Issue 70

Frattura ed Integrità Strutturale, 70 (2024); International Journal of the Italian Group of Fracture

Table of Contents

H. Siguerdjidjene, A. Houari, K. Madani, S. Amroune, M. Mokhtari, B. Mohamad, C. Ahmed, A. Merah, R.D.S.C. Campilho https://youtu.be/60wX5TknPLA Predicting damage in notched functionally graded materials plates through extended finite element method based on computational simulations …………………………………..…………. 1-23 S. K. Shandiz, H. Khezrzadeh, S. E. Azam https://youtu.be/8SxruxRwaq8 Introduction and application of a drive-by damage detection methodology for bridges using variational mode decomposition …………………………………………………….….. 24-54 T. Pham-Bao, V. Le-Ngoc https://youtu.be/1X8albuh9Rc Correlation coefficients of vibration signals and machine learning algorithm for structural damage assessment in beams under moving load ……..…………………………………………... 55-70 P. Kulkarni, S. Chinchanikar https://youtu.be/KlgHwUjxuG4 Modeling turning performance of Inconel 718 with hybrid nanofluid under MQL using ANN and ANFIS …...………………………………………………………………......... 71-90 K. Dileep, A. Srinath, N.R. Banapurmath, M. A. Umarfarooq, Ashok M. Sajjan https://youtu.be/qWoliXdwFkU Impact of hybrid nanoparticle reinforcements on mechanical properties of Epoxy-Polylactic Acid (PLA) Composites …................................................................................................................ 91-104 E.V. Feklistova, A.I. Mugatarov, V.E. Wildemann https://youtu.be/kd4Imak8yMY Numerical study of the influence of the parameters of statistical distribution of the structural elements’ ultimate strength on deformable bodies’ fracture processes …………………………. 105-120 M. Verezhak, A. Vshivkov, M. Bartolomei, E. Gachegova , A. Mayer, S. Swaroop https://youtu.be/JeSl4lFMQ14 Application of deep learning for technological parameter optimization of laser shock peening of Ti 6Al-4V alloy ……................................................................................................................... 121-132

I

Frattura ed Integrità Strutturale, 70 (2024); International Journal of the Italian Group of Fracture

D. Kosov, A. Tumanov, V. Shlyannikov https://youtu.be/abcp8kFJ_Qo ANSYS implementation of the phase field fracture approach …………………….……...… 133-156 P. Sahadevan, C.P. Selvan, A. Lakshmikanthan, A. Bhaumik, A. F. Cuautle https://youtu.be/pvvcB7RBFOM Effect of printing process parameters on tensile strength and wear rate of 17-4PH stainless steel deposited using SLM process ……………………………………………….................... 157-176 A. Chulkov, A. Moskovchenko, V. Vavilov https://youtu.be/eRITTw-OQ8k Enhancing generalizability of a machine learning model for infrared thermographic defect detection by using 3d numerical modeling ………………..……………………………....………. 177-191 A. Baryakh, A. Tsayukov https://youtu.be/unHOmhNinCE Elastic-viscoplastic deformation models of salt rocks ……………………………………… 192-209 F. Greco, L. Leonetti, P. Lonetti, P. Nevone Blasi, A. Pascuzzo, G. Porco https://youtu.be/waOuQjaEDsM An interface-based microscopic model for the failure analysis of masonry structures reinforced with timber retrofit solutions ……...................................................................................................... 210-226 V. Tomei, E. Grande, M. Imbimbo https://youtu.be/NrLvsW8YyyU Optimization of the internal structure of 3D-printed components for architectural restoration …... 227-241 N. Motgi, S. Chinchanikar https://youtu.be/_hQ93b2PSqA Tool wear evaluation of self-propelled rotary tool and conventional round tool during turning Inconel 718 ………………………...………………………………………………. 242-256 G. Costanza, I. Porroni, M. E. Tata https://youtu.be/tsqg3UgNqOY Exploring the elastocaloric effect of Shape Memory Alloys for innovative biomedical devices: a review …………………………………………………………………………….... 257-271 O. Naimark, V. Oborin, M. Bannikov https://youtu.be/z1Zy-u1A_Xg Self-similarity of damage-failure transition and the power laws of fatigue crack advance …...….... 272-285 H. A. Mohamed, H. Hassan, M. Zaghlal, M. A. M. Ahmed https://youtu.be/04u-p5V22aI Rubberized reinforced concrete columns under axial and cyclic loading …...….............................. 286-309 V. Dohan, S.-V. Galatanu, L. Marsavina https://youtu.be/FPNsr1Pk3_I Mechanical evaluation of recycled PETG filament for 3D printing …...…................................. 310-321

II

Frattura ed Integrità Strutturale, 70 (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)

Filippo Berto Jianying He

(Università di Roma “Sapienza”, Italy)

(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

(Military University of Technology, Poland)

III

Frattura ed Integrità Strutturale, 70 (2024); International Journal of the Italian Group of Fracture

Stavros Kourkoulis Carlo Mapelli Liviu Marsavina

(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)

Qingchao Li Paolo Lonetti

(Henan Polytechnic University, China)

(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

(Aircraft Research and Design Centre, Hindustan Aeronautics Limited Bangalore, India)

IV

Frattura ed Integrità Strutturale, 70 (2024); International Journal of the Italian Group of Fracture

Vasile Nastasescu Stefano Natali Pavlos Nomikos

(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)

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)

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

Special Issue

New Trends in Structural Health Monitoring

Yashar Eftekhar Azam

(University of New Hampshire, UK)

Parsa Ghannadi

(Iran)

Seyed Sina Kourehli

(Azarbaijan Shahid Madani University, Iran)

V

Frattura ed Integrità Strutturale, 70 (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)

VI

Frattura ed Integrità Strutturale, 70 (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.eu 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)

VII

Frattura ed Integrità Strutturale, 70 (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; 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! Ciao Francesco Iacoviello Frattura ed Integrità Strutturale Editor in Chief  Mechanical Engineering;  Mechanics of Materials.

VIII

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

Predicting damage in notched functionally graded materials plates through extended finite element method based on computational simulations Hakim Siguerdjidjene Materials, Processes and Environment (UR/MPE), Faculty of Technology, University M’hamed Bougara of Boumerdes, City Frantz Fanon, 35000 Boumerdes, Algeria h.siguerdjine@univ-boumerdes.dz, http://orcid.org/0000-0003-1428-6786 Amin Houari Laboratory of Motor Dynamics and Vibroacoustics (LDMV), Department of Mechanical Engineering, M’hamed Bougara University of Boumerdes, Boumerdes, Algeria Department of Mechanical Engineering, LMSS, University of Djillali Liabes, Sidi Bel Abbes, Algeria a.houari@univ-boumerdes.dz, http://orcid.org/0009-0004-2617-2182 Kouider Madani Department of Mechanical Engineering, LMSS, University of Djillali Liabes, Sidi Bel Abbes, Algeria koumad10@yahoo.fr, https://orcid.org/0000-0003-3277-1187 Salah Amroune Mechanical Engineering Department, Faculty of Technology, University Mohamed Boudiaf of M’sila, Algeria salah.amroune@univ-msila.dz, http://orcid.org/0000-0002-9565-1935 Mohamed Mokhtari Department of Mechanical Engineering, RTF, National Polytechnic School of Oran, Algeria mohamed.mokhtari@yahoo.fr, http://orcid.org/0000-0002-2014-1172 Barhm Mohamad Department of Petroleum Technology, Koya Technical Institute, Erbil Polytechnic University, 44001 Erbil, Iraq barhm.mohamad@epu.edu.iq, https://orcid.org/0000-0001-8107-6127 Chellil Ahmed Laboratory of Motor Dynamics and Vibroacoustics (LDMV), Department of Mechanical Engineering, M’hamed Bougara University of Boumerdes, Boumerdes, Algeria a.chellil@univ-boumerdes.dz, https://orcid.org/0000-0001-9467-4214 Abdelkrim Merah Materials, Processes and Environment (UR/MPE), Faculty of Technology, University M’hamed Bougara of Boumerdes, City Frantz Fanon, 35000 Boumerdes, Algeria

LTSE, Faculty of Physics, USTHB, Bab Ezzouar 16111, Algiers, Algeria abdelkrimerah@univ-boumerdes.dz, https://orcid.org/0000-0003-1376-5400

1

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

Raul D.S.G. Campilho CIDEM, ISEP – School of Engineering, Polytechnic of Porto, Porto, Portugal rds@isep.ipp.pt, https://orcid.org/0000-0003-4167-4434

Citation: Siguerdjidjene, H., Houari, A., Madani, K., Amroune, S., Mokhtari, M., Mohamad, B., Ahmed, C., Merah, A., Campilho, R.D.S.C., Predicting damage in notched functionally graded materials plates through extended finite element method based on computational simulations, Frattura ed Integrità Strutturale, 70 (2024) 1-23.

Received: 19.04.2024 Accepted: 30.06.2024 Published: 03.07.2024 Issue: 10.2024

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 . FGM (Functional Graded Materials), USDFLD (User-Defined Field Variables), XFEM (Extend Finite Element Method), Crack growth, Damage Prediction

I NTRODUCTION

D

ue to their many advantages, functionally graded materials (FGMs) of the metal/ceramic type have known a wide use in various disciplines, in particular in high technology applications, justifying why their analysis currently presents a recent and important research axis. Their particularity compared to other materials is the change in their mechanical properties depending on the choice of the designer and without the presence of interfaces. In fact, these materials show a continuous property variation leading to structures with an optimized design in strength and functionality. The development of these materials (FGMs) has pushed their use in different fields such as nuclear reactors, medical and piezoelectric devices, and biological systems [1], due to their structural properties such as heat transfer, electrical conductivity, and others. When a geometric discontinuity arises within the FGM material, the direction of material gradation is conditioned by the size and position of the notch. These geometric discontinuities are the main actor in the strength and mechanical behaviour of these structures. In fact, these constitute the areas of stress concentrations and the areas of crack initiation until total damage to the structure. The stress concentration factor around a notch in a plate is studied by several researchers such as Shen et al. [2]. Recent numerical work by the isoparametric finite element method has been proposed by Kong et al. [3] and Gong et al. [4] to model structures in FGM. Other works, such as Wang et al. [5], have studied the effect of gradation on the stress concentration factor and have shown that this factor is completely different from that of a homogeneous material. Kubair et al. [6] also analysed, by the finite element method, a structure with a circular notch under a uniaxial load. O n the other hand, the evaluation of the stress concentration around rectangular notches has been highlighted in the work of Dave and Sharma [7]. Graded structures according to different shapes of the notch have also been the objective of several researchers’ efforts by the analysis of the stress concentration factor. Recently, Enab et al. [8] analysed the stress concentration in a structure with an elliptical notch while Cardenas-Garcia et al. [9] studied the variation of the radial and tangential thermal stresses and strains around a circular hole in an FGM structure. Yang et al. [10] also determined the stress concentration in

2

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

three dimensions of rectangular holes. Mohammadi et al. [11] used the Frobenius series solution to study the stress concentration factor around a circular notch under uniform biaxial stress. In recent years, the XFEM technique has gained widespread popularity to analyse elastic, plastic, and fatigue crack propagation problems in conjunction with fracture mechanics and damage mechanics (Bansal et al. [12], Singh et al. [13] and Xu and Yuan[14]). In order to simulate the phenomenon of damage in FGM materials, several approaches have been proposed by researchers such as in the work of Asadpoure and Mohammad [15], Khazal et al.[ 16], Ueda [17], and Lee et al. [18], also by numerical methods [19-20] and experimentally [21, 22]. The use of the XFEM technique has many advantages in numerical computation, such as the elimination of remeshing and the introduction of enrichment functions in the structure. In addition, some researchers [23, 24] have applied the concept of FGM to study the mechanical behaviour of nanomaterials using a non-local model or a gradient elasticity model. Recently, Zivelonghi et al. [25] simulated the ductile fracture of an FGM structure based on CDM (continuum damage mechanics) and on its microstructure using the ABAQUS computer code. On the other hand, Gunes et al. [26] analysed the response impact under a reduced speed for FGM Al-SiC. An investigation was conducted to analyse the elastic-plastic behaviour of FGM material shells under various mechanical loadings by Huang et al. [27], Zhang et al. [28], and Jrad et al. [29]. By a gradation in the plane, Amirpour et al. [30] studied the model of damage in elastoplastic materials based on irreversible thermodynamics. The same analysis was carried out by Feulvarch et al. [31], on an FGM structure under buckling loading. Depending on the percentage and direction of gradation relative to the applied load, cracks in FGM materials behave in different ways. The initiation and evolution of a crack in FGM structures require a broad knowledge of the fracture behaviour. Actually, the crack in an FGM structure depends on several intrinsic or extrinsic parameters. All these parameters must be taken into account in the numerical simulation, where the crack trajectory is conditioned by the material gradation direction in FGM modelling. Jin et al. [32] have employed a cohesive zone model to examine the relationship between crack propagation and applied load for a Ti/TiB specimen. Khatri et al. [33, 34] analysed the growth of cracks in a notched isotropic plate using the XFEM technique. The authors examined the effect of the crack parameters and the notch radius. Hirshikesh et al. [35] also studied a field formulation for fracture in FGM materials. Another possible alternative is to locate the crack initiation using a damage-based approach. By these laws, one can model the degradation of the material and their critical zone [36]. Ritchie et al. [37] found that the initiation of a sudden rupture occurs when the value of the principal stress reaches its critical value. The maximum normal stress criterion was introduced by Erdogan and Sih [38]. It is based on the knowledge of the stress field at the crack tip. Another criterion such as the maximum energy restitution rate is proposed by Hussain et al. [39], showing that the crack propagates in the direction where the rate of restitution of the strain energy is maximum. In numerical predictions, the use of the USDFLD subroutine implemented in the ABAQUS code is one of the most efficient and frequently employed methods by many researchers, especially in FGM materials, such as Bouchikhi et al. [40], to determine the integral-J in a 2D structure of FGM type (TiB / Ti) and Mars et al. [41] for the analysis of the elastoplastic behaviour of FGM using user material subroutine (UMAT) and USDFLD. Martinez et al. [42] and Burlayenko et al. [43] investigated the crack propagation behaviour in FGM structures subjected to thermal shock conditions. Amirpour et al. [30] analysed damage in FGM structures with in-plane material properties variation using an elasto-plastic damage model. A novelty of this work is the integration of a power law-based gradation in the FGM structure, where the thickness of the material plays a significant role. This approach allows for a more realistic representation of the FGM properties throughout the structure. Additionally, the combination of the Tamura-Tomota-Ozawa (TTO) model homogenization model for determining plasticity properties and the XFEM technique for crack initiation and propagation analysis provides a comprehensive understanding of the elastoplastic behaviour and damage progression in the FGM plate. The numerical linkage of FGM properties with the model geometry enhances the accuracy of the analysis and facilitates exploration of the effects of parameters such as the notch diameter and volume fraction exponent on crack localization and propagation. Validation of this numerical model reinforces the credibility and reliability of the obtained results. The boundary conditions for the plate in tension are reproduced as follows (Fig. 1b): Embedding of the plate underside: U 1 =U 2 =U 3 =0, UR 1 =UR 2 =UR 3 =0 and uniaxial tensile stress of 250 MPa applied to the other side of the plate cross-section. The choice of this value is sufficient to induce damage in the FGM (Al/SiC) plate. A G EOMETRIC MODEL OF THE FGM PLATE plate with central circular notch in FGM (Al/SiC) was considered, presented in the form of the set of surfaces (Fig.1a), of dimensions 125 mm in length, 25 mm in width, and 2 mm in thickness. The structure presents a central circular notch of variable radius from r=1.66 mm up to 5 mm with a pitch of 0.84 mm. Each surface exhibits mechanical properties as being a homogeneous and isotropic material.

3

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

Tab. 1 provides the properties of the two constituents (Al/SiC) in the FGM material. The finite element model is built with C3D8R elements, which present the best choice for the numerical modelling of the plate and are used in numerous examples of FGM modelling [29].

(a) (b) Figure 1: FGM structure modelled as a plate with a centrally located circular notch (Al/SiC) a) 3D, b) 2D

Property

Metal (Al) 67000 MPa

Ceramic (SiC) 302000 MPa

Young’s Modulus E Poisson’s Ratio ν

0.33

0.17

Yield Stress Y σ

95.1 MPa

-

Ultimate Stress UT σ

160

1400

Fracture energy

Ic G

6.17 kJ/m 2 1000 MPa

0.065 kJ/m 2

Tangent modulus H p

-

Ratio of stress to strain transfer “q” 4800 MPa *

Table 1: Material properties of Aluminum alloy and SiC [44].

Figure 2: Linear distribution of Young's modulus (SDV1 of the unit MPa) through the thickness   h z of the plate with central circular notch in FGM (Al/SiC) via the USDFLD subroutine with power law.

G RADIENT OF FGM PROPERTIES

T

raditionally, the properties of FGM are determined through experimental means. However, in ABAQUS, the graded mechanical properties of the FGM can be defined using a user subroutine, such as user material subroutine UMAT or USDFLD, which are called at the integration points. If a UMAT subroutine is utilized, the constitutive mechanical behaviour of the material must be programmed separately, making it impractical to use the pre-existing material models already available in ABAQUS. Consequently, the material gradient is incorporated by utilizing a user subroutine called USDFLD. This method allows the elastic properties of the material to be defined based on a field variable programmed

4

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

within the USDFLD subroutine. To ensure continuity of forces at the interfaces, a USDFLD subroutine is implemented in ABAQUS to specify the material properties of the FGM based on the coordinates of the integration points in the finite element model. In this particular study, integration points were systematically arranged throughout the thickness direction of the FGM plate, which consists of a metal/ceramic composite. The distribution of material properties across the thickness H   z of the FGM plate, according to a power law distribution, is shown in Fig. 2.     m c m c P z P P .V P    (1) fraction exponent, and the subscripts c and m represent the ceramic and metal constituents, respectively. Most existing studies on FGMs commonly employ the simple mixing rule to obtain effective material properties. Various functions such as power law (P-FGM), sigmoid (S-FGM), or exponential functions (E-FGM) are utilized to determine the volume fraction distribution function and equivalent material properties of FGMs. While the mixture laws are practical and straightforward, these do not provide information regarding the size, shape, and distribution of particles at the microstructural level. In the finite element method, the properties of the material are defined by the integration points through the USDFLD user subroutine. However, this method presents challenges in achieving exact solutions due to the unmanageable distribution of integration points. To address this limitation, this study proposes a technique based on a geometric model in which the distribution of material properties occurs at the surface level, with integration points located on the surfaces. By adopting this approach, the aim is to improve the element's performance in terms of the continuity of material property distribution and stress continuity at interfaces, thereby enabling accurate calculation of resulting stresses. This method seeks to obtain a volume fraction function that closely aligns with experimental observations using the finite element method. The FGM is developed without discontinuity (no layers to avoid residual stresses). So, geometrically the thickness of the plate in functional gradient materials can be formed by an assembly of an infinite number of surfaces (Fig.1a), in which each surface has its own mechanical properties. For this purpose, a surface method is proposed by subdividing the interval H , 2 2 H       of the plate: P represents the effective material property, m V 1 2        n z H  is the metal volume fraction, n is the non-negative volume

H

  i 1 . i e   

(2)

z

2

where z i : coordinate of the surface relative to the global reference, 1, , i m   : position of the surface in the FGM plate, H e N  : distance between two surfaces successively, m = N +1: number of surfaces of the plate, and N : number of finite element layers.

To determine Eqn. (1), it is assumed that the surface is located exactly on the point of integration (Fig. 3). In addition, by the Simpson method, the point ( i =1) is located exactly on the lower surface of the plate, and the type of Gauss integration, the point ( i =1) is located near the lower surface [29]. So, the gradation of the FGM properties followed by the thickness is obtained by the points of the integrations:

e



(3)

h

g

n

g

where g h is the distance between the two successive integration points, and g n is the number of layers between integration points.

5

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

(a)

(b)

(c)

(d) Figure 3: Reference geometry of the FGM and Gauss points. a) Graded surface b) Property variation along coordinate (z), c) homogeneous elements [45], d) The element SOLID-FGM and Gauss points. So, Eqn. 1 becomes:

 e h     g 1

z

(4)

g

g

2

n

z 1

    

  

  g

    

  m

g



 

 

P z

P z P z

P z

(5)

m

1

e

2

where

1 g g m n   represents the total number of integration points within the interval H , . g 1, 1 g m    is the position number of integration point in the interval H , g h must be carefully chosen to minimize the error in the numerical results, especially when the number of integration points is minimal. Where   P z g denotes the effective material property of the FGM. Note that   1 P z and   P z m are respectively the properties of the top and bottom faces of the interval H .

6

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

To obtain an accurate analysis of the FGM structure using C3D8R finite elements of the plate, the integration points through the thickness of the FGM plate are ordered continuously. As a result, multiple layers of elements are necessary in the finite element model to ensure an equal number of integration points throughout the thickness, as illustrated in Fig. 4. In the elastoplastic behaviour of an FGM with a metal matrix reinforced with ceramic particles, isotropy and homogeneity are assumed for each surface. For this reason, the TTO model is used to determine the plastic properties of the FGM, which will be entered directly into ABAQUS.

A: 21 Layers

A:14 Layers

A:07 Layers

A

Figure 4: Description of the mesh density for an FGM plate according to thickness.

The TTO model is a metal alloy homogenization method and is used to evaluate locally effective elastoplastic parameters of the FGM (Al/SiC) compound. In the TTO model, the material mixture is treated as elastoplastic with linear isotropic hardening, for which the stresses and strains are related to the constitutive forces , m c   , , m  c  [32, 46] by the relation:

σ σ V 



 

σ V

and ε ε V

ε V

(6)

m m c c

m m c c

In the TTO model, an additional parameter q is introduced, which represents the ratio of stress to strain transfer:



  m c σ σ / ε ε  c



  

 

(7)

 

, 0 q

q



m

The TTO model utilizes the elastoplastic properties of the metal constituent, which are determined by the modulus of elasticity E(z) of the FGM, the initial yield stress   0 Y σ z , and the tangent modulus   H z p . These properties are described by the following relationships:

  

   

  

q E q E  

q E q E  









 

c

c



m m c .E V E . 1 V /     m

.V 1 V  

(8)

E z

m

m

m

m

  

   

  

q E q H  

q E q H  









 

c

c



m m c .H V E . 1 V /     m

.V 1 V  

(9)

H z

p

m

m

m

m

 

  

q E E q E E 

  0 z

.(1 ) m c







 

(10)

V V

Y

0 Y m

m

m

 

c

m

m  

is the initial yield stress of metal,

m H is the tangent modulus of metal,

and

where

m E

σ

σ

/

0 Y m

0 Ym

0

 

are the initial yield strains of the metal and FGM Al/SiC. The Poisson’s ratio   ν z of the FGM

  z

 



 

Y z

/E z

Y

0

0

just follows a rule of mixtures in the TTO model:

7

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

 





(11)

ν z

ν V

ν V

m m c c

XFEM IMPLANTATION FORMULATION FOR ELASTOPLASTIC ANALYSIS IN FGM

A

n analytical and numerical formulation was developed for the prediction of damage and crack propagation in FGM plates, especially in elastic-plastic behaviour [47]. The choice of the XFEM technique used in the calculations is based on several parameters, such as the damage criterion used, the gradation of properties in the structure (FGM), and crack propagation. Governing equations Let Ω be a three-dimensional domain of a continuous medium in two distinct parts, with boundary Γ , consisting of Γ t and Γ , as shown in Fig.5. The boundary condition of displacement is imposed on Γ u , the traction is applied on Γ t and, given the presence of a cracked surface, the elastoplastic equilibrium equations with boundary conditions are written as follows [48].

(12)

σ b 0 in Ω   

in Γ t t 

σ .n

(13)

0 in Γ c 

(14)

σ .n

where n is the unit outward normal,  is the Cauchy stress tensor, b is the body forces per unit volume, and t ̅ represents the traction vector. In the present investigation, small strains and displacements were considered. The kinematics equations therefore consist of the strain–displacement relation:   u ε ε u u and u u on Γ s    (15) where ∇ s is the symmetric part of the gradient operator, u is the displacement field vector.

t

Γ ௧ Γ ௖௥ Ω

z

Γ ௨

x

y

Figure 5: Schematic representation of homogeneous cracked body

Elastoplastic formulation In the context of an FGM material consisting of a metal matrix reinforced by ceramic particles, it is assumed that each surface is both isotropic and homogeneous when exhibiting elastic-plastic behaviour. To model the elastoplastic behaviour of the FGM, the von Mises criterion and an isotropic hardening variable are employed. The rate expression for the stress strain relationship of the elastoplastic material can be represented by the following equation:   e p d σ Cd ε C d ε d ε    (16)

8

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

where C is elastoplastic constitutive matrix of FGM. The change in strain is assumed to be divisible into elastic ( e d ε ) and plastic components ( p d ε ) for any increment of stress such that: (17) The strain tensor corresponds to the symmetric part of displacement gradient for an isoparametric element: e p d ε d ε d ε   where B is the strain-displacement matrix of shape function derivatives. The elastic component of increment strain can be written as:   1 e d ε C d σ   (19) where C is the constitutive matrix for three-dimensional linear elastic FGM. For the von Mises yield criteria, the yield surface can be described by:   eq r , σ σ 0 f R     (20) ε Bu sym   u (18)

3 2



 ,



f R  : is the yield function, and eq σ

where

: is the von Mises equivalent stress of FGM, where S is the

S.S

deviatoric part of the Cauchy stress tensor:

1

 

S

ij σ σ δ 3 m

(21)

ij

ij

where m σ is the mean stress, δ ij is the Kronecker delta, and r  is the radius of the yield surface, defined by:

 

   

(22)

R p

r

Y

0

where Y0 σ is the initial yield stress of the FGM, and   R p H .p P  is the isotropic hardening (p is the internal variable corresponding to isotropic hardening and H P is the hardening module). The rule of normality makes it possible to establish the following complementary relations:



f 3 S σ 2 σ

p



  

(23)

d ε

d λ .N

and N

eq

where N: is the gradient of the yield function with respect to the stress tensor, and p d ε : is the plastic strain rate. If the scalar (p), which is the accumulated plastic strain, is defined as the integration of the rate of the accumulated plastic strain during an iterative procedure .

0 t d d    p

 

(24)

p t

where τ is an integration operator, and the rate itself is defined as:

2 3

p p

p d 

(25)

d ε .d ε

9

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

The scalar d λ is the plastic multiplier, which is equal to the rate of the accumulated equivalent plastic strain. Given that p d ε represents the plastic strain rate, it can be demonstrated using Eqn. 23 and the von Mises equivalent stress eq σ , that the plastic multiplier is equivalent to the rate of the accumulated plastic strain:

   

   

   

2 3d λ S 3d λ S

3 S S

2





dp

d λ

(26)

3 2 σ

2 σ

2

σ σ

eq

eq

eq eq

dp d λ 

(27)

The plastic multiplier d λ in Eq. 22 is determined by using the consistency condition, to obtain the following expression:

 

 

 

   



f

f

f

f R R p

R



dp d σ   

dp 0 , and 

R p  

(28)

d σ



σ

p

σ

p

Eq. 26 can be written as:



dR d

eq

Nd σ H d λ 0 and H  

 

(29)

P

P

p eq

dp d ε

Using Eq. 17 and Eq. 19 in Eq. 22 leads to:



f

  1 C 



d σ d λ 

d ε

(30)





C T N :

After pre-multiplying both sides of Eq. 30 by

T T N C d ε N d σ N C d λ N   T

(31)

T

T N C d λ N

(32)

eq N C d ε H .d ε P 



The following expression is obtained for the plastic multiplier:

N:C.d ε

p eq  

(33)

d λ d ε

N:C:N H 

P

p d ε d λ .N  into Eq. 17, the elastoplastic tangent modulus is derived

Finally, by substituting the expression of the plastic part

as:



  N:C N:C N:C:N H  



ep C C

 

(34)

P

At last, the updated stresses ( σ ) at the conclusion of the time step ( ∆ t) can be expressed as follows:

n 1 n σ σ d σ   

(35)

The subscript (n+1) indicates the values corresponding to the end of the time increment.

10

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

3D XFEM formulation for FGM The enrichment functions in the XFEM technique require a high number of integration points to obtain satisfactory results. The XFEM technique is implemented in the ABAQUS code [49] and is not compatible with user-defined finite elements in graded materials (FGM). Indeed, for the XFEM technique to be implemented by the user it is necessary to use a USDFLD subroutine in the ABAQUS code to simulate the force equilibrium with linear finite elements in three dimensions. The weak form of the governing equation for a solid mechanics problem can be expressed as:

.

.

.

  u C u d     : :







(36)





b ud

t ud

Ω

. Ω

. Γ

Ω

Ω

Γ

The above equation can be written in the form of the finite element procedure, as [12]:

.

.

.







T B d

T N bd

T N t d

(37)

Ω  



Ω

. Ω

i

i

Ω

Ω

Γ

By substituting the trial and test functions from Eqn. (36) into Eqn. (37) and leveraging the flexibility of nodal variations, the ensuing discrete system of linear equations is attained:     e f u e h k      (38) The element stiffness matrix, denoted as e k , and the external load vector, denoted as e f , are provided as follows:   e 1 2 3 4 f f f f f f f u a b b b b i i i i i i  (39)     e . Ω k B C B d Ω , , , T e ij i j k and u a b                  (40)

.

.





u

(41)



Ω Γ N td 

N bd

f

i

i

i

Ω

Γ









.

.

 

 

  H x H x 

  H x H x 





a

(42)





N

bd

N

td

f

Ω

Γ

i

i

f

f

i

i

f

f

i

Ω

Γ

    j N F x bd Ω i

    j N F x td Γ i

.

.





b α

( α 1 4)  

(43)





f

i

Ω

Γ

 and

i N are standard shape functions, and B i

In which t is the external force, b is the body force, C is the elasticity matrix,

B j  are the matrix of shape function derivatives. In numerical predictions of the three-dimensional crack behavior by the XFEM technique, the elastoplastic behavior is given by Eqn. (44), and the displacement field variable u(x) for a domain that includes a crack is represented as follows [50]:

  

   

4

  N x u   i i 

i  

 

    N x H x a

  N x F x b   

u x h

j i

(44)





f

i

i

j

i M 

i N 

i N 

j 1 

d

p

In the given context, M represents the set of nodes within the mesh, and i u represents the classical degree of freedom at node i.   i N x refers to the classical finite element shape functions associated with node i. Additionally, a and b are supplementary degrees of freedom introduced to the common degrees of freedom.   H x f denotes the Heaviside function, and   j F x represents a specific set of four tip enrichment functions, as outlined in reference [48]:

11