Issue 37

Vol. X, Issue 37, J ul y 2016

ISSN 1971 -

8993

FRATTURA ED INTEGRITÀ STRUTTURALE

FRACTURE AND STRUCTURAL INTEGRITY

THE INTERNATIONAL ]OURNAL OP GRUPPO ITALIANO FRATTURA (IGF)

www. gru pp ofrattura.it

Frattura ed Integrità Strutturale, 37 (2016); International Journal of the Italian Group of Fracture

Table of Contents

A. Eberlein, H. A. Richard Crack front segmentation under combined mode I- and mode III-loading …………………………. 1 C. Madrigal, A. Navarro, C. Vallellano Plastic flow equations for the local strain approach in the multiaxial case ………………………… 8 P. Bernardi, R. Cerioni, E. Michelini, A. Sirico Numerical simulation of early-age shrinkage effects on RC member deflections and cracking development 15 A. Shanyavskiy, A. Toushentsov Multiaxial fatigue of in-service aluminium longerons for helicopter rotor-blades …………….……..... 22 B. Jo, Y. Shim, A. Raji ć , S. Sharifimehr, A. Fatemi Deformation and fatigue behaviors of carburized automotive gear steel and predictions …...………….. 28 J. Vázquez, S. Astorga, C. Navarro, J. Domínguez Analysis of initial crack path in fretting fatigue ………………………………………...……... 38 C. Brugger, T. Palin-Luc, P. Osmond, M. Blanc Ultrasonic fatigue testing device under biaxial bending ……………………….…........................... 46 C. Riess, M. Obermayr, M. Vormwald The non-proportionality of local stress paths in engineering applications ...………………………… 52 D. Angelova, R. Yordanova, S. Yankova Analysis of fatigue behaviour of stainless steels under hydrogen influence ………….……………… 60 H. A. Richard, A. Eberlein 3D-mixed-mode-loading: material characteristic values and criteria’s validity …….............................. 80 I. Llavori, M.A. Urchegui, W. Tato, X. Gomez An all-in-one numerical methodology for fretting wear and fatigue life assessment ………………...… 87 J. Albinmousa Investigation on parametric representation of proportional and nonproportional multiaxial fatigue responses ………....……………………………………………………………………... 94

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Fracture and Structural Integrity, 37 (2016); ISSN 1971-9883

K. Yanase, B.M. Shönbauer, M. Endo High cycle torsional fatigue properties of 17-4PH stainless steel …………………………………. 101 Les P. Pook, F. Berto, A. Campagnolo Coupled fracture modes under anti-plane loading …………………………...………………… 108 M. Mokhtarishirazbad, P. Lopez-Crespo, B. Moreno, D. Camas, A. Lopez-Moreno, M. Zanganeh Experimental and analytical study of cracks under biaxial fatigue ........................................................ 114 V. Anes, L. Reis, M. de Freitas On the assessment of multiaxial fatigue damage under variable amplitude loading………………….. 124 M. Vieira, M. de Freitas, L. Reis, A. M. R. Ribeiro, M. da Fonte Development of a Very High Cycle Fatigue (VHCF) multiaxial testing device …………………… 131 M. A. Meggiolaro, J. T. P. de Castro, Hao Wu A multiaxial incremental fatigue damage formulation using nested damage surfaces …........………… 138 M. Margetin, R. Ďurka, V. Chmelko Multiaxial fatigue criterion based on parameters from torsion and axial S-N curve ………......…….. 146 J. Kramberger, M. Šori, M. Šraml, S. Glodež Computational simulation of biaxial fatigue behaviour of lotus-type porous material ……………….. 153 N. R. Gates, A. Fatemi Interaction of shear and normal stresses in multiaxial fatigue damage analysis ……………..……… 160 N. R. Gates, Ali Fatemi, N. Iyyer, N. Phan Fatigue crack growth behavior under multiaxial variable amplitude loading .......................................... 166 P.S. van Lieshout, J.H. den Besten, M.L. Kaminski Comparative study of multiaxial fatigue methods applied to welded joints in marine structures ……… 173 V. Shlyannikov, A. Zakharov, R. Yarullin A plastic stress intensity factor approach to turbine disk structural integrity assessment ……………... 193 C.M. Sonsino, R. Franz Multiaxial fatigue of cast aluminium EN AC-42000 T6 (G-AlSi7Mg0.3 T6) for automotive safety components under constant and variable amplitude loading …........……………………………… 200 L. Susmel, D. G. Hattingh, M. N. James, E. Maggiolini, R. Tovo Designing aluminium friction stir welded joints against multiaxial fatigue …......................………… 207 S. Vantadori, A. Carpinteri, G. Fortese, C. Ronchei, D. Scorza, F. Berto Two-parameter fracture model for cortical bone …………………………………………........………… 215 M. Kurek, T. Łagoda, A. Carpinteri, S. Vantadori Estimation of fatigue strength under multiaxial cyclic loading by varying the critical plane orientation ... 221 G. Beretta, V. Chaves, A. Navarro Biaxial fatigue tests of notched specimens for AISI 304L stainless steel …........…………………... 228

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Frattura ed Integrità Strutturale, 37 (2016); International Journal of the Italian Group of Fracture

Y. Hos, M. Vormwald Growth of long fatigue cracks under non-proportional loadings – experiment and simulation …............. 234 T. Wang, L. Susmel Estimation of fatigue lifetime for selected metallic materials under multiaxial variable amplitude loading 241 D. Angelova, R. Yordanova, T. Lazarova, S. Yankova On fatigue behavior of two spring steels. Part I: Wöhler curves and fractured surfaces …........……… 249 D. Angelova, R. Yordanova, T. Lazarova, S. Yankova On fatigue behaviour of two spring steels. Part II: Mathematical models ….........................………… 258 D. Angelova, R. Yordanova, A. Georgiev, S. Yankova On monitoring of mechanical characteristics of hot rolled S355J2 steel …........…………………..… 265 Petar Velev, Rayna Bryaskova Novel magnetic composites based on water soluble unsaturated polyester resin and iron oxide nanoparticles ……………………………………………………………..........………… 272 D. Krastev, V. Paunov, B. Yordanov Recast layers on high speed steel surface after electrical discharge treatment in electrolyte …...................... 280 I. Mazur, T. Koinov Quality Control system for a hot-rolled metal surface …........…………………………………… 287 E. Mihailov, P. Popgeorgiev, M. Ivanova An effect of heat insulation parameters on thermal losses of water-cooled roofs for secondary steelmaking electric arc furnaces …........……………………………………………………………….. 297 U. Muhin, S. Belskij, E.Makarov, T. Koynov Simulation of accelerated strip cooling on the hot rolling mill run-out roller table …........……….…… 305 U. Muhin, S. Belskij, E.Makarov, T. Koynov Application of between- stand cooling in the production hot – rolled strips …........……………….… 312 U. Muhin, S. Belskij, T. Koynov Study of the influence between the strength of antibending of working rolls on the widening during hot rolling of thin sheet metal ….............................................................................................………… 318 C. Patil, H. Patil, H. Patil Experimental investigation of hardness of FSW and TIG joints of Aluminium alloys of AA7075 and AA6061 …........…………………………………………………………………… 325 G. Cricrì, M. Perrella Modelling the mechanical behaviour of metal powder during Die compaction process …........………… 333 Q. Like, D. Jun Analysis on the growth of different shapes of mineral microcracks in microwave field …........………… 342 Z. Hongping, Z. Lili, X. Zhengbing Analysis of time-dependent reliability of degenerated reinforced concrete structure …............………… 352

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Fracture and Structural Integrity, 37 (2016); ISSN 1971-9883

M. S. Raviraj, C. M. Sharanaprabhu, G. C. Mohankumar Experimental investigation of effect of specimen thickness on fracture toughness of Al-TiC composites ... 360 R. Sepe, G. Lamanna, F. Caputo A robust approach for the determination of Gurson model parameters ……………..........………… 369 N. Zuhair Faruq An elasto-plastic approach to estimate lifetime of notched components under variable amplitude fatigue loading: a preliminary investigation ……………..........…………………………………......... 382

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Frattura ed Integrità Strutturale, 37 (2016); International Journal of the Italian Group of Fracture

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 Andrea Carpinteri

(Università di Parma, Italy; Multiaxial Fatigue and Fracture ) (University of Toledo, USA; Multiaxial Fatigue and Fracture ) (University of Seville, USA; Multiaxial Fatigue and Fracture )

Ali Fatemi

Carlos Navarro Pintado

(University of Chemical Technology and Metallurgy, Sofia, Bulgaria; Fracture Mechanics in Central and East Europe )

Donka Angelova

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

(Helsinki University of Technology, Finland)

Robert O. Ritchie Ashok Saxena Darrell F. Socie

(University of California, USA)

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

(University of Illinois at Urbana-Champaign, USA)

Shouwen Yu

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

Cetin Morris Sonsino

Ramesh Talreja David Taylor

Editorial Board Stefano Beretta

(Politecnico di Milano, Italy)

Nicola Bonora

(Università di Cassino e del Lazio Meridionale, Italy)

Elisabeth Bowman Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Eugenio Giner Tommaso Ghidini Daniele Dini

(University of Sheffield) (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

(Università della Calabria, Italy) (Università della Calabria, Italy)

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Fracture and Structural Integrity, 37 (2016); ISSN 1971-9883

Liviu Marsavina

(University of Timisoara, Romania) (University of Porto, Portugal)

Lucas Filipe Martins da Silva

Hisao Matsunaga

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

Mahmoud Mostafavi

Marco Paggi Oleg Plekhov

(IMT Institute for Advanced Studies Lucca, Italy)

(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) (University of Belgrade, Serbia) (Università di Parma, Italy) (Università di Parma, Italy)

Marco Savoia

Aleksandar Sedmak Andrea Spagnoli Sabrina Vantadori Charles V. White

(Kettering University, Michigan,USA)

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Frattura ed Integrità Strutturale, 37 (2016); International Journal of the Italian Group of Fracture

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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Fracture and Structural Integrity, 37 (2016); ISSN 1971-9883

Great news for our journal!

D

ear friend, in this Editorial it is necessary to spread some great news:

First of all, we are publishing two great special issues in this volume. I am sure you will appreciate both the papers focused on the Multiaxial Fatigue and Fracture (Guest Editors: Andrea Carpinteri, Ali Fatemi, Carlos Navarro Pintado) and the papers focused on the Fracture Mechanics in Central and East Europe (Guest Editor: Donka Angelova), and I wish to warmly thank all the Guest editors for their efforts to organize and publish a high level special issue in our journal. Secondly, I wish to thank all the authors that submitted their papers according to the “normal” submission procedure. Since our first issue, we never had a papers submission problem but … now we can affirm that the number, and the quality, of the submissions is really interesting, allowing to the journal to follow a correct selection procedure (always trying to help the authors to get the paper publication). Last, but not least… indexing. First great news: we are now indexed in ISI Web Of Science (ESCI-Emerging Sources Citation Index). Second great news: Scimago Journal & Country Rank has been updated with the 2015 values … we further improved our “numbers” and now, in the Mechanical Engineering topic, we are really near to the Q2 quartile (4/1000). Only the parameters that are normalized with the published papers number show a little decrease, but this is due to the strong increase of the published papers in 2015 (we published 215 papers, to be compared to 93 papers in 2014). These great results are obviously due to a great group of friends that constantly supports all the IGF initiatives, to the help of the Advisory Editorial Board and of the Editorial Board members, to the Associate Editors and to the Guest Editors and, obviously, to all the authors that believed in our journal submitting their papers. Now, we are approaching the tenth year of the Journal life… althought the Journal is really young, in this ten years it achieved great results, but I am sure that with the help of our community … the best has yet to come!! Ad maiora!

Francesco Iacoviello F&IS Chief Editor

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

Focussed on Multiaxial Fatigue and Fracture

Crack front segmentation under combined mode I- and mode III loading

A. Eberlein University of Paderborn, Institute of Applied Mechanics, Pohlweg 47-49, 33098 Paderborn, Germany eberlein@fam.upb.de, http://mb.uni-paderborn.de/fam/ H. A. Richard University of Paderborn, Institute of Applied Mechanics, Pohlweg 47-49, 33098 Paderborn, Germany richard@fam.upb.de, http://mb.uni-paderborn.de/fam/

A BSTRACT . This article approaches the topic of crack initiation and crack growth behaviour under combined mode I- and mode III-loading conditions. Such loading combinations especially lead to a crack, which unscrew out of its initial orientation and segments into many single cracks respectively facets. This characteristic depicts the crucial difference to a crack growth under pure mode I-loading, pure in-plane shearing (mode II) as well as 2D-mixed-mode-loadings. Since this stepped fractured surfaces thus far are proved little and therefore their characterisation remains to be done, a facets quantification using some characteristic dimensions will be performed within this article. After the description of experiments for facet creation the facet’s quantification using the crack profile near the initial position each facet will be analysed concerning characteristic dimensions.

Finally the findings will be illustrated and discussed in this contribution. K EYWORDS . 3D-mixed-mode; Facets; Fatigue; Fracture; CTSR-specimen.

I NTRODUCTION

A

by today unsolved and long existing research matter in fracture mechanics is the characterisation of crack initiation and growth behaviour under combined mode I- and mode III-loading. This loading combination specially leads the initial crack to twist out of its previous direction and separate at once into multiple daughter cracks, afterwards called facets. Building on the researches and findings from Sommer [1], Knauss [2] as well as Pons and Karma [3] within this article a quantification of facets will be presented and discussed. The purpose is to get new insights and facts about facets creation and initiation.

E XPERIMENTS FOR F ACET C REATION

C

reating facets experiments under mixed-mode I + III-loading were performed using the CTSR-specimen and corresponding loading device [4, 5]. A detailed explanation of the experimental procedure and CTSR-specimen’s geometry follows below.

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

CTSR (Compact Tension Shear Rotation) -specimen Referring to the AFM-specimen a new specimen, so-called CTSR-specimen (Fig. 1) has been developed [4]. Relevant specimen’s dimensions are listed in the chart on the right hand side of Fig. 1. Hereby the specimen thickness t is a compromise between a thick specimen with a high torsional stiffness and high testing load levels. The appropriate loading device is shown in Fig. 2. In combination with the new specimen this loading device enables any combination of mixed mode-loading including pure mode I-, pure mode II- and pure mode III-loading. A detailed explanation and illustration of adjusting the mixed-mode-loading can be found in [4, 5, 6].

specimen length l

103 mm

specimen width w

55 mm

initial crack length a 0

27.5 mm

specimen thickness t

15.5 mm

Figure 1 : CTSR-specimen with characteristic dimensions.

Experimental Procedure under combined Mode I-Mode III-loading For the creation of facets crack growth experiments with changing loading directions were performed. After a crack growth of a crack length a = 3.5 mm under mode I-loading condition with a constant cyclic comparative stress intensity factor Δ K V the loading direction by turning the loading device and rotating the specimen was changed into mixed mode I + III-loading with Δ K I ≠ 0 and Δ K III ≠ 0 (Fig. 2).

Figure 2 : Mixed-mode I + III-loading by shifting the loading device.

Then the tests started again under constant cyclic load range Δ F set so, that the cyclic comparative stress intensity factor Δ K V before and after the loading direction was identically. Crack length-Cycle Curves and Fractured Surfaces Typical a - N -curves resulting from such experiments with changing loading directions are shown in Fig. 3. The mixed mode I + III-loading was adjusted by varying both loading angles α and β . A crack length a of 3.5 mm under mode I loading, due to that high cyclic stress intensity factor Δ K V = Δ K I , is reached after ca. N = 70,000 cycles. Afterwards changing the loading direction the crack growth delays. Significant crack growth retardations were noticed by stress intensity factor ratios 1 < K III / K I < ∞.

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

Figure 3 : Crack growth retardation after changing the loading direction from mode I to mixed-mode I + III. On the one hand the retardation effects are caused by the new crack growth direction due to mode III-loading part. Hereby the crack twists at an angle ψ 0 out of its previous orientation. Additionally the crack separates in many facets, which influence the crack growth rate too. Characteristic fractured surfaces with facet formation after changing the loading direction from mode I-loading to mixed-mode I + III-loading are pictured in Fig. 4. This row on fractured surfaces shows the facet formation depending on mode III-part on the total stress intensity factor K III /( K I + K III ). The figures indicate that facet formation begins at a specific mode III-part on the total stress intensity factor of K III /( K I + K III ) = 0.37. Within this experimental research no facets were observed below that ratio. The first fractured surface on the left hand side of Fig. 4, captured by mode III-part on the total stress intensity factor of K III /( K I + K III ) = 0.26, shows that the crack continuously and smoothly without any facet initiation changes its direction. Furthermore, the crack front is still coherent. With increasing mode III-part on the mixed-mode I + III-loading the facets’ shape changes clearly.

Figure 4 : Facet formation with increasing mode III-part.

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

To get a better knowledge of such facet formation under mixed-mode I + III-loading for its consideration in existing hypotheses for crack growth prediction under 3D-mixed-mode-loadings a facet quantification was performed, which is presented and discussed in the next section.

C HARACTERIZATION OF C RACK FRONT SEGMENTATION

T

he first step to understand the crack growth behaviour under mode I-mode III-loading conditions is to quantify the crack front segmentation respectively the facet formation. Therefor some characteristic dimensions and angles were defined. Definition of characteristic Dimensions First of all, the geometry of each facet will be simplified to a circular shape. Then due to the non-planar shear stress τ z facets initiate twisted under a facet angle ψ F as Fig. 5 a) shows. Reflecting the work of Lin et al. [7] this facet quantification distinguishes between two facet types – ascending facets f as indicated in Fig. 5 b) by red lines – and falling facets f fa indicated in Fig. 5 b) by dashed lines, which finally connects the ascending facets f as . Ascending facets f as initiate induced by a local opening mode-loading [8] whereas falling facets f fa form in a bridging region B making a connection to each f as facet.

Figure 5 : Definition of characteristic dimensions for facet quantification: a) Schematic facet formation at the crack front due to τ z; b) Facet’s geometry and characteristic dimensions in the y-z-plane The f fa facets are unfavourable oriented to a local opening mode-loading. Consequently, another local mechanisms, like local friction or plasticity [7], are probably responsible for their formation. So higher energies respectively loads for the creation of f fa facets are required. As a conclusion such facet formation proceeds at a later crack growth stage as the initiation of f as facets [3, 7]. Other characteristic dimensions for facet quantification are the projected facet length d , the facet distance c and the width e of the bridging region B (see Fig. 5 b)). Approach for Quantification of Facet’s Geometry For facet quantification the fractured surfaces were analysed microscopically. Hereby the crack’s profile was measured close to the initial notch that is after a short crack extension Δ a . Fig. 6 illustrates a typical crack’s profile of a mode III fractured surface. The measurement plane of crack’s profile, indicated by the arrow, lies in a distance of about Δ a ≈ 285 µm from the wire eroded notch. In the front view (indicated by the red arrow) the crack’s profile looks as in Fig. 6 b) shown. Thereby the f as facets, which were considered for the quantification, are marked in the graph. Furthermore, it is visible that in the middle of the specimen the biggest facets creates. The analysis of all f as facets reveals an average projected facet length of d = 1.23 mm and an average distance of c = 1.63 mm. The biggest facet angles ψ F exhibit the f as facets f as,3 , f as,4 and f as,5 (see Fig. 6 b)). The angles ψ F lie within an expected range between 42.3° and 49.3°. Starting from the middle of the specimen to the specimen borders a decreasing facet angle was noted. The reason for this is the decreasing shear stress τ z and an increasing mode II-part by moving from the middle of the specimen to the border. Due to no pure mode III-loading condition facets near the specimen border initiate under smaller twist angles. The measurement of the bridging regions B exposed an average width e of 342 µm. Such a systematic analysis approach for facet quantification was performed for all fractured surfaces within this experimental research. In the next section the results of facet quantification are shown and discussed.

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

Figure 6 : Analysis of facet formation: a) Fractured surface under pure mode III-loading; b) Crack’s profile: measurement of each formed facet. Results of Facet’s Quantification The results of facet’s quantification are illustrated in Fig. 7. Due to the fact that within this experimental research no facets below K III /( K I + K III ) of 0.37 initiated, the number of facets in Fig. 7 a) for lower K III /( K I + K III )-ratios is zero. When facets create their quantity decreases with increasing mode III-part to an average number of five facets at pure mode III loading. Moreover, an increasing projected facet length d and facet distance c can be detected by means of fractured surfaces (shown in Fig. 7 b)). At pure mode III-loading an average projected facet length d of 2.5 mm and an average facet distance c of 3.4 mm result. Since the shear stress τ z declines by moving along the specimen thickness from the middle of the specimen to the border, the conditions for pure mode III-loading are mostly given only in a limited range around the centre plane [9]. Therefore, for the measurement of the facet angles ψ F only three facets around the centre plane of the specimen thickness are considered. Fig. 7 c) displays the results of the measured facet angles. In contrast to the hypothesis by Richard for the crack twisting angle ψ 0 the measured facet angles partially are ca. 10° smaller as the hypothesis predicts. However, the measured facet angle ψ F for pure mode III-loading coincides very well with the hypothesis by Richard. The determination of the facet angles respectively the crack twisting angles is principally very difficult. The measured deviations can be formed e. g. by local plastic deformations of the facets while final rupture. In addition a strong correlation between the projected facet length d , the facet distance c and the bridging width e (see Fig. 7 d)) of the bridging regions B (see Fig. 7 e)) with the measured facet angles ψ F exist. Such a correlation between this characteristic dimensions Lin et al. found too [7]. This correlation they explain on the basis of energy’s balance consideration. The result of the bridging regions analyses is pictured in Fig. 7 d). Here the characteristic width e increases with rising mode III-part. The magnitude of the bridging width e is in comparison with the projected facet length d and the facet distance c significantly smaller and is in a direct contact with both values. At pure mode III-loading the bridging width e is almost 1 mm. Further the fractured surfaces with higher mode III-loading parts in the bridging regions B exhibit no fatigue characteristics. Instead the fractured surfaces show a classical strength failure due to cleavage fracture as well as shear fracture. Based on this experimental knowledge of facet’s creation and crack front segmentation under combined mode I mode III-loading the next step will be an establishing of a criterion for crack growth initiation under mixed-mode I + III loadings. In this field today only a few approaches subsist, which are subjected to many assumptions and restrictions [7, 10, 11, 12, 13].

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A. Eberlein et alii, Frattura ed Integrità Strutturale, 37 (2016) 1-7; DOI: 10.3221/IGF-ESIS.37.01

Figure 7 : Overview of facets quantification’s results (averages): a) Number of facets depending on mode III-part; b) Projected facet length d and facet distance c depending on mode III-part; c) Facet angle ψ F depending on mode III-part in contrast to the crack twisting angle ψ 0 by Richard [6]; d) Bridging width e of regions B depending on mode III-part; e) Bridging regions B of a pure mode III fractured surface

R EFERENCES

[1] Sommer, E., Formation of fracture ‘lances’ in glass, Engng. Frac. Mech., 1 (1969) 539–546. [2] Knauss, W.G., An observation of crack propagation in anti-plane shear, Int. J. Frac., 6 (1970) 183-187. [3] Pons, A.J., Karma, A., Helical crack-front instability in mixed-mode fracture, Nature, 464 (2010) 85-89. [4] Schirmeisen, N.-H., Risswachstum unter 3D-Mixed-Mode-Beanspruchung, VDI-Verlag, Düsseldorf, (2012). [5] Eberlein, A., Einfluss von Mixed-Mode-Beanspruchung auf das Ermüdungsrisswachstum in Bauteilen und Strukturen. VDI-Verlag, Düsseldorf, (2016). [6] Richard, H.A., Schramm, B., Schirmeisen, N.-H., Cracks on Mixed-Mode loading – Theories, experiments, simulations, Int. J. Fat., 62 (2014) 93-103.

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[7] Lin, B., Mear, M.E., Ravi-Chandar, K., Criterion for initiation of cracks under mixed-mode I + III loading, Int. j. Frac., 165 (2010) 175-188. [8] Pollard, D.D., Segall, P.E., Delaney, P.T., Formation and interpretation of dilatant echelon cracks, Geol. Soc. Am. Bull., 93 (1982) 1291-1303. [9] Kullmer, G., Richard, H.A., Wang, C., Eberlein, A., Numerische Untersuchungen zur Ermittlung der Rissablenkungs- und Rissverdrehungswinkel bei allgemeiner Mixed-Mode-Belastung, DVM-Bericht 245, Bruchmechanische Werkstoff- und Bauteilbewertung: Beanspruchungsanalyse, Prüfmethoden und Anwendungen, Deutscher Verband für Materialforschung und –prüfung e.V. Berlin (2013) 59-68. [10] Cambonie, T., Lazarus, V., Quantification of the crack fragmentation resulting from mode I + III loading, Proc. Mater. Science 3 (2014) 1816-1821. [11] Pham, K.H., Ravi-Chandar, K., Further examination of the criterion for crack initiation under mixed-mode I + III loading, Int. J. Frac. 189 (2014) 121-138. [12] Leblond, J.-B., Lazarus, V., Karma, A., Multiscale cohesive zone model for propagation of segmented crack fronts in mode I + III fracture, Int. J. Frac. 191 (2015) 167-189. [13] Ronsin, O., Caroli, C., Baumberger, T., Crack front echelon instability in mixed mode fracture of a strong nonlinear elastic solid, Europhys. Lett. 105 (2014) 34001.

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

Focussed on Multiaxial Fatigue and Fracture

Plastic flow equations for the local strain approach in the multiaxial case

C. Madrigal, A. Navarro, C. Vallellano Dpto. Ing. Mecánica y Fabricación, Escuela Técnica Superior de Ingeniería Avda. Camino de los Descubrimientos, s/n. 41092, Seville University of Seville navarro@us.es

A BSTRACT . This paper presents a system of plastic flow equations which uses and generalizes to the multiaxial case a number of concepts commonly employed in the so-called Local Strain Approach to low cycle fatigue. Everything is built upon the idea of distance between stress points. It is believed that this will ease the generalization to the multiaxial case of the intuitive methods used in low cycle fatigue calculations, based on hysteresis loops, Ramberg ‐ Osgood equations, Neuber or ESED rule, etc. It is proposed that the stress space is endowed with a quadratic metric whose structure is embedded in the yield criterion. Considerations of initial isotropy of the material and of the null influence of the hydrostatic stress upon yielding leads to the realization of the simplest metric, which is associated with the von Mises yield criterion. The use of the strain ‐ hardening hypothesis leads in natural way to a normal flow rule and this establishes a linear relationship between the plastic strain increment and the stress increment. K EYWORDS . Low cycle fatigue; Plastic Flow Rule; Kinematic Hardening; Non-proportional Loading; Multiaxial Fatigue The Local Strain Method constitutes nowadays a standard tool for fatigue life predictions in many industries. It has been incorporated in commercial software [7, 8] and it is very well described in textbooks [9, 10]. The extension of the Local Strain Method to the multiaxial case requires at least three main steps. The first one is the development of plastic flow rules which reproduce the way we operate with hysteresis loops, cyclic curves, memory effect and so on in the simple uniaxial case. The second step would be the development of multiaxial Neuber-type rules for dealing with inelastic strains at notches. This relies heavily on the use of a theory of plasticity and hence on the previous step. There are already a W I NTRODUCTION e are trying to develop a theory of cyclic plasticity which allows fatigue designers to make calculations for multiaxial loads in a way as similar as possible to which they do when using the well-known Local Strain methodology for uniaxial low cycle fatigue problems. We would like to define concepts that translate to multiaxial loadings in a simple manner the tools of that trade, namely, the use of the Cyclic Stress-Strain Curve and hysteresis loops, the invocation of the memory rule when hysteresis loops are “closed”, the extension of the Neuber or ESED rules to multiaxial loading, etc. We have found it useful to base our theory in the idea of distance between stress points and to calculate these distances by using the expression for the yield criterion [1-6].

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

number of proposals in this respect [11, 12]. The third step is probably the most difficult and it is the area where more work has been invested so far: the multiaxial cycle counting and fatigue life criteria. There are too many of them to single any one out. A comparison of several criteria is provided in [13]. They need the stresses and strains as inputs and therefore they also depend on the two previous steps. We are concerned here with the first step. Our theory does not make use of yield or loading surfaces that move about in stress space, a common ingredient of existing cyclic plasticity theories. It uses the concept of distance in a stress space endowed with a certain metric measurable from the yield criterion. The full mathematical details of the method have been given elsewhere [1-6] and we would just like here to provide a first insight of this idea of distance in the stress space and show some comparison with experimental results. To keep the discussion at the simplest possible level we will restrict the treatment given here to the case of combined tension and torsion loading. he local strain method revolves around a simplified description of the stress-strain behaviour. A very characteristic feature of the calculations of plastic strains in low cycle fatigue problems is the clear distinction between loading and unloading . In the uniaxial case, one speaks of loading when the stress goes up in the cycle of applied stress and of unloading when it goes down. During the first quarter of the very first cycle, we “move” along the cyclic curve (dashed line in Fig. 1) until unloading starts, marking the first point of load reversal (point A). We then “depart” from the cyclic curve and switch to the hysteresis loop. After a while moving along the descending branch of the hysteresis loop another point of load reversal (point B) will be reached and we will leave the current branch of the loop being traversed and start a new branch going up, and so on. One of the key elements in the simulation of the    behaviour at a notch for variable amplitude loading is the correct application of the memory effect (see [9, chapters 12-14] and [10, chapter 5]), both for closing hysteresis loops and for switching the axes where Neuber’s hyperbolas are drawn for each load excursion. This is shown to occur in Fig. 1 as one moves, for example, from point D to point E. After reaching point E the strain is then decreased to point F, following the path determined by the hysteresis loop shape. Upon re-loading, after reaching point E E', the material continues to point A along the hysteresis path starting from point D, proceeding just as if the small loop E-F-E had never occurred. The same thing happens in the loop B-C-B. As we will point out later on, this memory rule is a simplified representation of the so-called kinematic hardening. T P LASTIC STRAINS CALCULATIONS

Figure 1 : Uniaxial memory effect.

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

As can be seen, the application of the memory effect depends on a precise control of the distance or separation, in terms of stress, between the successive points of load reversal. Thus, for example, when the stress is descending from C, the memory effect is invoked at B' when the distance between the current stress point and C becomes equal to the distance previously established between B and C. Distances between stress peaks and valleys are kept in a stack for comparison and this kind of comparison (at the applied stress level) are really the basis of the cycle counting methods, such as the well-known Rainflow algorithm. It is not at all clear how we can perform these checks in a multiaxial situation, where some of the components of stress may be increasing while others are decreasing. The question then is how we reckon distances in the multiaxial case and how we apply the memory effect. We have proposed a way to answer these questions by looking at the yield criterion in a particular way. The development of the theory is still ongoing and the reader interested in the rigorous derivation is directed to [1-6]. To put it in a nutshell, we believe the yield criterion defines the metric of the stress space and we show how to obtain all the equations of plasticity from this idea. The metric is the mathematical device that allows one to calculate distances and angles in a vector space, in the stress space in our case. How does it work? First, we treat the stress and strain tensors as vectors, just listing all the components in succession. Let’s illustrate the procedure with a relatively simple example. Consider a tension-torsion experiment: a thin-walled cylindrical specimen is subjected to combined tension and torsion under strain controlled conditions. There are only two components of the stress vector σ different from zero in this case, the longitudinal stress  and the shear stress  . Let’s assume the material follows the von Mises yield criterion. The Mises yield locus is a circle of radius 2 k or 2 3 Y in the deviatoric plane, where k and Y denote the yield stresses in pure shear and uniaxial tension (or compression) respectively (see [14, p. 62]). Then, we define the magnitude of the stress vector in the following way, which allows us to say simply that yielding begins when the length or magnitude of the stress vector attains the critical radius

    2 2 2 2 2 3

σ

(1)

We notice that we are not using the usual Euclidean norm in Eq. (1), for the coefficients of this quadratic form, which is what it is called the metric 1 of the space, are not equal to unity. Mathematically, this signals that our space is not Euclidean, which means, loosely speaking, that the basis vectors are not orthogonal. They form an oblique basis. We calculate angles between vectors by means of the familiar dot product, but we have to realize that with the metric chosen the rule is a little different from the usual one. Thus for two stress vectors      1 1 1 , σ and      2 2 2 , σ ,

2 3

 σ σ



  1 2



  1 2

2

(2)

1 2

and the angle  between the two vector follows from

  1 2 1 2 σ σ σ σ



(3)

cos

What about the plastic strain vector? Do we use the same rule as in the stress case? Not really, because plastic strain components are slightly different. Look at the dot product in Eq. (2). Can we apply this to calculate the plastic work? The result would be:         2 2 3 p σ dε p p p dW d d (4) This is obviously not correct. The plastic work is simply

1 Eq. 1 is really an integrated form of the metric since the metric is in fact defined in terms of differentials

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

    p

p

  d

p dW d

(5)

How come? Remember the oblique basis business? When we have an oblique basis, we also have a reciprocal basis. The plastic strain components have to do with this reciprocal basis . In more precise mathematical terms, stresses and plastic strains behave as dual vector spaces . If we have a vector expressed in the original basis and another vector expressed in the reciprocal basis, since the vectors of both bases are, so to speak, orthogonal, then it turns out that their dot product has exactly the form given in Eq. (5). Thus everything is all right. So, finally, how do we calculate the magnitude of plastic strain vectors? One of the nice surprises of the Mises’ metric in Eq. (1) is that the rule for calculating the norm of the plastic strain vector turns out to be the usual Euclidean formula:

             2 2 2 2 p p      

3 2

p

 ( ) p

p

p

p

2

2

2

2

ε





   ( )

 ( ) ( )  

(6)

Please note that in the tension-torsion experiment, while there are only two components of the stress tensor different from zero (  and  ), there are more than two components of plastic strain that must be taken into account, for the hoop and the radial strains are not zero on account of the fact that plastic deformation preserves volume. So if  p is the axial plastic strain, the hoop and radial plastic strains both equal   2 p . This comes out nicely from the general equations given in [2-6]. The usual strain hardening hypothesis, which assumes that the radius of the Mises circle (the so-called equivalent stress) is a function only of the generalized or equivalent plastic strain increment (see [14, p. 68]) now takes the form     σ ε p H d (7) where the integral is taken along the strain path starting at some initial state.   H is a function characteristic of the metal concerned that must be determined experimentally. It is usually a steadily increasing function, for most metals harden when deformed plastically. Under this condition, the function   H has an inverse,    1 H whose derivative   Φ  relates the length of plastic strain vector to the increment of the magnitude of the stress vector    Φ ε σ σ p d d (8) We call this new function,   Φ σ the hardening modulus . It can be derived empirically from conventional uniaxial cyclic tests. This rule (8) implies that plastic deformation only occurs when there is a positive increment in the magnitude σ of the stress vector σ . That is, hardening only depends on the increment of the distance. This fact naturally leads to the normality flow rule. Then, the strain increment vector is given by     Φ ε ε n σ σ n p p d d d (9) where n is the normal unit vector to the yielding surface, i.e., the iso-distance surface in our view, at the stress point. The normal vector is thus given by the gradient of the magnitude σ of the stress vector,

     σ



2 3

(10)

n

1

σ

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

    τ σ



(11)

n

2

2

σ

and it is easy to see that substituting back in (9) we obtain precisely the Prandtl-Reuss equations. Now, situations where load reversals occur are obviously more complicated. We have found that the definition of distance, or rather separation, between stress points after load reversals, from the point of view of plasticity, must involve the angle formed by the lines joining the current stress point and the point of load reversal and this with the origin. However, given the introductory nature of this presentation, this will not be discussed here further. The reader is kindly directed to our last publication [6] to see how this leads to an alternative representation of kinematic hardening and how a multiaxial memory rule can be defined in a rather intuitive way. The flow equations derived can be seen to involve explicitly the points of reversal in each loading segment and there is no need to use yield surfaces moving around in stress space. Everything is controlled by distance. his section discusses the application of the proposed model to experimental results reported by Lamba and Sidebottom [15] on a tubular specimen of oxygen-free high-conductivity (OFHC) copper subjected to combined tension and torsion. The specimen was used for investigating the subsequent strain hardening behaviour after shear strain cycling through the strain control program shown in Fig. 2. The cyclic path sequence was 0-1-0-1-0-2-0-2-etc. All paths started at the crossing point, in the left top corner of the figure. Prior to this strain history the specimen had been cycled under shear strain control until it cyclically stabilized and then along a 90° out-of-phase path until it re stabilized. A comparison made between uniaxial and out-of-phase hardening cycling showed that the cyclic hardening produced by out-of-phase cycling was appreciable greater. T A PPLICATION TO EXPERIMENTAL RESULTS

Figure 2 : Imposed strain history for the subsequent strain hardening experiment. (The cyclic path sequence was 0-1-0-1-0-2-0-2-etc).

This experiment was employed by Lamba and Sidebottom [15] to show the erasure of memory effect observed if the material had been stabilized by 90° out-of-phase strain cycling. According to the authors, as long as the subsequent strain paths remain in the region enclosed by the out-of-phase cycling, one “big” strain cycle with the same or slightly lesser maximum strain as that imposed for the out-of-phase cycling always brings the material to one particular plastic state. The importance of this observation is that the entire strain hardening behaviour can be studied with just one specimen as long as it is subjected to one “big” cycle between each pair of strain paths.

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C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02

The stress-strain response to the path in Fig. 2 appears in Fig. 3, along with the predicted results. In the calculations reported here only the path sequence in Fig. 2 has been considered. The out-of-phase hardening has been taken into account by obtaining the model parameters, namely the metric constants and the hardening modulus function from the stress response and from the stable effective cyclic stress-strain curve derived in a 90° out-of-phase experiment, respectively.

Figure 3 : Stress response to strain path in Fig. 2.

The model predictions quantitatively agree with the experimental results. The strain and stresses at the end of each path are shown in Tab. 1, along with the model predictions. The model’s average error is around 7,5 %, which is much lower than similar studies reported in the literature [16, 17].

ε

γ

σ (MPa) 0.0

τ (MPa) -94.5

σ (MPa) 0.0

τ (MPa) -98.8

Strain Path

Experimental Results

Model Predictions

Error (MPa)

Error (%)

Error (MPa)

Error (%)

Point

(%)

(%) -1.1

0

0.0

-

0.0

4.3

-4.6

1

0.4

-0.8

80.9

-86.7

77.9

3.0

3.7

-90.5

3.8

-4.4

2

0.5

-0.4

99.7

-80.5

108.5

-8.8

-8.9

-81.9

1.4

-1.7

3

0.6

0.2

142.4

-64.2

152.2

-9.8

-6.9

-60.3

-4.0

6.2

4

0.4

0.9

198.6

-8.8

178.2

20.4

10.3

-7.1

-1.6

18.6

5

0.2

1.1

162.9

44.3

170.7

-7.8

-4.8

49.9

-5.6

-12.6

Table 1 : Comparison of experimental results and model predictions.

A CKNOWLEDGEMENTS

he authors would like to thank the Spanish Ministry of Education for its financial support through grant DPI2014-56904-P. T

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