtended

Finite Element Method

831

P. Martinuzzi, L. Pellet

Growth and interaction of high temperature fatigue cracks nucleated from multiple holes

under small or large scale yielding

839

F. Salgado, A. Köster, V. Maurel, L. Rémy

Optical and infrared vision non-destructive techniques: integration as a means for the

defects detection on impacted composite materials

841

A. Bendada, S. Sfarra, M. Genest, D. Paoletti, S. Rott, E. Talmy, C. Ibarra-Castanedo, X. Maldague

Microstructure and loading effects on fatigue crack growth paths in engineering alloys

849

A.G. Gavras, D.A. Lados

Comparative study of crack path evolution under bending fatigue strain of two

corrosion resistant materials

859

H. Sedjal,F. Hellal

Approximate modelling approaches for estimating the parameters of fatigue

867

T. Ait Saadi, D. Fournier, A. Berred, B. Sadeg, B. Ait Saadi

NURBS-basedgeometric fracture growth representation

875

A. Paluszny, R.W. Zimmerman

vii

Evaluation of the ModeI plastic zone size at the crack tip using R K P Mand F E M

885

M. Hajali, C. Abishdid

Determination of the Stress Intensity Factor at the single edge crack tip using R K P M

895

M. Hajali, C. Abishdid

Characteristics of crack interior initiation and early growth originated from inclusion

for Very-High-Cycle Fatigue of high strength steels

905

Y. Hong, Z. Lei, C. Sun, A. Zhao

Advances in fatigue crack growth modeling

913

H. Sehitoglu, P. Chowdhury, G. Pataky, R. Rateick, H.J. Maier

Numerical simulation of fatigue crack growth in heterogeneous material

915

M. Kikuchi, Y. Wada, K. Suga, Y. Li

Cracks path growth in turbine blades with T B C under thermo – mechanical cyclic loadings

923

T. Sadowski, P. Golewski

Determination of the fatigue life of welded various steels by using Finite Element Method

933

T.E. Ozdemir, H. Cetinel

Numerical stability of plane crack paths under ModeI loading conditions

943

R. Thumser

An experimental investigation on crack paths and fatigue behaviour of riveted lap joints

in aircraft fuselage

951

A. Skorupa, M. Skorupa, T. Machniewicz, A. Korbel

Three-dimensional stress distributions ahead of sharply radiused V-notches in finite thick plates

959

M. Zappalorto, P. Lazzarin

Fatigue crack paths in cast Inconel 713LCwith/without Al diffusion coating under various

loading and temperature conditions

967

J. Pokluda, K.Obrtlík, K.Slámeþka, J.Horníková, S.Pospíšilová, M.Kianicová, T.Podrábský

Determination of the cyclic plastic zone using ECCI-Technique

975

J. Bär

983

Size effect in the damage behaviour of short fibre reinforced composites

I. Scheider, T. Xiao, N. Huber, J. Mosler

Crack orientation in a complete contact fretting-fatigue problem

991

E. Giner, M. Sabsabi, P. Dasí, F.J. Fuenmayor

Where is the stretch zone? - 3DS E M- a powerful tool to analyze fracture

999

S. Henkel, A. Weidner, T. Mottitschka, C. Segel, H. Biermann

viii

1007

Crack propagation in a railway wheel rim in a case of rectilinear ride

P. Navratil, P. Skalka, P. Damborsky, M. Kotoul

X-FEMbased modelling of complex mixed modefatigue crack propagation

1015

H. Minnebo, S. André, M. Duflot, T. Pardoen, E. Wyart

Effect of ultrasonic peening on initiation and propagation of fatigue cracks in welded elements

1023

Y. Kudryavtsev, J. Kleiman, V. Knysh, S. Solovei

On the crack path in a structure of non uniform stiffness

1025

M. Grasso, A. De Iorio, F. Penta, G.P. Pucillo

Crack growth evolution from a notch

1033

C. Navarro, J. Vázquez, J. Domínguez

Towards a probabilistic concept of the Kitagawa-Takahashi diagram

1041

A. Fernández-Canteli, R. Brighenti, E. Castillo

Growth and paths of small cracks initiated from inclusions in ultra high cycle fatigue

1049

A. Roiko, J. Solin

Crack paths in a superalloy in aged condition

1057

F. Kuhn, F. Zeismann, A. Brückner-Foit, K. Kadau, P. Gravett

Determination of crack propagation direction using approach based on generalized

linear elastic fracture

1065

L. Náhlík, P. Huta, M. Ševþík, L. Sestáková

Lowcycle fatigue of pseudoelastic NiTi alloys

1073

C. Maletta, E. Sgambitterra, F. Furgiuele, R. Casati, A. Tuissi

A method for quantitative fatigue fracture surface analysis

1081

N. Ranganathan, N. Sedghi, D. Joly, T.D. Do, R. Leroy, F. Chalon, P. Feraud

On the crack path of rolling contact fatigue cracks in a railway wheel steel

1089

R. Roberti, M. Faccoli, G. Cornacchia, A. Ghidini

Fretting fatigue strength/life estimation considering wear process and Critical Distance Stress Theory

1097

M.A. Bin Ab Wahab, T. Hattori, M. Yamashita

1105

Growth of a doubly periodic array of fatigue cracks in anisotropic elastic medium

V. Bozhydarnyk, H. Sulym, Ia. Pasternak

Crack propagation in a composite laminated plate under bending

1113

V. Bozhydarnyk, Vasyl Shvabyuk, Ia. Pasternak, Volodymyr Shvabyuk

Hot tensile behaviour and cavitation analysis in as-cast and solutionized Al-5.5Mg-Zn alloys

1121

P. Leo, E. Cerri, S. Spigarelli

ix

Effect of temperature and microstructure on hot ductility properties of a boron steel

1131

A. Dimatteo, G. Lovicu, M. DeSanctis, R. Valentini

Crack formation and crack path in CFRPmachining

1139

R. Rentsch

Author Index

1147

x

Review of fatigue crack growth under non-proportional

loading

M.Vormwald1and P. Zerres2

1 Material Mechanics Group, Technische Universität Darmstadt, Petersenstraße 12,

64287 Darmstadt, Germany, e-mail: vormwald@wm.tu-darmstadt.de

2 E L A N - A U SGYmbH, Channel 2, Harburger Schloßstraße 24, 21079 Hamburg,

Germany, e-mail: patrick.zerres@elan-ausy.com

ABSTRACT.Cyclic non-proportional loading is common experimental practise for

investigations of large structures like vehicles. Numerical analysis of local non

proportional loading conditions is also a well established field of research and

application. However, theoretical and practical support is rare for evaluating the

growth of fatigue cracks under non-proportional cyclic loading conditions. At least

seven influence factors – most of them not yet throroughly understood – are listed and

discussed in the paper: the mode-mixity, the material’s influence including its

anisotropy if existant, the degree of cyclic plastic deformation and its direction ahead of

the crack tip, the crack closure phenomenon, the related mean stress effect, the

component’s geometry in general and especially the variable mode-mixity along a crack

front. Two crack propagation mechanisms must be considered: The tensile stress

dominated, mode II minimising mechanism and the shear stess dominated mechanism.

Transition mode-mixities are observed. Some successful explanations of experimental

findings have been published, however, a generally accepted and validated formulation

of a crack driving force parameter is out of sight.

I N T R O D U C T I O N

Most of the engineering structures and components are subjected to fatigue load

conditions, which are a combination of various load sequences originating from

different sources. Only in rare cases, a correlation of these load sequences may be

observed. For ground vehicles, for example, the excitation provided by the roadway

surface is uncorrelated with load sequences from manoeuvring, be it curving or

acceleration and braking. After an onset of fatigue damage – for metallic materials this

generally means the initiation of a fatigue crack – the crack is cyclically loaded in a way

such that at the crack front non-proportional mixed-mode situations will exist.

In experimental investigations of the fatigue strength of such structures, it is common

state of the art to reproduce the action of varios load sequences in their realistic

interconnection in a laboratory. Chassis suspension test systems, for example, with

1

twelve or more actuators simultaneously loading the structure is a commonsight in

vehicle test laboratories.

In numerical fatigue life assessments, methods for dealing with the initiation of

fatigue cracks are available even for the complicated non-proportional cases of

combined cyclic loading. The accuracy of the fatigue life estimates obtained by

applying these methods and the associated software tools is still under thourough

investigation. Nevertheless, the engineers responsible for the fatigue strength of the

structures are supported by these helpful numerical tools. The theoretical and practical

support immediately stops as soon as the growth of fatigue cracks under non

proportional cyclic loading conditions is a matter of concern. A great discrepancy exists

between experimental and numerical feasibilities of performing a proof of structural

durability.

The topic of non-proportional mixed-mode fatigue crack growth has become a field

of scientific interest. The intention of this paper is to provide a collection of references

to already investigated cases, the experimental observations and the analytical and

numerical models developed therein.

M I X E D - M OFDREA C T U RCREITERIA

An early hypothesis for mixed-mode fracture was published by Erdogan and Sih [1].

Their maximumtangential stress criterion postulates that a mixed-mode loaded crack

extends in the direction perpendicular to the maximumtangential stress ahead of the

crack tip. The stress involved is usually calculated for linear elastic conditions and only

the near the crack tip asymptotic, singular stress field is exploited. Shih [2], however,

extended the maximumtangential stress criterion to elastic-plastic analysis for strain

hardening material.

Sih [3] further proposed the strain energy density criterion according to which crack

extension in the direction of the minimumstrain energy density is assumed. Another

energy-based approach was developed by Hussain et al. [4] who made the maximum

energy release rate of a kinked crack responsible for fracture propagation. All of the

hypotheses listed so far predict very similar directions of a growing crack under mixed

mode I and II conditions. In the case that any of the aforementioned hypotheses is

applied for fatigue crack growth analysis, the crack path is predicted such that the mode

II loading at the crack tip is minimised.

A completely different path is obtained by the maximumshear stress criterion [5,6].

This criterion is especially useful in some cases when a crack subjected to mixed-mode

I and II loading may remain or turn to propagate in a direction collinear with the plane

of the maximumshear stress rather than the plane plane perpendicular to the maximum

normal stress. Such a fatigue crack growth behaviour is observed, for example, during

stage I of microstructurally short cracks as well as under enforced severe cyclic plastic

deformation of notched axis-symmetric shafts under torsion.

2

This initial list of most relevant mixed-mode fracture criteria is concluded with

reference to extensive literature surveys of mixed-mode fatigue crack growth under

proportional [7,8,9] and non-proportional loading [10].

O B S E R V A T I O NOSN F A T I G U EC R A C K SG R O W I NUGN D E RN O N

P R O P O R T I O NLAOLA D I N G

Superimposing non-proportional mixed-mode conditions may be performed in

innumerably different ways. In academic studies on the subject the available test rig is

the limiting feature for the choice of non-proportional load sequences. Having only a

uniaxial testing machine at hand, all what can be achieved is to change the crack tip

loading mode (or the mode-mixity) abruptly by changing the specimens’ fixing

conditions. Investigations on mode I pre-cracked specimens which are sujected to a

mode-mixity, tan ) = 'KII / 'KI, different from zero may be seen as being

investigations on non-proportional mixed-mode loading concerning the early growth

after the mode-mixity change.

Abrupt change of the mode-mixity

The crack growth rate for the non-proportional first cycle after a mode-mixity change is

experimentally inaccessible. The investigations focus on the crack deflection angle from

the direction of the pre-crack grown under pure modeI. Since the early investigation of

Iida and Kobayashi [11] the majority of experimental results [7,8] show a crack turning

or kinking towards a path minimising 'KII which may be well described by the

maximumtensile stress criterion. However, Roberts and Kibler [12] found cases for

which the maximumtensile stress criterion was not valid. For high mode-mixities co

planar (with the original Mode I pre-crack) fatigue crack growth was observed. In

descriptions of this observations, the maximumshear stress criterion must be called.

Besides the mode-mixity, this behaviour seems to be dependent on the material under

investigation. No clear classification is available today with respect to which materials

show preferred obedience to a mixed-mode criterion deviating from the popular

maximumtensile stress criterion and its close relatives, strain energy density and energy

release rate criterion. A third factor influencing the fatigue crack growth behaviour must

be emphasised: The co-planar, nearly maximumshear stress driven fatigue crack growth

behaviour is observed preferrably for higher stress intensity factor ranges. At the same

time, this means that larger and more extended cyclic plastic deformations occur in the

vicinity of the crack tip. Plastic deformations in metals – when observed on the

microscopic scale – are dislocation motions in planes with high shear stresses. It seems

that these planes provide the opportunity for crack extension. Even under the

conventional mode I fatigue crack growth situation, the micro mechanism of crack

extension is explained by shear bands deviating from the mode I plane. Under mode I

conditions two symmetric (to the mode I plane) shear systems are competing. After a

deflection to the one side, the other, originally symmetric shear system’s intensity

increases and forces the crack tip to move back towards the symmetry plane. On the

3

macro scale, this fatigue crack growth mechanism results in the rough fracture plane.

Under mixed-mode conditions a symmetry-return mechanism may also appear after an

initial kinking leading to the fatigue crack growth path described by the maximum

tensile stress criterion. Eventually, at high mixed-mode stress intensity ranges, one

shear plane remains dominating and the fatigue crack stays in this plane. For thes cases,

even crack growth rate data can be aquired. A mixed modestress intensity factor range

'KMMwas introduced replacing the conventional stress intensity factor range in a crack

growth rate equation (see the overview in reference [10]) according to:

˜ '

q K

(1)

1 m

'

K K '

M M I m

II

m

Values of m = 2 together with q = 2 were suggested as well as m = 4 together with q = 4

or q = 8, or all of the parameters adjusted to experimental evidence [8].

In some cases, delayed deflection after a short distance of co-planar, shear mode

driven propagation was observed, see for example Gao et al. [13]. Opposite (or in

addition) to what was said to the data from reference [12], the co-planar growth was

observed in the near threshold region. In the discussion of the obtained results a fourth

item with strong influence on the fatigue crack growth behaviour is outlined: The crack

closure mechanism must be considered. Especially the roughness induced closure in

high mode-mixity situation gives rise to a mode II crack tip shielding for low stress

intensity ranges. Large plastic deformations in combination with wear at the fracture

surface may – on the other hand – remove and crack closure associated with mode II.

As it is the case in modeI fatigue crack growth, crack closure and mean stress effect are

closely related phenomena and therefore all effects should be discussed against the

background of the meanstress or stress ratio effect, R = Kmin / Kmax.

In a recent investigation by Highsmith [10] the abrupt change of mode-mixity was

not only performed from pure mode I to proportional mode II and mode I loading.

Highsmith used mode I pre-cracked thin-walled tube specimens with the pre-crack

oriented perpendicular to the specimen axis. With a testing equipment able to apply

cyclic tension and torsion independently of each other on the specimen, real long

ranging non-proportional load sequences can be applied. The material was the Nickel

based super alloy Inconel 718. It was tested at room temperature and the maximum

stress intensity factors at the pre-crack front were in the order of 10 to 25 MPa.m0.5.

Positive stress ratios were applied, R = 0.1 in most cases with some results also for R =

0.6. In his accompanying study on proportional tension-torsion loading, he discovered

the transition from the maximumtensile stress dominated crack deflection to the nearly

co-planar maximumshear stress dominated crack growth occuring at a mode-mixity of

44°. The situation of two competing criteria may be visualised in an interaction diagram

according to Figure 1.

4

K

II

Maximumshear

stress criterion

Transition

Maximumtensile stress cri r on

K

I

Figure 1. Schematic visualisation showing the competition of the maximumtensile and

the maximumshear stress criterion.

In non-proportional mixed-mode loading the mode-mixity varies during a cycle.

Highsmith presents his results in diagrams showing the deflection angle as a function of

the mode-mixity. In Figure 2, the reproduction of some Highsmith’s results, in

especially the results for constant torsion and cyclic tension, show that none of the

criteria mentioned so far is able to predict the correct deflection angle. This statement

holds for both options, inserting the mode-mixity at the maximumload or for the

ranges.

Highsmith also tested a few through-crack round specimens. The main difference

between a thin-walled tube and the round shaft with a through-(pre-)crack is that the

mode-mixity varies considerable along the crack front in the latter case. This issue

inserts a sixth factor of influence on the non-proportional (but also on the proportional)

mixed-mode fatigue crack growth: At the different positions along the crack front, the

crack may obey to different criteria and therefore it may follow different deflection

angles. The final fracture surface appears either strongly warped or it shows a step-like

joining of cracks initially grown in different planes according to different criteria.

Moreover, the out-of-plane constraints differ between the specimens in that the thin

walled tube creates a plane-stress situation and in the through-crack round specimen a

plane-strain situation prevails. If the starter crack is cut in a direction which deviates

from being perpendicular to the specimen axis, mode III non-proportional conditions

may be investigated, too. Observations on this loading case are unaware to the present

authors.

5

860°

Maximumshear

stress criterion

40°

o n a ng l e

20°

40°

80°

60°

Modemixity )

t i

e c

D 20° e f l -20°

Experiments

-40°

-60°

Maximumtensile

-80°

stress criterion

Figure 2. Crack deflection angles over the range of mode-mixity for constant torsion

with superimposed cyclic tension after Highsmith [10].

Highsmith concludes there is no single formulation at hand to predict crack direction

for all cases. A best-practise approach is proposed which is based on the stress intensity

factors at an infinitesimally small kink crack’s tip, k1 and k2.

cos

(2)

2 -

3 2 2 -

-

k

§ ¨ ©

K

cos

K

sin

· ¸ ¹

1

I

2

II

2 1cos s i n 3 c o s 1 k 2 K2 K - - - I II

(3)

- is the kink angle. Since crack deflection angles fall between

In the Eqs. (2) and (3),

the angles of maximumkink tip stress intensity factor, ki,max, and the maximumkink tip

stress intensity factor range, 'ki, a crack driving force combining the influence of both

was suggested:

k

(4)

i k k ' ' ˜

1,max w w i i

A fitting parameter, w, appears in Eq. (4) which is allowed to take different values for

the two cases, i = 1 (tensile stress dominated), and i = 2 (shear stress dominated). A

transition criterion similar to what is shown in Fig. 1 completes the approach.

Highsmith formulated his concept against the background of his overview on

published results. The reference to two more summarypapers by Liu [14] and Bold [15]

6

is given here, both focussing on rolling contact fatigue which is a special type of non

proportional mixed mode loading. Some of these papers are discussed in a later

paragraph; the others are referenced here for an attempt for – however, unreachable –

completeness [16-26]. Highsmith did not intend to compare crack growth rates when

applying Eq. (4) in connection with a crack growth law for pure-mode loading. He only

restricted his work to the description of deflection angles which, however, is an

important pre-requisite for any fatigue crack growth modeling. In the following

amendments to Highsmith’s overview are listed.

Effect of interposed modeII load cycles on modeI fatigue cracks

It is well known that mode I overloading leads to a crack retardation or arrest [27,28]

under pure mode I loading. Investigating the effect of mode II overloads – as a special

case of non-proportional loading – Nayeb-Hashemi and Taslim [29] found that in

contrast to mode I overloads, mode II overloads give rise to short time crack growth

acceleration with no retardation afterwards. In contrast to their observations, Gao and

Upul [30], whoapplied ten overload cycles instead of only one, Srinivas and Vasudevan

[31], Sander and Richard [32] as well as Dahlin and Olsson [33,34,35] observed an

retardation of the mode I crack after the mode II overload, see Fig. 3. Based on the

results shown in Fig. 3, Dahlin and Olsson [33,34,35] stated that there should be a

certain threshold value for the mode II overload, below which no decrease of the

subsequent mode I crack growth rate occurs. They found the crack closure due to the

mode II displacement of the crack-surface roughness which causes mismatch between

the upper and lower crack faces to be the main reason for the decrease in crack growth

rate.

mm/cycle

KII = 20 MPa.m0.5

th r a t e d i f f e r e n c e

0

30 MPa.m0.5

40 MPa.m0.5

-20

gwro

C r a c k

-40

60 m m

20

40

Crack length

Figure 3. Change in modeI crack growth rate due to a single modeII load cycle of three

different magnitudes ('KI = 20 MPam0.5, R=0.1) after Dahlin [35].

7

Changing the overload from purely mode II to Mixed-mode, Sander and Richard [32]

and Srinivas and Vasudevan [31] observed that, by fixing the size of the transient plastic

zone, the crack growth rate retardation decreases with an increasing modeII portion.

Dahlin and Olsson [36] extendend their study to investigate not only the case of

mode II overloads, but also periodic modeII cycles during mode I loading. They stated

that 'KI, the R-ratio of the modeI loading, the magnitude of the modeII loadcycles and

the frequency of the mode II cycles, i.e. the number of mode II loadcycles per mode I

loadcycles, are the main parameters concerning the influence of the sequential mode II

cycles on the crack growth behavior. They found two mechanisms resulting from the

mode II loadcycles: mode II-induced crack closure, which results in a reduction of the

crack propagation rate, and a mechanism that increases the growth rate temporary at

every mode II load. For a crack with high mode I R-ratios, i.e. the crack is open during

the whole mode I loadcycle, the decreasing effect of the mode II load on the crack

growth rate vanishes. They also stated that the crack path deviation from the mode I

crack path is only significant at high modeII frequencies, as shown in Fig. 4.

Direction of KII loading

Crack

Figure 4. Crack path of a specimen under cyclic modeI loading with gradually

increasing modeII frequency after Dahlin and Olssen [36].

As a limiting case of the aformentioned loading Doquet and Pommier [37] studied

sequentially applied mode I and mode II load cycles in ferritic-pearlitic

steel. They

observed coplanar growth for mode-mixities ) between 45° and 76°, see Figure 5.

For a mode-mixity of 76° they found that the crack growth rate is the sum of each

contribution. However, for higher mode-mixities they found a synergetic effect

resulting in a faster growth compared to the simple sum of both contributions. This

effect was also observed by Wonget al. [38,39] for a rail steel. Doquet and Pommier

explained the different bifurcation behavior between their and Wong’s results, who

observed deflection to tensile dominated crack propagation at mode-mixities below 63°,

by the different loading conditions in both tests.

8

ModeI precrack

'KI = 'KII 'KI=0.75'KII ' K I

= 0.5'K II

'KI = 0.25'KII

Notch

200 P m

Figure 5. Crack path development in sequential modeI+II loading with

'KII = 20 MPa.m0.5 after Doquet and Pommier[37].

Effect of superimposed static component on fatigue crack growth

Another special case of non-proportional loading, which was often investigated in

research, is the superposition of a static and a cyclic load component. Here, the

following kinds of non-proportional loading can be distinguished:

1. Cyclic modeI loading and static modeII load

2. Cyclic modeII loading and static modeI load

3. Cyclic proportional loading and static modeI or II load

4. Cyclic modeI or II loading and static Mixed-modeload

Plank and Kuhn [40] studied the crack growth behavior under load cases (1), (2) and

(4) on different aluminum alloys using compact tension shear specimen [41]. Within

their studies they distinguish two modes of propagation, as shown in Fig. 6; a mode I

controlled deviation named tensile mode and a coplanar mode II controlled crack

growth namedshear mode.

ModeI controlled

Tensile mode

growth

Static modeI

-

precrack

Cyclic modeII

ModeII controlled

Shear modegrowth

Static modeI

precrack

Cyclic modeII

Figure 6. Different modes of stable crack growth after Plank and Kuhn [40].

9

They stated that stable shear mode crack growth can only be observed under cyclic

mode II with superposed static mode I, while for cyclic mode I with superposed static

mode II as well as for cyclic mixed-mode with superposed static mode I only stable

tensile mode I crack growth is observed. Moreover, once the crack has turned into

tensile mode it will not reverse to shear mode any more. Based on their experimental

findings, they stated that two loading parameters and one material-specific parameter

are decisive for the kind of crack propagation. For the initiation of shear mode crack

growth, the effective range of the mode II stress intensity factor 'KII,eff must exceed a

certain threshold value 'KII,th,sm, which is material-specific. Furtermore, the 'KII-value

on the starter crack has to be larger than the mode I range 'k1 an the infinitesimally

short kink crack’s tip. They also found 'KII,th,sm to be indirectly proportional to the

average grain size.

Concerning the crack growth rates they found that under cyclic modeII a static mode

I load leads to a significant increase in crack growth rate, because of the reduction of

crack face contact. This result is confirmed by several other researchers, e.g.

[42,43,44,45]. Moreover, at identical cyclic stress intensity factors, the crack growth

rate is higher, if the crack is growing in shear modethan in tensile mode, what would be

in accordance with Eq. (1). The introduction of a static mode II component on cyclic

mode I loading leads to a decreasing crack growth rate and therefore to longer fatigue

lives. Plank and Kuhn [40] explained this behaviour by increasing friction between the

two crack faces.

Effect of phase shift on fatigue crack growth

A case of non-proportional loading, which has become much attention, is the out-of

phase loading. Especially the 90° out-of-phase loading shows significant differences

compared to the proportional loading. In Table 1 a comparison of fatigue lives under

proportional and non-proportional loading is presented.

Referring to the results of the publications cited in Table 1, a phase shift from in

phase loading to out-of phase loading leads to a increase in fatigue life, if the test are

performed in a stress or load controlled condition. This increase is caused by a smaller

increase in local deformations (plastic ratcheting) [9]. In contrast to this, under strain

controlled test, the fatigue life decreases under out-of-phase loading compared to in

phase loading. However, the material also plays an important role. Whatwas said above

holds for ductile materials wheras for semi-ductile materials a life reduction in strain

controlled condition could not be observed. The total life discussed here is the sum of

the life to inititate a crack of technical size and subsequent fatigue crack growth.

Whether or not these general statements on the total life can be transformed unaltered to

the crack growth life alone has not yet been investigated. One aspect of this practically

relevant distinction is that the fatigue cracks have to initiated in the test specimens

applying the same load sequence as it is used in the crack growth investigation. In these

cases, the academic abrupt initial mode changes hardly appear. A variety of such

experimental results have been published by Brüning et al. [59,60,61]. In these

experiments on thin-walled tubes under non-proportional tension and torsion, the wall

10