# Issue 68

Frattura ed Integrità Strutturale (Fracture and Structural Integrity): issue 68 (April 2024)

Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Table of Contents

Ch. F. Markides, S K. Kourkoulis https://youtu.be/gMy-3uo-x2A

Revisiting classical concepts of Linear Elastic Fracture Mechanics - Part II: Stretching finite strips weakened by single edge parabolically-shaped notches ………………………………... 1-18 H. M. F. Mostafa, A. A. Mahmoud, T. S. Mustafa, A. N. M. Khater https://youtu.be/1vC_ixMgMMI Enhancement of punching shear behavior of reinforced concrete flat slabs using GFRP grating …. 19-44 A. Belguebli, I. Zidane, A. H. Amar, A. Benhamou https://youtu.be/x82-a6S0iRA Numerical investigation of an extra-deep drawing process with industrial parameters: formability analysis and process optimization …………………………………………...…………. 45-62 E. M. Strungar, D. S. Lobanov, E. A. Chebotareva, Y. V. Kochneva https://youtu.be/S9cENcv6Kp4 Analysis of mechanical behavior of fiber-glass plastic with hole pattern using digital image correlation and acoustic emission methods ………………………………………...……... 63-76 G. S. da Silveira, C. N. Zenatti, G. M. S. Gidrão, R. M. Bosse, P. R. Novak https://youtu.be/ATvcQu7FeXs Exploring strength and ductility responses of beam-column joints composed UHPC and UHPFRC employing concrete damaged plasticity ........................................................................ 77-93 M. C. Chaves, D. Castro, A. D. Pertuz Comas https://youtu.be/PylLJgbdmTY Uniaxial fatigue study of a natural-based bio-composite material reinforced with fique natural fibers ………………………………………………………………………………. 94-108 S. Cecchel, R. Ferraresi, M. Magni, L. Guerini, G. Cornacchia https://youtu.be/2q6nPxju2UM Evolution of prototyping in automotive engineering: a comprehensive study on the reliability of Additive Manufacturing for advanced powertrain components ……............................................. 109-126

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

V. S. Uppin, P. S. Shivakumar Gouda, I. Sridhar, A. Muddebihal, M. A. Umarfarooq, A. Edacherian https://youtu.be/8OCymabiIcY Interleaving Carbon-Glass veil in glass epoxy composite for improved mode-I fracture toughness – A hybrid approach ………………………………………………………………...… 127-139 K. W. Nindhita, A. Zaki, A. M. Zeyad https://youtu.be/L8qTEwE0jdM Effect of Bacillus Subtilis Bacteria on the mechanical properties of corroded self-healing concrete ... 140-158 P. V. Trusov, E. A. Chechulina, R. M. Gerasimov, V. E. Vildeman, M. P. Tretyakov https://youtu.be/XuXU00dA-x0 Using the wavelet transform to process data from experimental studies of the discontinuous plastic deformation effect ……………………………………………………………………. 159-174 L. M. Torres Duarte, G. M. Domínguez Almaraz, H. M. Venegas Montaño https://youtu.be/yLU0LOHuxMM Ultrasonic fatigue testing of AISI 304 and 316 stainless steels under environmental and immersion conditions ……………………………………………………………...….. 175-185 Z. Moqadaszadeh, H. Salavati, M. Rashidi Moghaddam https://youtu.be/FursWjr1wT8 In-plane mixed-mode brittle fracture assessment of Harsin marble using HCSP specimen ……... 186-196 S. H. Moghtaderi, A. Jedi, A. K. Ariffin, P. Thamburaja https://youtu.be/lisgrKePYA4 Application of machine learning in fracture analysis of edge crack semi-infinite elastic plate ……. 197-208 A. Aabid, M. Baig, M. Hrairi, J. S. Mohamed Ali https://youtu.be/t2de3-VogUs Effect of fiber orientation-based composite lamina on mitigation of stress intensity factor for a repaired ……………………………………………………………………………. 209-221 P. Kulkarni, S. Chinchanikar https://youtu.be/GlBBf3x6tPM Machining effects and multi-objective optimization in Inconel 718 turning with unitary and hybrid nanofluids under MQL …………………………………………………....….... 222-241 V.-H. Nguyen https://youtu.be/5Zw0NZpgiB4 An investigation of cracks caused by concrete shrinkage and temperature differences in common reinforced concrete bridge structures ……………………………………………………... 242-254 M. Sokovikov, S. Uvarov, M. Simonov, V. Chudinov, O. Naimark https://youtu.be/bc-7lZ8Aw_g Metastability, adiabatic shear bands initiation and plastic strain localization in the AMg6 alloy under dynamic loading ………………….……………………………………………. 255-266

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

A. Fedorenko, D. Firsov, S. Evlashin, B. Fedulov, E. Lomakin https://youtu.be/WyrwEcgLEOo A method for rapid estimation of residual stresses in metal samples produced by additive manufacturing ……….…………….……………………………………………….. 267-279 F. E. Altunok, G. De Pasquale https://youtu.be/HWQD3K3l9go Numerical Cohesive Zone Modeling (CZM) of Self-Anchoring AM Metal-CFRP joints …….. 280-295 B. Spisák, S. Szávai, Z. Bézi, R. Erdei https://youtu.be/eATNZfneb9M Modelling of crack propagation in miniaturized and normal SENB specimens based on local failure criterion …………………………………………………………………….... 296-309 A. Aabid, M. A. Raheman, M. Hrairi, M. Baig https://youtu.be/pmSn7UZ8RQM Improving the performance of damage repair in thin-walled structures with analytical data and machine learning algorithms …………………………………………………………... 310-324 E.V. Feklistova, A.I. Mugatarov, V.E. Wildemann, A.A. Agishev https://youtu.be/3CDzFTZ2LuA Fracture processes numerical modeling of elastic-brittle bodies with statistically distributed subregions strength values …………….………………………………..……………… 325-339 M. Sarparast, M. Shafaie, M. Davoodi, A. Memaran Babakan, H. Zhang https://youtu.be/oVnEehHuirI Predictive modeling of fracture behavior in Ti6Al4V alloys manufactured by SLM process …… 340-356 M. Matin, M. Azadi https://youtu.be/hIUZXZGcz5Q Shapley additive explanation on machine learning predictions of fatigue lifetimes in piston aluminum alloys under different manufacturing and loading conditions …………………….... 357-370 C. Bleicher, S. Schoenborn, H. Kaufmann, M. Alizadeh-Sh https://youtu.be/o0n_b9NdKJ4 On the stress- and strain-based fatigue behavior of welded thick-walled nodular cast iron ……… 371-389 V.O. Alexenko, S.V. Panin, S.A. Bochkareva, T. Defang, I.L. Panov https://youtu.be/TGwea8nivWE Ultrasonic welding of lap joints of PEI plates with PEI/CF-fabric prepregs ………....……… 390-409 S. Kotrechko, O. Zatsarna, O. Filatov, V. Bondarchuk, G. Firstov, V. Dubinko https://youtu.be/uPeQEV46TnI Phenomenon of ignition and explosion of high-entropy alloys of systems Ti-Zr-Hf-Ni-Cu, Ti-Zr Hf-Ni-Cu-Co under quasi-static compression …………………………………....……… 410-421

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

U. De Maio, D. Gaetano, F. Greco, R. Luciano, A. Pranno https://youtu.be/5Decn0nCjSM Degradation analysis of dynamic properties for plain concrete structures under mixed-mode fracture conditions via an improved cohesive crack approach ………....……………………... 422-439 S. K. Kourkoulis, E. D. Pasiou, I. Stavrakas, D. Triantis https://youtu.be/l_InDjizg7I Assessing structural integrity of non-homogeneous systems by means of Acoustic Emissions and Non-Extensive Statistical Mechanics ………....………………………………………... 440-457

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Editorial Team

Editor-in-Chief Francesco Iacoviello

(Università di Cassino e del Lazio Meridionale, Italy)

Co-Editor in Chief Filippo Berto

(Università di Roma “La Sapienza”, Italy; Norwegian University of Science and Technology (NTNU), Trondheim, Norway)

Sabrina Vantadori

(Università di Parma, Italy)

Jianying He

(Norwegian University of Science and Technology (NTNU), Trondheim, Norway)

Section Editors Sara Bagherifard Vittorio Di Cocco Stavros Kourkoulis

(Politecnico di Milano, Italy)

(Università di Cassino e del Lazio Meridionale, Italy) (National Technical University of Athens, Greece) (National Technical University of Athens, Greece)

Ermioni Pasiou

(Perm federal research center Ural Branch Russian Academy of Sciences, Russian Federation)

Oleg Plekhov

Ł ukasz Sadowski Daniela Scorza

(Wroclaw University of Science and Technology, Poland)

(Università di Parma, Italy)

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Tel Aviv University, Israel) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (Politecnico di Torino, Italy)

Leslie Banks-Sills Alberto Carpinteri Andrea Carpinteri Giuseppe Ferro Youshi Hong M. Neil James Gary Marquis Liviu Marsavina Thierry Palin-Luc Robert O. Ritchie Yu Shou-Wen Darrell F. Socie Ramesh Talreja David Taylor Cetin Morris Sonsino Donato Firrao Emmanuel Gdoutos Ashok Saxena Aleksandar Sedmak

(Democritus University of Thrace, Greece) (Chinese Academy of Sciences, China)

(University of Plymouth, UK)

(Helsinki University of Technology, Finland)

(University Politehnica Timisoara, Department of Mechanics and Strength of Materials, Romania) (Ecole Nationale Supérieure d'Arts et Métiers | ENSAM · Institute of Mechanics and Mechanical Engineering (I2M) – Bordeaux, France)

(University of California, USA)

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

(University of Belgrade, Serbia)

(Department of Engineering Mechanics, Tsinghua University, China)

(University of Illinois at Urbana-Champaign, USA)

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

John Yates

(The Engineering Integrity Society; Sheffield Fracture Mechanics, UK)

Regional Editorial Board Nicola Bonora

(Università di Cassino e del Lazio Meridionale, Italy)

Raj Das

(RMIT University, Aerospace and Aviation department, Australia)

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Dorota Koca ń da Stavros Kourkoulis Carlo Mapelli Liviu Marsavina

(Military University of Technology, Poland) (National Technical University of Athens, Greece)

(Politecnico di Milano, Italy)

(University of Timisoara, Romania) (Tecnun Universidad de Navarra, Spain)

Antonio Martin-Meizoso Mohammed Hadj Meliani

(LPTPM , Hassiba Benbouali University of Chlef. Algeria) (Indian Institute of Technology/Madras in Chennai, India)

Raghu Prakash

Luis Reis Elio Sacco

(Instituto Superior Técnico, Portugal) (Università di Napoli "Federico II", Italy) (University of Belgrade, Serbia) (Tel-Aviv University, Tel-Aviv, Israel)

Aleksandar Sedmak

Dov Sherman Karel Sláme č ka

(Brno University of Technology, Brno, Czech Republic) (Middle East Technical University (METU), Turkey)

Tuncay Yalcinkaya

Editorial Board Jafar Albinmousa Mohammad Azadi Nagamani Jaya Balila

(King Fahd University of Petroleum & Minerals, Saudi Arabia) ( Faculty of Mechanical Engineering, Semnan University, Iran) (Indian Institute of Technology Bombay, India) (Università di Cassino e del Lazio Meridionale, Italy) (Institute of sciences, Tipaza University center, Algeria) (GM Institute of Technology, Dept. Of Mechanical Engg., India)

Costanzo Bellini

Oussama Benaimeche

K. N. Bharath

Alfonso Fernández-Canteli

(University of Oviedo, Spain) (University of Mascara, Algeria)

Bahri Ould Chikh

Angélica Bordin Colpo

(Federal University of Rio Grande do Sul (UFRGS), Brazil)

Mauro Corrado

(Politecnico di Torino, Italy)

Dan Mihai Constantinescu

(University Politehnica of Bucharest, Romania)

Abílio de Jesus

(University of Porto, Portugal) (Università della Calabria, Italy) (University of Belgrade, Serbia)

Umberto De Maio

Milos Djukic

Andrei Dumitrescu

(Petroleum-Gas University of Ploiesti, Romania)

Devid Falliano

(Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Italy)

Leandro Ferreira Friedrich

(Federal University of Pampa (UNIPAMPA), Brazil)

Parsa Ghannadi Eugenio Giner

(Islamic Azad university, Iran)

(Universitat Politècnica de València, Spain) (Université-MCM- Souk Ahras, Algeria) (Middle East Technical University, Turkey) (Hassiba Benbouali University of Chlef, Algeria) (Università di Roma “La Sapienza”, Italy)

Abdelmoumene Guedri

Ercan Gürses

Abdelkader Hocine Daniela Iacoviello

Ali Javili

(Bilkent University, Turkey) (University of Piraeus, Greece) (Federal University of Pampa, Brazil)

Dimitris Karalekas

Luis Eduardo Kosteski

Sergiy Kotrechko Grzegorz Lesiuk

(G.V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine, Ukraine)

(Wroclaw University of Science and Technology, Poland)

(Henan Polytechnic University, China)

Qingchao Li Paolo Lonetti

(Università della Calabria, Italy)

Tomasz Machniewicz

(AGH University of Science and Technology) (Università Politecnica delle Marche, Italy)

Erica Magagnini Carmine Maletta

(Università della Calabria, Italy) (Università Roma Tre, Italy) (University of Porto, Portugal) (University of Porto, Portugal) (University of Bristol, UK)

Sonia Marfia

Lucas Filipe Martins da Silva

Pedro Moreira

Mahmoud Mostafavi

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Madeva Nagaral Vasile Nastasescu Stefano Natali Pavlos Nomikos

(Aircraft Research and Design Centre, Hindustan Aeronautics Limited Bangalore, India) (Military Technical Academy, Bucharest; Technical Science Academy of Romania)

(Università di Roma “La Sapienza”, Italy)

(National Technical University of Athens, Greece)

(Karpenko Physico-Mechanical Institute of the National Academy of Sciences of Ukraine, Ukraine)

Hryhoriy Nykyforchyn

Marco Paggi

(IMT Institute for Advanced Studies Lucca, Italy) (Università di Cassino e del Lazio Meridionale, Italy)

Gianluca Parodo Arturo Pascuzzo

(Università della Calabria, Italy)

Hiralal Patil

(GIDC Degree Engineering College, Abrama-Navsari, Gujarat, India)

Alessandro Pirondi Andrea Pranno Zoran Radakovi ć D. Mallikarjuna Reddy

(Università di Parma, Italy) (Università della Calabria)

(University of Belgrade, Faculty of Mechanical Engineering, Serbia) (School of Mechanical Engineering, Vellore Institute of Technology, India)

Luciana Restuccia

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

Mauro Ricotta

Giacomo Risitano Camilla Ronchei

Hossam El-Din M. Sallam

(Jazan University, Kingdom of Saudi Arabia) (Università di Roma "Tor Vergata", Italy)

Pietro Salvini Mauro Sassu Raffaele Sepe

(Università di Cagliari, Italy) (Università di Salerno, Italy)

Dariusz Skibicki Marta S ł owik Luca Sorrentino Andrea Spagnoli Cihan Teko ğ lu Dimos Triantis Andrea Tridello

(UTP University of Science and Technology, Poland)

(Lublin University of Technology, Poland)

(Università di Cassino e del Lazio Meridionale, Italy)

(Università di Parma, Italy)

(TOBB University of Economics and Technology, Ankara, Turkey)

(University of West Attica, Greece) (Politecnico di Torino, Italy) (Università di Pisa, Italy) (Universidade de Brasília, Brasilia) (Kettering University, Michigan,USA)

Paolo Sebastiano Valvo Cristian Vendittozzi

Charles V. White Andrea Zanichelli Shun-Peng Zhu

(Università di Parma, Italy)

(University of Electronic Science and Technology of China, China)

Special Issue

Russian mechanics contributions for Structural Integrity

(Mechanical Engineering Research Institute of the Russian Academy of Sciences, Russia) (Institute of Continuous Media Mechanics of the Ural Branch of Russian Academy of Science, Russia)

Valerii Pavlovich Matveenko

Oleg Plekhov

IGF27 - 27th International Conference on Fracture and Structural Integrity

Special Issue

Sabrina Vantadori Daniela Scorza Enrico Salvati Giulia Morettini Costanzo Bellini

(Università di Parma, Italy) (Università di Parma, Italy) Università di Udine (Italy) (Università di Perugia, Italy)

(Università di Cassino e del Lazio Meridionale, Italy)

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Special Issue

Damage Mechanics of materials and structures

Shahrum Abdullah

(Universiti Kebangsaan, Malaysia) (Universiti Kebangsaan, Malaysia)

Salvinder Singh

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

Frattura ed Integrità Strutturale is an Open Access journal affiliated with ESIS

Sister Associations help the journal managing Algeria: Algerian Association on Fracture Mechanics and Energy -AGFME Australian Fracture Group – AFG Czech Rep.: Asociace Strojních Inženýr ů (Association of Mechanical Engineers) Greece: Greek Society of Experimental Mechanics of Materials - GSEMM India: Indian Structural Integrity Society - InSIS Israel: Israel Structural Integrity Group - ISIG Italy: Associazione Italiana di Metallurgia - AIM Italy: Associazione Italiana di Meccanica Teorica ed Applicata - AIMETA Italy: Australia:

Società Scientifica Italiana di Progettazione Meccanica e Costruzione di Macchine - AIAS Group of Fatigue and Fracture Mechanics of Materials and Structures

Poland: Portugal:

Portuguese Structural Integrity Society - APFIE Romania: Asociatia Romana de Mecanica Ruperii - ARMR Serbia:

Structural Integrity and Life Society "Prof. Stojan Sedmak" - DIVK Grupo Espanol de Fractura - Sociedad Espanola de Integridad Estructural – GEF

Spain: Turkey: Ukraine:

Turkish Solid Mechanics Group

Ukrainian Society on Fracture Mechanics of Materials (USFMM)

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Frattura ed Integrità Strutturale, 68 (2024); 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 (January, April, July, October). 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 gruppofrattura@gmail.com. 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 usually completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

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

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

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Frattura ed Integrità Strutturale, 68 (2024); International Journal of the Italian Group of Fracture

FIS news

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ear friends, Frattura ed Integrità Strutturale (Fracture and Structural Integrity) now changes the publishing timing. All the papers are now published immediately after the proofs acceptance and the Visual Abstracts uploading. The "traditional" publishing deadlines (January, April, July and October) will be used to collect the papers published in the three months before the deadline and to publish the browsable version. Don’t forget the next IGF event: The 8th International Conference on Crack Paths (CP2024; https://crackpaths.org/) . The Conference will be held in Rimini (Italy) and online in September 10-12, 2024 (https://www.crackpaths.org). This Conference follows the Conferences in Parma in 2003 and 2006, Vicenza in 2009, Gaeta in 2012, Ferrara in 2015, Verona in 2018 and online in 2021. The deadlines are: - Always open : Registration - 01.01.2024 to 15.06.2024 : Abstracts submission - 15.06.2024 : Acceptance notification - 15.08.2024 : Early bird registration and payment - 10.09.2024 to 12.09.2024: Conference - 30.09.2024 : Papers submission (after the Conference) - 15.10.2024 : Papers acceptance

Francesco Iacoviello Frattura ed Integrità Strutturale Editor in Chief

P.S. Don’t forget to join the new discussion platform we recently activated… the FIS BLOG: https://fisfracture.blogspot.com/

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Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 68 (2024) 1-18; DOI: 10.3221/IGF-ESIS.68.01

Revisiting classical concepts of Linear Elastic Fracture Mechanics - Part II: Stretching finite strips weakened by single edge parabolically shaped notches Christos F. Markides National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mechanics, Zografou Campus, 5 Heroes of Polytechneion Avenue, 157 73, Attiki, Greece markidih@maill.ntua.gr, http://orcid.org/0000-0001-6547-3616 Stavros K. Kourkoulis National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mechanics, Laboratory of Strength and Materials, Zografou Campus, 5 Heroes of Polytechneion Avenue, 157 73, Attiki, Greece stakkour@central.ntua.gr, http://orcid.org/0000-0003-3246-9308

Citation: Kourkoulis, S.K., Markides, Ch.F, Revisiting classical concepts of Linear Elastic Fracture Mechanics-Part II: Mode I stress concentrations in single and double notched strips, Frattura ed Integrità Strutturale, 68 (2024) 1-18.

Received: 14.12.2023 Accepted: 05.01.2024 Published: 11.01.2024 Issue: 04.2024

Copyright: © 2024 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

K EYWORDS . Linear Elastic Fracture Mechanics, Parabolically-shaped notches, Single edge notched strip, Mode-I loading, Stress Concentration Factor, Stress Intensity Factor, Complex potentials.

I NTRODUCTION

his is the second part of a short three-paper series, the aim of which is to revisit some classical concepts of Linear Elastic Fracture Mechanics (LEFM). In the first paper of the series [1] some controversial issues were discussed, related to: (i) the unnatural overlapping of the lips of ‘mathematical’ cracks in ‘infinite plates’ (a phenomenon that is predicted by the solution of the respective first fundamental problem of LEFM, in case it is formalistically applied), and, (ii) the closely associated issue of negative mode-I Stress Intensity Factors (SIFs). It could be counterargumented of course T

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Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 68 (2024) 1-18; DOI: 10.3221/IGF-ESIS.68.01

that concepts like ‘infinite plate’ and ‘mathematical’ cracks are of limited applicability in practical engineering problems (where neither the size of actual structural members is infinite nor the discontinuities are ‘mathematical’ cuts with singular point tips). In spite of this skepticism, it was highlighted in ref. [1] that the solutions of LEFM, if properly adjusted, can provide interesting results, also, for actual structures. In this context, an attempt is described here to relief the suffocating assumptions of ‘mathematical’ discontinuities and ‘infinite’ media, by considering a uniaxially stretched plane strip of finite dimensions weakened by a notch of parabolic shape. The distinction between cracks and notches (either sharp or blunt) concerns the engineering community long ago. In fact, it can be stated that the magnificent history of Fractures Mechanics started with the pioneering study of Inglis, related to the stress concentration at the apex of elliptical holes [2]. The importance of flaws in the form of ‘mathematical’ cracks (rather than of notches) was highlighted a few years later [3], when Gri ffi th performed his well-known experiments with specimens made of glass [4]. Based on the data gathered, he concluded that the “... weakness (of the specimens tested) is due almost entirely to minute cracks in the surface ...” [5]. From this instant on, the main challenge was the description of the stress fields that are developed in the immediate vicinity of the tips of ‘mathematical’ cracks or sharp notches (V-shaped notches) and around the ‘crowns’ of blunted notches (either U- or hyperbolically-shaped). The main difference between the two cases is that around the tip of sharp discontinuities the stress field components attain infinite values while around the crown of blunt notches the stresses remain bounded. Although it may be considered a paradox, the problem of the stress field around sharp discontinuities was proven to be easier (at least under the assumption of linear elasticity), and its solution was essentially facilitated by the concept of the Stress Intensity Factor, introduced by Irwin [6]. Using the SIF concept and taking advantage of Westergaard’s work [7], Irwin provided, already at the late fifties, compact and relatively flexible equations for the stress components around the tips of ‘mathematical’ cracks as the first terms of a series expansion. Around the same period, Williams [8, 9] published his seminal papers with the familiar equations for the stress field around sharp V-notches, in terms of eigenfunction series expansions. Almost simultaneously, Neuber [10] dealt with the stress concentration factors for notched bodies, of various geometries under various loading schemes, adopting Airy’s biharmonic potential functions. For the next decades Neuber’s books became reference points for engineers dealing with problems of practical interest. Equations similar to the ones introduced by Williams [8, 9] were presented almost thirty years later by Carpenter [11], who adopted the technique of complex potentials, developed by Kolosov [12] and Muskhelishvili [13]. On the other hand, the respective problem for blunt discontinuities was proven rather tougher. Even today, contributions providing reliable, full-filed, closed form solutions of the problem are mostly welcome. Creager and Paris are, perhaps, the first ones to tackle the problem, in their effort to confront stress corrosion cracking, i.e. “... growth of cracks due to the combined and interrelated action of stress and environment ...” [14]. According to their approach, corrosion cracking is responsible for the generation of discontinuities in the form of “... an elliptical or hyperbolic cylinder ... in which the radius of curvature at the tip is small in comparison to the major dimensions of the void ...” and not in the form of the “... usual plane ending with zero radius of curvature ...” [14]. They concluded that the respective stress field in the immediate vicinity of the crown of the blunt notch can be still described by means of a ‘generalized’ SIF, assuming that the Cartesian reference system is translated from the tip of the ellipse to its focal point. Some twenty years later, simplified expressions for the respective stress field components were obtained by Glinka [15], in terms of the stress concentration factor, the radius of the crown of the notch and the distance from it (keeping the exponents of the distance from the tip of the notch unaltered). Although it was reported that these simplified formulae provided results in good agreement with the respective ones obtained numerically by means of the Finite Element technique, Lazzarin and Tovo [16] indicated that Glinka’s statement about the ‘universality’ of the notch stress fields are incompatible to Williams’ [8, 9] conclusions concerning the dependence of the singularity degree on the opening angle of the notch. In other words, Lazzarin and Tovo [16] suggested that Glinka’s formulae are valid exclusively for notches with zero opening angles. Some years later, the problem was tackled, also, by Nui et al. [17], who presented a solution for the stress field in a finite plane weakened by a single edge V-shaped notch with rounded tip. The solution was achieved using a modified Schwartz-Cristoffel transformation in parallel with Muskhelishvili’s technique [13]. The last decade of the 20 th century is signaled by the contributions of Lazzarin’s scientific team, which, are, perhaps, the most influential ones on the issue of the stress fields around either blunt or sharp notches. They dealt not only with the description of the stress components but they contributed, also, in the direction of understanding the conditions causing fracture of notched structural members either under static or fatigue loading schemes. Having as starting point the study of welded lap joints under fatigue conditions [18], they presented flexible approximate solutions for the components of the stress field in the vicinity of open notches, adopting the complex potential’s technique [16]. Later, their approximate solutions [16] were improved, taking into account, also, rounded V-shaped notches with large opening angles [19]. In general, they considered a broad variety of geometrical configurations [20] (including even finite domains [21]), discussing their potential influence on the fatigue limit predictions. In addition, they contributed significantly to the development of

2

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 68 (2024) 1-18; DOI: 10.3221/IGF-ESIS.68.01

the ‘generalized stress intensity factor’ concept, both for rounded V-shaped notches [22] and, also, for U-shaped ones [23]. Later on, their attention was focused on the fracture of notched members, both experimentally and analytically, by means of proper fracture criteria, either energy- or stress-based [24-28]. Concluding this short review, it becomes evident that the interest on the issue of the stress field around various types of notches is continuous and uninterrupted since it was discussed for the first time, due to its paramount importance for the solution of practical engineering problems. The topic is even today under intensive study. In most cases the stress field used is that introduced by the scientific team of late Professor Lazzarin [19, 22]. Indicatively only, one could mention the contributions by Chen and Fan [29], who attempted estimation of fracture loads for blunt U-shaped notches by means of two fracture criteria, adopting the equations for the stress field provided by Lazzarin and Filippi [22]. Recently, Ghadirian et al. [30] determined the mode-I fracture toughness of rock like materials using the notched Brazilian disc configuration and the stress field introduced by Filippi et al. [19]. The same field was used by Sangsefidi et al. [31], who determined the mixed-mode fracture toughness of rocks, again by means of the Brazilian disc test with specimens weakened by U-shaped notches. Nowadays, hybrid schemes are extensively used, combining successfully analytical solutions with experimental protocols and numerical tools [32-34]. In the light of the above discussion, it can be definitely stated that the question concerning the stress field in the vicinity of the crown of notched members is by no means closed. In this context, an alternative approach is presented in this study for the analytical confrontation of the problem, based on a proper conformal mapping and the complex potentials technique [13]. The novelty of the present approach is its capability to deal with finite domains and parabolically shaped notches, independently of the specimen-notch relative dimensions. Moreover, the formulae provided for the components of the stresses are full-field and of closed form. Results of the present solution were comparatively considered against the ones obtained by the respective solution by Filippi et al. [19], for similar geometrical configurations and loading schemes. The agreement is proven quite satisfactory, at least from the qualitative point of view. The same is true for the comparison of the results of the present solution against those of a numerical project that is in progress [35].

T HEORETICAL CONSIDERATIONS

The problem he first fundamental problem of plane elasticity of a finite strip (length: 2b, width: 2h), weakened by a stress-free edge notch, of parabolic shape L, is analytically explored in this study. The strip is uniaxially stretched by means of a uniform stress distribution σ xx = σ o ( σ ο >0) that is applied all along its width 2h, as it is shown in Fig.1. The axis of symmetry of the parabola is, also, axis of symmetry of the strip and it coincides with the y-axis of the Cartesian reference system xOy, while the load is applied along the x-axis of the system. T

y

L

c

x

2h

2b

Figure 1: The configuration of the problem.

Assuming further that the material of the strip is linearly elastic, isotropic and homogeneous, Muskhelishvili’s complex potentials technique is adopted. In this context, use is made of the following analytic function ω ( ζ ) [13]:

2 z ω ( ζ ) i( ζ i α ) ; z x iy i( ξ i η i α ) ; x 2( α η ) ξ ; y ξ ( α η ) 2 2 2

(1)

This function maps conformally the region of the plane z=x+iy=re i θ that lies outside of the parabola L with equation:

3

(2)

2

x 2 α α

y

to the mathematical lower half plane of the complex variable ζ = ξ +i η . In other words, orthogonal parabolas in the z plane (with their focus at the origin O of the (x, y) Cartesian reference system), defining a curvilinear reference system ( ξ , η ) in the z plane (Fig.2a), correspond to orthogonal lines in the mathematical lower half plane (Fig.2b). In this way, the vertex (or base-point or tip), z(0, – α 2 ) of the parabola L, corresponds to the point ζ (0, 0) in the mathematical plane. A finite part, of dimensions 2bx2h, of the z plane (bounded by red lines in Fig.2a), containing a finite part of the parabola L, represents the region of the edge notched strip under study.

y

L

0 5 10

[cm]

1

η

ξ

z = ω ( ζ )

z

o

o

c

r

0

a

a

x

o x

o x

2

-1

ζ

-20 -15 -10 -5

2h

z = ω ( ζ )

d f

d f

-2

-3

2b

g

g

[cm]

a d f g

g f d a

m

m

m

m

-4

-20 -15 -10 -5 0 5 10 15 20

-4-3-2-1 0 1 2 3 4

(a) (b) Figure 2: The conformal mapping of the actual plane z with the notched strip (a) on to the mathematical lower half plane ζ (b). For the sake of generality, the x-axis is not considered as symmetry axis of the strip, but rather it is located at a distance c from its upper side. In this context, the two end points of the parabolically shaped notch are (–x o , c), (x o , c) (Fig.2a), while the notch itself is reflected to the linear segment (marked red in Fig.2b) defined by the two end points (– ξ ο , 0), ( ξ ο , 0) in the mathematical plane ζ = ξ +i η . Outline of the method To obtain the solution for the stress field developed in the notched stretched strip, an intact strip of the same dimensions 2bx2h stretched by the uniformly distributed tensile stresses σ ο >0, is considered first. Obviously, at any point of this intact strip (and, therefore, at any point of an internal parabolic locus L), the stress field is described by a uniaxial tensile stress, σ xx = σ ο >0, while it holds that σ yy = σ xy =0 (Fig.3a). The solution of this auxiliary problem in terms of the variable ζ in the mathematical plane will be determined in terms of the functions Φ 1 ( ζ ), Ψ 1 ( ζ ). As a next step, consider the opposite stress field, applied at all the points of the internal locus L, i.e., σ xx = – σ ο , analyzed into components σ ηη , σ ξη in the ( ξ , η ) curvilinear system, mentioned previously (Fig.3b). Let us now set these σ ηη , σ ξη as the boundary stresses on the parabolic notch L of a notched strip 2bx2h with stress free sides (Fig.3c), and denote by Φ 2 ( ζ ), Ψ 2 ( ζ ) the functions solving the specific problem. Then, the solution for the edge notched strip into question, i.e., the finite strip subjected to uniform tension σ o on its sides and weakened by a stress free parabolically-shaped notch L, is obtained by simply superposing the previous two solutions ( Φ 1 ( ζ ), Ψ 1 ( ζ )) and ( Φ 2 ( ζ ), Ψ 2 ( ζ )), as it is shown schematically in Fig.3d).

(3)

2 Φ ( ζ ) Φ ( ζ ) Φ ( ζ ), Ψ ( ζ ) Ψ ( ζ ) Ψ ( ζ ) 1 2 1

The above procedure and the particular solutions Φ 1 ( ζ ), Ψ 1 ( ζ ) and Φ 2 ( ζ ), Ψ 2 ( ζ ), are explicitly described in next sections. In order for the above described method to be better understood it is deemed necessary, to highlight some crucial points of the superposition procedure proposed. The underlying idea is to somehow reach the configuration and solution of a parabolically notched strip by ‘introducing’ the notch to the respective intact stretched strip (Fig.3a). This is here achieved by imposing an opposite stress field – σ o at the points of the internal locus L of the intact stretched strip. In fact, such an imposition of – σ o on the locus L of the intact stretched strip (Fig.3b), cancels the tensile stress σ o along L, rendering the

4

area enclosed by L and the upper boundary of the strip stress free (or, equivalently, this area is ‘neutralized’). Actually, this can be conceived as if the specific area had been removed from the strip, thus, transforming it into the notched one.

y

y

c

x

L

c

2h

1 1 ( ), ( )

x

2b

L

(a)

(b)

y

y

L

c

c

x

x

L

2h

2h

1 ( ) ( ) ( ) ( ) ( ) ( ) 2

( ), ( )

2

2

1

2

2b

2b

(c)

(d)

Figure 3: (a) The problem of the stretched intact strip under direct uniaxial tension and the stress field along the internal parabolic locus L; (b) The opposite stress field applied at all the points of the internal locus L; (c) The problem of the notched strip under the opposite stress field on the boundary of the notch L and stress free sides; (d) The stretched, parabolically notched strip obtained by the superposition of the problems (a) and (c). The question arising now is how could this ‘neutralization’ be implemented? A direct approach is by using the principle of superposition, holding for a linearly elastic behavior of the strip, i.e., by superposing to the initial problem of stretching of the intact strip a problem that induces a – σ o stress along L. But the – σ o stress field acting along the internal locus L could only be achieved by considering the problem of the intact strip acted by σ xx =– σ o on its loaded sides, which upon being superimposed to the stretched strip (by σ xx = σ o ) would lead to the unstressed strip (i.e., there would be no problem). For this reason, it was necessary to consider the notched strip with stress free sides, loaded along its notch L by the boundary stresses σ ηη , σ ξη (Fig.3c), which correspond to the – σ o stress field (Fig.3b). Then, the superposition of the problems shown in Figs.3a and 3c, suffices for both the zeroing of stresses along L in the intact strip (indirectly transforming it into the notched one), and, also, for the achievement of the required loading scheme (i.e., that of a tensile uniform stress σ o at the loaded boundaries of the strip). Obviously, the above-described procedure is essentially a superposition of complex potentials, Φ 1 (z)+ Φ 2 (z), Ψ 1 (z)+ Ψ 2 (z), namely, the perturbation to the stress field due to Φ 1 (z), Ψ 1 (z) that is caused by Φ 2 (z), Ψ 2 (z), which are responsible for the stress vanishing along L. Furthermore, this procedure is well conceived as an extension of the respective one introduced in Muskhelishvili’s [13] milestone book. Indeed, according to Muskhelishvili, the configuration of a perforated infinite strip is achieved by considering the intact strip, an internal boundary L in the location of the perforation, and, then by mapping conformally the area outside of L onto the infinite plane with the unit hole, and, finally, by demanding L to be free from stresses (thus, rendering the intact infinite strip a perforated one). The above assumptions and the respective analytical solutions introduced in next sections are validated by comparing their outcomes against those provided by well-established solutions, for similar geometries and loading schemes, as well as against a numerical solution in progress. Stretching of the intact strip It is easily seen that the complex potentials solving the problem of stretching an intact strip 2bx2h by a stress σ o , which is uniformly applied along its vertical edges (Fig.3a), read as:

5

σ

σ

Φ (z) Φ ( ω ( ζ )) Φ ( ζ )

, Ψ (z) Ψ ( ω ( ζ )) Ψ ( ζ ) 4 ο 1 1 1

(4)

ο

1

1

1

2

Indeed, substituting Eqns.(4) into the familiar formulae providing stresses in terms of Φ 1 (z), Ψ 1 (z) in the actual domain, i.e.:

(5)

1

i σ

2 Φ (z) z Φ (z) Ψ (z)

xx σ σ

4 Φ (z), σ

yy

1

yy

xy

1

1

(where denotes the real part; prime denotes the first order derivative and over-bar denotes the conjugate complex value [13]), it follows that at every point of the intact strip it holds that σ xx = σ ο >0, σ yy = σ xy =0, as it was expected, and, obviously, the same stresses appear all along the internal parabola L of the strip (Fig.3a). Consider now the opposite stress field, i.e.:

(6)

xy σ σ , σ σ xx o yy

0

Substituting from Eqns.(6) in the transformation formulae [13]:

(7)

2i β

σ σ σ σ , σ σ

2i σ ( σ σ

xy 2i σ )e

ηη

ξξ

yy

xx

ηη

ξξ

ξη

yy

xx

one obtains the respective stress components in the ( η , ξ ) curvilinear system, rotated by an angle β with respect to the x-axis (Fig3.b), as: ηη ξξ o σ σ σ (8)

(9)

2i β

ηη σ σ

2i σ σ e

ξξ

ξη

o

Taking into account that [13]:

Eqns.(1)

ω ( ζ ) ω ( ζ )

ζ i α ζ i α

2i β

(10)

e

Eqn.(9) becomes:

ζ i α ζ i α

(11)

2i σ σ

ηη σ σ

ξξ

ξη

o

Adding Eqns.(8) and (11), and recalling that ζ = ξ +i η , yield:

2

ξ

i ξ ( η α )

(12)

σ

ξη i σ σ

ηη

o 2

2

ξ

( η α )

In particular, for η =0, Eqn.(12) provides (after separating the real from the imaginary part) the σ ηη ,L and σ ξη ,L components of the stress field at the points of the internal parabola L, due to the opposite stress field of Eqns.(6), as:

2

ξ

αξ

, σ σ ξη ,L

σ

σ

(13)

ηη ,L

o

o

ξ α 2

2

ξ α 2

2

The notched strip with stress free sides loaded by certain stresses along the parabolic notch L Consider now the first fundamental problem of a strip 2bx2h with stress free linear edges, baring a parabolically shaped edge notch L acted by the previous stresses σ ηη ,L and σ ξη ,L of Eqns.(13) (Fig.3c). The solution of this problem, in terms of

6

the variable ζ = ξ +i η (via the mapping of Eqns.(1)), is expressed in terms of Φ 2 ( ζ ), Ψ 2 ( ζ ). Taking into account Eqns.(1) for η =0, and Eqns.(13), the boundary condition for the stresses on the parabolic notch L reads, in terms of the two functions Φ 2 and Ψ 2 , as follows:

2

( ξ i α )

(14)

( ξ i α ) Φ ( ξ ) ( ξ i α ) Φ ( ξ )

Φ ( ξ ) ( ξ i α ) Ψ ( ξ )

ο σ ξ

2

2

2

2

2

Following Muskhelishvili [13], the solution to the functional equation of Eqn.(14) is:

ξ

ξ

( ζ i α ) 2( ζ i α ) 2

σ ξ d ξ ζ i α ξ ζ ζ i α o

1

o σ ξ d ξ

1

o

o

(15)

Φ ( ζ )

Φ ( ζ )

, Ψ ( ζ )

Φ ( ζ )

2

2

2

2

2 π i( ζ i α )

ξ ζ

2 π i( ζ i α )

ξ

ξ

o

o

After some relatively lengthy algebra it is obtained that:

ζ ξ ζ ξ

σ

(16)

Φ ( ζ )

ζ log

2 ξ

o

ο

2

ο

2 π i( ζ i α )

ο

2

2 6 α ζ i α ζ ξ ζ 3

2

2

2

( ζ i α )

σ

3i αζ

2i αζ 5 α

ζ ( ζ i α )

1

1

(17)

Ψ ( ζ )

log

ξ

o

ο

2

ο

3

3

2( ζ i α ) ζ ξ ζ ξ 2

2( ζ i α )

2 π i

ζ ξ

ο

ο

ο

Stretching of the strip with the stress free edge notch L Substituting from Eqns.(4), (16) and (17) in Eqn.(3), the solution of the problem in question (i.e., that of stretching a finite strip 2bx2h with the stress free edge notch L, (Fig.3d or Fig.1)), is obtained as:

ζ ξ ζ ξ

σ

σ

(18)

o

Φ ( ζ )

ζ log

2 ξ

o

ο

ο

4 2 π i( ζ i α )

ο

2

2 6 α ζ i α ζ ξ ζ 3

2

2

2

( ζ i α )

σ σ

3i αζ

2i αζ 5 α

ζ ( ζ i α )

1

1

(19)

o

Ψ ( ζ )

log

ξ

o

ο

ο

3

3

2( ζ i α ) ζ ξ ζ ξ 2

2( ζ i α )

2 2 π i

ζ ξ

ο

ο

ο

Inversing the transformation of Eqns.(1) yields:

ζ i α iz

(20)

Substituting for ζ from Eqn.(20) in Eqns.(18) and (19), the solution in terms of the variable z=x+iy=re i θ is obtained as:

σ σ

i

i α i α

iz ξ iz ξ

(21)

o

Φ (z)

i α

iz log

2 ξ

o

ο

ο

4 2 π i z

ο

σ σ 3 α z 4i α

3

2

iz 4 α

i α i α

iz ξ iz ξ

o

Ψ (z)

i log

2 i ξ

o

ο

ο

3 2

3 2

2 4 π

z

z

ο

(22)

2

3 iz 4i α 2 ξ z

5 α z iz iz 8 α

ο

2

2 ο

iz

i α

ξ

The stress field in the stretched notched strip At one’s convenience, the stress components in the stretched, notched strip (Fig.1 or Fig.3d), may be expressed either in the Cartesian (x, y) or the curvilinear ( ξ , η ) reference sytem, via the well-known formulae [13]:

7

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