Issue 69

Frattura ed Integrità Strutturale, 69 (2024); International Journal of the Italian Group of Fracture

Table of Contents

S. Cao, A. A. Sipos https://youtu.be/d42nLG_sLzg

About measuring the stress intensity factor of cracks in curved, brittle shells …………………. 1-17 C. Bellini, V. Di Cocco, F. Iacoviello, L.P. Mocanu, L. Sorrentino, R. Borrelli, S. Franchitti https://youtu.be/OlzrYUJYGno Titanium/FRP hybrid sandwich: in-plane flexural behaviour of short beam specimens ……….. 18-28 K. J. Anand, T. Ekbote https://youtu.be/WmoMYDRu4cE Optimization of clamshell content for improved properties in bamboo-epoxy composites ……..…. 29-42 A. Anjum, M. Hrairi, A. Aabid, N. Yatim, M. Ali https://youtu.be/cpGwo0F5ZbU Civil structural health monitoring and machine learning: a comprehensive review …...……......... 43-59 S.V. Slovikov, D.S. Lobanov, E.A. Chebotareva, V.A. Melnikova https://youtu.be/X0rEqkNhCiU The influence of technological defects on the mechanical behavior of CFRP during buckling under compression based on DIC data and acoustic emission …............................................................ 60-70 S. D. Raiyani, P. V. Patel, S. Suriya Prakash https://youtu.be/iOR8XuLaIc4 Effectiveness of partial wrapping of stainless-steel wire mesh on compression behavior of concrete cylinders ……………………………………………………………………………. 71-88 A. K. Almeida, F. S. Brandão, L. F. F. Miguel https://youtu.be/Eno_uKaICe8 Methodology to minimize the dynamic response of tall buildings under wind load controlled through semi-active magneto-rheological dampers ……................................................................. 89-105 M. Semin, L. Levin, S. Bublik, A. Brovka, I. Dedyulya https://youtu.be/5f1ne_c1dKQ Influence of soil salinity on the bearing capacity of the frozen wall ………………………...… 106-114

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Frattura ed Integrità Strutturale, 69 (2024); International Journal of the Italian Group of Fracture

O. Staroverov, A. Mugatarov, A. Sivtseva, E. Strungar, V. Wildemann, A. Elkin, I. Sergeichev https://youtu.be/vD4dGMlEogE Fatigue behavior of pultruded fiberglass tubes under tension, compression and torsion ................... 115-128 M. P. Khudyakov, S. A. Rusanovskiy, N. A. Kapustina https://youtu.be/tuwq9vbaLL0 Experimental study and mathematical modelling of face milling forces of high-strength high viscosity shipbuilding steel ……………………..……………………………....………. 129-141 D. Leonetti, B. Schotsman https://youtu.be/0cZFe-txGBc Experimental investigation on the fatigue and fracture properties of a fine pearlitic rail steel …… 142-153 M. B. Prince, D. Sen https://youtu.be/7M_PaOY7QDM A numerical study on predicting bond-slip relationship of reinforced concrete using surface based cohesive behavior ……................................................................................................................ 154-180 M. J. Khadim, A. J. Abdulridha https://youtu.be/G3Tacm80oPo Enhancing the flexural performance of lightweight concrete slabs with CFRP Sheets: an experimental analysis ……………………………………………………………….... 181-191 S. Eleonsky, V. Pisarev, E. S. Statnik, A. I. Salimon, A. M. Korsunsky https://youtu.be/j6kXih0kvAA Residual stress determination by blind hole drilling and local displacement mapping in aluminium alloy aerospace components ……………………………………………………………. 192-209 T. B. Prakash, M. Gangadharappa, S. Somashekar, M. Ravikumar https://youtu.be/wUX4Kum9Uq0 Impact of nanoparticles (B4C-Al2O3) on mechanical, wear, fracture behavior and machining properties of formwork grade Al7075 composites ……………………………………….... 210-226

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Frattura ed Integrità Strutturale, 69 (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 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)

Dorota Koca ń da Stavros Kourkoulis

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

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Frattura ed Integrità Strutturale, 69 (2024); International Journal of the Italian Group of Fracture

Carlo Mapelli Liviu Marsavina

(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)

(Federal University of Pampa (UNIPAMPA), Brazil)

Leandro Ferreira Friedrich

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 Madeva Nagaral Vasile Nastasescu

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

IV

Frattura ed Integrità Strutturale, 69 (2024); International Journal of the Italian Group of Fracture

Stefano Natali Pavlos Nomikos

(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)

Abdul Aabid Shaikh

(Prince Sultan University, Saudi Arabia)

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)

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, 69 (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 Australia: 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:

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, 69 (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, 69 (2024); International Journal of the Italian Group of Fracture

FIS news

D

ear friends, I wish to thank you for all the appreciation I received concerning the new publishing timing (paper publication immediately after the proofs acceptance and the Visual Abstracts uploading). The "traditional" publishing deadlines (January, April, July and October) allow for the collection of the papers published in the three months before the deadline and for publishing the browsable version.. Concerning the browsable version, I am delighted to inform you that also this service is particularly appreciated: considering only the last six months, we collected more than three hundred thousand views! … and all these “views” are additional to our journal's “traditional” users. To improve the readability and the usefulness of the papers published in Frattura ed Integrità Strutturale - Fracture and Structural integrity, we defined the following limits for the number of references and self-references:  25 references max for a research paper (with max 5 self-references);  50 references max for a review paper (with max 8 self-references);  no limits for an invited review (with no limits for self-references). In June, the new evaluation of Scimago was published. I am happy to inform you that Frattura ed Integrità Strutturale – Fracture and Structural integrity has been confirmed in the Q2 quartile for the following categories:  Civil and Structural Engineering; Finally, 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 on 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 - 30.06.2024 : Abstracts submission - 30.06.2024 : 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  Mechanical Engineering;  Mechanics of Materials. We are grateful for the efforts of the authors, reviewers and editorial board members: they are the authors of this amazing result! But… do not hesitate to use the paper published in FIS for your refs (especially the papers published in the last two years)… this will help the journal to maintain the Q2 evaluation and, maybe … obtain something better!

Francesco Iacoviello Frattura ed Integrità Strutturale Editor in Chief

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

About measuring the stress intensity factor of cracks in curved, brittle shells

Siwen Cao Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Budapest, Hungary cao.siwen@edu.bme.hu, https://orcid.org/0009-0000-5323-2612 András A. Sipos* Department of Morphology and Geometric Modeling, Budapest University of Technology and Economics, Budapest , Hungary HUN-REN-BME Morphodynamics Research Group, Budapest, Hungary siposa@eik.bme.hu, http://orcid.org/0000-0003-0440-2165

Citation: Cao, S., Sipos, A.A., About measuring the stress intensity factor of cracks in curved, brittle shells, Frattura ed Integrità Strutturale, 69 (2024) 1-17.

Received: 19.02.2024 Accepted: 05.04.2024 Published: 13.04.2024 Issue: 07.2024

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 . Curved shell, Stress intensity factor, Digital Image Correlation method, Williams expansion, non-developable surface

I NTRODUCTION racture of thin curved shells has severe consequences in the broad field of engineering. From pipelines and pressure vessels [1-5], to masonry vaults and concrete shells [6-8], the dominant membrane behavior makes curved shells an efficient structure in countless applications. Since the membrane behavior dominantly balances the external actions, the thickness of curved shells tends to be small, meaning that in case of fracture, the entire cross-section becomes cracked instantaneously [9-11]. This situation calls for experimental, analytical, and numerical investigations of crack propagation; however, most of the results of classical fracture mechanics tackle planar media . Specifically, in the realm of linear elasticity, the Stress Intensity Factor (SIF) is used to characterize the singular stress distribution F

1

S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

around the crack tip. Keeping the assumption on linear elasticity, the SIF can be relatively easily obtained from displacement measurements on the specimen undergoing fracture. The displacement field in the vicinity of the crack tip can be reliably recorded by Digital Image Correlation (DIC) techniques [12-14], which, due to its simple setting, seems to be gradually eradicating traditional measurement techniques, such as the strain gage method [15] or the photomechanical methods [16 22], regardless a quasi-static or a dynamic problem is studied. Here, we focus on cracks emerging under quasi-static action. Nonetheless, mechanical assumptions are needed to approximate the SIF from the recorded displacement data. The SIF associated with cracking modes I and II is traditionally derived by the plane stress assumption. Beyond techniques based on the J-integral [23-25], the application of the Willams expansion [26] is widely adopted. On the one hand, it is consistent with linear fracture mechanics; on the other hand, it operates directly on the displacement field recorded in the vicinity of the crack tip. The truncated Williams series fitted to the displacements delivers the SIF as the first-order coefficient in the expansion. The higher-order terms in the expansion might be associated with non-linearities [27-29], but in an experiment, they also reflect the noise of the testing procedure. Depending on the number of terms in the truncated Williams expansion and the number of data acquisition points, the method leads to an overdetermined system of linear equations, where the best-fit solution is sought. Beyond classical least-square techniques [30,31], there are approaches matched to the finite element method (DIC-FEM)[32] and the extended finite element method (HAX-FEM)[33]. In the case of curved surfaces, the curvature has a non-vanishing effect on the stress distribution, and this contribution is found to be so significant that methods assuming a planar medium fail to recover the SIF faithfully [34,35]. Approaches to developable surfaces exist [36], but a general solution for the problem is still missing. This paper introduces a new method to obtain the SIF from experimental data of cracks in weakly curved, brittle shells with a non-vanishing Gaussian curvature. In the case of curved shells, the stress in the surfaces depends on the surface’s curvature [37]. For shallow shells, this contribution can be easily accounted for; hence, the measured displacements can be readily transformed to an equivalent planar medium under plane stress. In the equivalent setting, the application of the Williams expansion is straightforward. As in most engineering applications, the investigated surfaces are weakly curved (i.e., their curvature is moderate), and the cracks are limited in length; we argue the new method is sufficient for most applications to predict the SIF from the measured data reliably. Specifically, the SIF is obtained via the first-order coefficients of the best-fit Williams expansion. While verifying the method’s reliability in experimental work, the tension problem of circumferential cracks in cylindrical shells has been repeated [38], and the obtained test results are compared to theoretical and numerical predictions. Similarly, results on spherical domes are compared against theoretical predictions in the literature. Finally, the convergence properties of the method are studied. T HEORETICAL CONSIDERATIONS he SIF in Mode 1 and 2 cracking characterizes the stress singularity around the crack tip. This singularity is traditionally studied in a plane stress setting, i.e., for a thin, planar medium with Young modulus E , Poisson ratio ν and thickness h , the T stress tensor and the e infinitesimal strain tensor read: ( ) ( ) 2 1 1 1 xx xy xx yy xy xy yy xy xx yy T T e e e Eh T T T e e e ν ν ν ν ν + −     = =     − + −     (1) T

2  ∂ ∂ ∂  +   ∂ ∂ ∂ u v x y x u

e

e

  

  

1 2

xx xy xy yy e

  

   

=

=

e

(2)

e

y ∂ ∂ ∂ ∂

∂ ∂

u v

v y

2

 +

x

where u (x,y) and v (x,y) are the in-plane displacement components. Following the lead of [37] in the case of a shell with moderate curvature, the classical Föppl-von Kármán (FvK) plate equations can be readily extended. Let W ( x,y ) denote the midsurface of the shell in the reference (unloaded) state, and let the vertical displacement component be w ( x , y ). In specific, stress T is formally identical to Eqn. (1), but the strain components of the curved shell read:

2

S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

2

u W w x x x ∂ ∂ ∂ ∂ ∂ ∂

1 2  ∂  +   ∂  w x

= +

e

(3)

xx

1 2   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + +   ∂ ∂ ∂∂ ∂∂ ∂∂ u v W w W w w w y x x y y x x y

e

(4)

xy

2

1 2 w y   ∂ +    ∂ 

v W w ∂ ∂ ∂ y y y ∂ ∂ ∂

= +

e

(5)

yy

Note that the shear strain e xy defined here is half of the engineering shear strain and w ≡ 0 recovers the classical plane stress setting. Similarly, W ≡ 0 leads to the FvK plate theory. Nonetheless, measurements can provide the values for u,v,w and W. In our work, we introduce two simplifying assumptions: i. based on the moderate curvature of the surface, we postulate that the distribution of w around the crack tip is close to linear; i.e., we approximate the non-linear function w ( x,y ) with its first-order truncated Taylor series. That is w ax by c ≅ + + (6) is postulated and the triple ( a,b,c ) is obtained from the measurements via a least-square fit. ii. locally, the surface is approximated with a paraboloid. These two assumptions yield, that we can introduce the displacements ( ū , v̄ ) of the equivalent planar problem, namely:

2 1 + +

1 2

(

)

, u u aW x y = +

a x aby

(7)

2

2 1 + + b y

1 2

(

)

, v v bW x y = +

abx

(8)

2

Substitution of Eqns. (7) and (8) into the expression in Eqn. (2) is identical to the spatial problem in Eqns. (3-5) if assumption (i) is followed. For the sake of completeness, we provide the W c and W s formulas for cylindrical and spherical specimens, respectively. In both cases, based on assumption ii. and R denoting the radius of the main circle, we have:

1

(

)

(9)

2

=−

, c W x y

x

R

2

1

1

(

)

(10)

2

2

=−

−

, s W x y

x

y

2 R R

2

In summary, the equivalent plane stress problem, characterized by ( ū , v̄ ), provides an identical growth rate of the stress (compared to the curved situation) because the stresses in the shallow shell in the membrane state in the vicinity of the crack tip resembles to a 2D plane stress crack tip, with Eqns. (7) and (8) providing the transformation between the two cases, making the method to a reliable predictor of the SIF. M ETHODOLOGY D-DIC (digital image correlation) is widely applied in the full-field measurements of deformation and strains in scientific and industrial conditions [39-41], and it can measure the displacement of shell structures like cylindrical structures and spherical structures, as Fig. 1 shows, the components of one 3D-DIC experiment include the specimen, two CCD cameras, and a computer. For improved visibility, the relevant region of the specimen surface is painted with artificial speckles; during the loading process, two CCD cameras capture images simultaneously. After the experiment, the 3

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

displacement field with components stored in vectors U , V , and W in a global frame at each time instant can be retrieved.

right CCD camera

computer

specimen

left CCD camera

Figure 1: Components of 3D-DIC and results of displacement field.

To calculate the stress intensity factor of a curved shell, as Fig. 2 (a) shows, the tangent to the surface is located at the crack tip. Denote the unit normal vector of the surface at the crack tip to k , the unit vector in the tangent plane directed along the extended crack to i , and set j = k × i , where ‘ × ’ denotes the cross product. The displacement component vectors U , V , W in the global basis ( x , y , z ) can be transformed to the local basis ( i , j , k ) via:

    u

          U V W

v A

(11)

  =     w

( , , ) ( , , ) x y z i j k →

where matrix A (x,y,z) → (i,j,k) is the transformation matrix from basis ( x , y , z ) to the basis ( i , j , k ), and u , v , w are the displacement component vectors in basis ( i , j , k ). The displacement component w is normal to the tangent plane spanned by i and j , which means the 2D displacement components are vectors u , v . In the following step, the elements of w is used to fit a plane and obtain the constants ( a,b,c ), as it is described in the previous section. In order to compute (u ,v) in Eqns. (7) and (8), we need W . It is either measured in the unloaded state, or in the case of simple geometries, it is known a-priori, as it is given for a cylinder (with a horizontal crack) in Eqn. (9) or for the sphere in Eqn. (10). The values of the equivalent displacements (u ,v) are stored in the vectors u and v .

curved surface

tangent plane

j

j

cracking direction

j

R S 1

i

R S 2

o

o

i

i

crack

θ

k

r

normal vector

a

data point

y

z

x

o

(a)

(b)

(c)

Figure 2: Projection of the displacements and selection of data points on the tangent plane. (a) placement of the tangent plane and the local basis ( i , j , k ). (b) the local basis and the applied polar coordinates at the crack tip. (c) Data selection ring with the radius R s . Then, with the equivalent 2D displacement component vectors u and v , the computation of the stress intensity factor can be carried out via the Williams expansion [23]. As Fig. 2 (b) shows, the i -axis of the local basis is aligned with the

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

extending direction, i.e., the tangent vector of the crack’s curve. The polar coordinates with angle θ and radius r are introduced. On the data selection ring around the crack tip, the displacement components u and v are determined from the displacement component vectors u and v by interpolation. Note that the number of data points D N and the ring radius R S are free parameters in the method and affect the convergence rate (see later). Then, the in-plane displacement field around any crack can be expressed with the help of the following Williams expansion:

2 n

cos( 2 2 2 n n n n n n θ − 2 2 2 n n n 2 2 2 n n n 2 2 2 θ θ θ + + + sin( sin( cos(

     

      

 

  

n

κ

θ

+ + −

−

( 1) cos

2)

u     =   v

∞ ∑

A



/2

n n

r

G

2

2 n

  

  

n

=

n

0

κ

θ

− − −

−

( 1) sin

2)

(12)

2 n

          



  

n

κ − − + −

θ

−

( 1) sin

2)

    

∞ ∑

B

/2

n n

+

r

G

2

2 n

  

 − + −

=

n

n

0

κ

θ

−

( 1) cos

2)

where G is the material’s shear modulus, u and v are the i and j -directed displacement components. κ =(3- ν )/(1+ ν ) for plane stress. A n and B n are the coefficients of the Williams expansion. In specific, four coefficients among A n and B n are essential for fracture mechanics [42]:

0 u G

2 +

=

A

0

K κ

1

I

=

A

1

π

2

0 v G 2

=

B

(13)

0

+

κ

1

K

II

=−

B B

1

( 2 2 / 1 c G π ϕ κ +

)

=−

2

Here u 0 and v 0 are the rigid body displacement, φ c is the rigid body rotation to the crack tip, and K I and K II are the stress intensity factors for mode I and mode II cracks, respectively. For details, we refer to [42-44]. In this paper, we study a problem where a mode I crack is dominant [38]; hence, we aim to approximate A 1 , and consequently, K I can be computed by Eqn. (13). The T N number of terms in the truncated Williams expansion should be sufficiently large to calculate the SIF with high precision. Nonetheless, the D N number of data points should be equal to or exceed ( T N +1) [42,23]: N N 1 D T ≥ + (14) Following Fig. 2 (c), the local coordinate basis (in particular, the location of the crack tip and the crack orientation) is detected and corrected by the user manually (since the notches are pre-cut, these parameters are easily determined.). The data points are selected on data-selecting rings surrounding the crack tip with different radii R S1 , R S2 , …etc. These radii should be sufficiently big to avoid the intensively nonlinear zone around the crack tip [45]. The size of the region varied from specimen to specimen and can be determined from the strain-field contour obtained by the DIC method. For any data point i , the coordinates θ i , and r i are given by the location of the point, and its displacements u i and v i are obtained from the DIC data. The truncated Williams expansion up to the term T N readily follows in a matrix form:

5

S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

(

)

(

)

(

)

( f r f r ( 1

)

(

)

(

)

θ θ

θ θ

θ θ

θ θ

θ θ

θ θ

 

              

f

r

f

r

f

r

f

r

f

r

, ,

, ,

, ,

, ,

, ,

, ,

 

A 1 1

B 1 1

A 1 1

B 1 1

A 1 1

B 1 1

N             =  1 2 D u u u

T

T

0

0

1

N

N

(

)

(

)

(

)

)

(

)

(

)

N                                  N 0 0 1 1 A B A B A B T T

f

r

f

r

f

r

f

r

f

r

B 2 2

A 2 2

B 2 2

A 2 2

B 2 2

A 2 2

T

T

0

0

1

1

N

N









 



(

)

(

)

(

)

(

)

(

)

(

)

θ θ θ

θ θ θ

θ θ θ

θ θ θ

θ θ θ

θ θ θ

f

r

f

r

f

r

B f r

f

r

f

r

, , ,

, , ,

, , ,

, , ,

, , ,

, , ,

  

D

N N N D D

D D

D D

D D

D D

D

B

A

B

A

A

(15)

T

T

N

0

N N

0

N N

1

N N

1

N N

N N

N 1              2 D v v v

( g r g r ( 0

)

( g r g r ( 0

)

( g r g r ( 1

)

( g r g r ( 1

)

(

)

(

)

g

r

g

r

B 1 1

A 1 1

B 1 1

A 1 1

B 1 1

A 1 1

T

T

N

N

)

)

)

)

(

)

(

)

g

r

g

r

B 2 2

A 2 2

B 2 2

A 2 2

B 2 2

A 2 2

T

T

0

0

1

1

N

N









 



  

(

)

(

)

(

)

(

)

(

)

(

)

θ

θ

θ

θ

θ

θ

A g r

g r

A g r

B g r

g

r

f

r

,

,

,

,

,

,



D D

D D

D D

D D

D D

D D

A

B

T

T

0

N N

B

N N

1

N N

1

N N

N N

N N

0

N

N

where the functions in the coefficient matrix are given by

n

/2

r

2 n

cos( 2 2 2 n n n n n n θ − 2 2 2 n n n 2 2 2 θ θ + + sin( sin(

  

   

  

  

( ) , r θ

n

κ

2) , θ 

=

+ + −

−

f

( 1) cos

A

/2 G

2

n

n

r

2 n

  

  

  

( ) , θ

n

κ − − + −

2) , θ 

=

−

B f r

( 1) sin

G

2

n

(16)

n

/2

r

2 n

  

  

  

  

( ) , θ

n

κ

2) , θ

=

− − −

−

A g r

( 1) sin

G

2

n

n

/2

cos( 2 2 n n

r

2 n

2 n

  

 





( ) , θ

n

+

−

θ

2) . θ 

κ

=

B g r

( 1) cos

 − + −   

G

2

n

Eqn. (16) can be written as:

N 2 ,1 D = U C

A

(17)

N N 2 ,2 2 2 +2,1 D T T + N

where vector A contains the unknown coefficients of the Williams expansion. If 2 D N >2 T N +2, then matrix C is rectangular, and the system is overdetermined. Utilizing the generalized inverse of C , vector A can be obtained via

1 − = A C C C U ( ) T T

(18)

In fact, Eqn. (18) yields the solution of Eqn. (17) with the minimal least-squares error.

C YLINDRICAL SHELLS ylinder shell structures are used in this chapter to execute and validate the method for the detection of SIF on curved surface shells. For the experiments, the tensile test of polymethyl methacrylate (PMMA) cylindrical shell specimens (shown in Fig. 3(a)) was carried out; the mechanical properties and dimensions are shown in Tab. 1. Through cracks of specimens were cut by 0.25 mm diameter diamond wire saws, and the length of the crack is denoted by its central angle 2 α (2 α =30 º , 60 º , 90 º , 120 º , 150 º , and 180 º , respectively). Fig. 3 (b) shows that the specimen is under displacement-controlled tension; the loading rate is kept at 0.25 mm/min using a Zwick/Roell Z-150 testing machine to prevent the interference of dynamic actions (i.e., a quasi-static load). Fig. 3 (c) shows the clamped support of the cylinder specimens. The upper and lower parts of the specimen are fixed by specially designed fixtures, which are fastened with two sets of hose clamps. Sand the ends of the specimen with sandpaper for firmer clamps. The length of the clamps are 15mm. The 3D deformation data on the surface of specimens are obtained by a 3D-DIC system produced by Correlated Solutions, Inc., the capture frequency of photos was 1 Hz. To reduce the environmental effect, the DIC-3D system calculated the average of 5 sets of photos at a time. The vertical distance between two cameras and the specimen is 0.6 m, and the angle C

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

between the two cameras is 6.68 º . The subset size was 11 × 11 pixels, the diameters of the speckles varied between 0.01~0.05mm. The test stopped when the specimen broke. Each test was repeated three times.

Poisson’s ratio ν

Modulus of elasticity E [MPa]

Length L [mm]

Thickness h [mm]

Radius R [mm]

3300 20.0 Table 1: The material parameters and geometric properties of the cylinder model 0.37 120.0 2.0

2 α

L

crack

L/2

R

(a) (c) Figure 3: Experimental setup. (a) shape and dimensions of the cylindrical shell structure (b) the loading and observing of the cylinder specimen. (c) the way of specimens clamping. Meanwhile, the numerical displacement field was simulated by Abaqus 2017 with a linear elastic constitutive equation. The dimensions and mechanical properties of the numerical models were identical to the experimental specimens; the setting of boundary conditions is shown in Fig. 4 (a), and all DoFs at the bottom of the cylinder are fixed. At the top, all DoFs, except the axial direction, are also fixed. The loading force varies from 0 N to 3000 N along the axial direction during 300 s. The finite element mesh type in Fig. 4 (d) is hexahedral (C3D8R), and the characteristic length of the mesh reads 1 mm. (b)

F

L /6

4 L /6

crack

L /6

(a) (b) Figure 4: Cylindrical shell numerical simulation (a) FEM cylinder model for numerical validation. (b) The FEM mesh model used in the numerical simulation. Fig. 5 (a) shows the curves of both K I and K II for increasing tensile stress from experimental specimens EA30, EA30-1,

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

and EA30-2 and the numerical specimen NA30. Nonetheless, the curve of the numerical model is a straight line for both K I and K II, but they exhibit fluctuations in the experiments. This happens because the deformation data from the numerical simulation is smooth. On the contrary, the experimental deformation data are affected by noise. Still, the SIF curves of experimental specimens agree with the trend of the numerical results. Due to the uniaxial tension, we expect a mode I crack. Our results agree with this expectation: the mode I stress intensity factor K I shows an upward trend, while the mode II stress intensity factor K II remains around zero.

Figure 5: Stress intensity factor results of cylindrical shell. (a)SIF (K I , K II )-stress curves at increasing tensile stress for specimens/model with center angle 2α=30 º. (b) Dimensionless SIF comparison of theory, experimental, and numerical simulation results for the cylindrical shell. The experimental, numerical, and theoretical predictions were compared via the dimensionless SIF F, defined as follows:

a σ π

I =K / F

(19)

Tab. 2 shows the results of all experimental specimens and numerical models, and Fig. 5 (b) depicts the dimensionless SIF comparison of theoretical, experimental, and numerical simulation results. From the table, the difference between experimental specimens, numerical models, and theoretical value with small center angle cracks is evidentially smaller than for the large center angle cracks, which trend is more apparent in Fig. 5 (b). The dimensionless SIF F curve of Forman, R. G. [38] changing by crack center angle 2 α in Fig. 5 (b) shows an upward trend. The experimental and numerical results are scattered around this theoretical solution. The average result of the repeated experiments (with the same crack center angle) is also added to Fig. 5 (b) to highlight the overall trend of all experimental and numerical outcomes. When the crack central angle is between 30 º and 120 º , the mean results of the experimental and numerical results are close to the theoretical curve. At 2α=150 º and 2α =180 º , all testing results are smaller than the theoretical value, which can be reflected by results from numerical simulation.

Crack center angle 2 α

30º

60º

90º

120º 2.23 1.57 1.41 1.80

150º 2.02 2.44 2.03 2.24

180º 2.48 2.59 2.46 2.73

1.15 1.08 0.83 1.15

1.18 1.03 1.75 1.32

1.34 1.70 1.20 1.50

Experimental

numerical

Forman, R. G.

1.10

1.27

1.50

1.89

2.53

3.56

Table 2: Dimensionless SIF results of cylindrical experimental specimens, numerical models, and Forman, R. G.’s research [38].

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

S PHERICAL SHELLS pherical shell structures are used in this chapter to verify the suitability of the displacement method for non developable surfaces. Erdogan, F. [46] research the SIF for the crack on a sphere loaded by a uniform membrane load. To compare the results of Erdogan’s research, numerical simulations are conducted by ABAQUS 2017 with a linear elastic constitutive equation. The numerical simulation model dimensions and mechanical properties are shown in Tab. 3, which refers to the properties of spherical shell experimental specimens and the research of Erdogan, F., and the max crack length of Erdogan, F.’s research is 78.97 º , the crack length 2 α of models is set as 16 º , 32 º , 48 º , 64 º , 80 º . The setting of boundary conditions is shown in Fig. 6(a); at the bottom of the spherical model, the DoF of the vertical direction is fixed, and the total membrane force perpendicular to the crack varies from 0 N to 300 N during 300 s. The finite element mesh type in Fig. 6 (b) is hexahedral (C3D8R), and the characteristic length of the mesh reads 1.25 mm. Modulus of elasticity E [MPa] Poisson’s ratio ν Thickness h [mm] Radius R [mm] 3300 0.33 5.0 100.0 Table 3: The material parameters and geometric properties of the spherical model. S

N N

2 α

R

(a)

(b)

Figure 6: spherical shell numerical simulation (a) FEM spherical model for numerical validation. (b) The FEM mesh model used in the numerical simulation. Crack center angle 2 α 16º 32º 48º 64º 80º numerical 1.90 1.90 2.65 3.64 4.63 Erdogan, F. 1.26 1.80 2.45 3.19 4.00 Table 4: Dimensionless SIF results of numerical spherical models, and the Erdogan, F.’s research

Figure 7: Dimensionless SIF comparison of theory and numerical simulation results for a spherical shell.

The results of dimensionless SIF detected by the new method are summarized in Tab. 4 and Fig. 7; the dimensionless SIF F of Erdogan, F. in the table is calculated by linear interpolation. According to Fig. 7 the method can be used for non

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S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01

developable surfaces; the result of dimensionless SIFs meets the result of Erdogan, F.’s well between 30 º to 60 º , and the values are all larger than Erdogan, F. for other crack lengths.

C ONVERGENCE FEATURES ased on the Methodology section, the F dimensionless SIF is affected by T N number of terms in the truncated Williams expansion, the D N number of data points, and the R s data selection radius. To investigate the influence of those factors on the convergence of F clearly, the experimental data of cylindrical shell structure specimens/models are used below. The E N exceed number of D N to T N is defined as: N N N 1 E D T = − − (20) Meanwhile, to separate each specimen clearly, all labels of specimens shown in Tab. 5. Crack center angle 2α 30º 60º 90º 120º 150º 180º Experimental specimen EA30_1 EA60_1 EA90_1 EA120_1 EA150_1 EA180_1 EA30_2 EA60_2 EA90_2 EA120_2 EA150_2 EA180_2 EA30_3 EA60_3 EA90_3 EA120_3 EA150_3 EA180_3 Numerical model NA30 NA60 NA90 NA120 NA150 NA180 Table 5: Label of cylindrical shell structure specimens/models. B

(a)

(b)

(c)

(d)

Figure 8: dimensionless SIF F surfaces with different R S , T N , and E N for specimens/model with crack center angle 2 α =30 º . (a) dimensionless SIF F surfaces of NA30 (b) dimensionless SIF F surfaces of EA30_1. (b) dimensionless SIF F surfaces of EA30_2. (d) dimensionless SIF F surfaces of EA30_3.

10