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7th International Conference on Structural Integrity and Durability (ICSID 2023)

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Procedia Structural Integrity 58 (2024) 1–2

© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers c Technical University of Kosice, Faculty of Manufacturing Technologies, Sturova 31, 080 01 Presov, Slovakia d ELKEME - Hellenic Research Centre for Metals S.A., Athens – Lamia National Road , 32011 Oinofyta Viotias, Greece e University of Bergamo, Department of Management, Information and Production Engineering, Viale Marconi 5, Dalmine 24044, Italy f Università di Cassino e del Lazio Meridionale, via G. DI Biasio 43, 03043, Cassino (FR), Italy © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Keywords: Preface; fatigue; fracture; structural integrity 1. Preface The 7 th International Conference on Structural Integrity and Durability, ICSID 2023, was organized by the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, and the Croatian National Group of the European Structural Integrity Society (ESIS), in Dubrovnik in Croatia from September 19 to 22, 2023. ICSID 2023 was organized as the Hybrid Conference i.e, a combination of on-site and online participation to provide convenience for participants. The Conference was held in the Centre for Advanced Academic Studies (CAAS) of the University of Zagreb, in the city of Dubrovnik. The magnificent old building of CAAS is situated in the center of Dubrovnik on the Croatian Adriatic Coast, in the vicinity of the most prominent historical places of the Old Town (https://icsid2023.fsb.hr/). Online presentations were held using a commercial meeting platform. In total 52 on-site and online presentations were given, including five plenary lectures. 7th International Conference on Structural Integrity and Durability (ICSID 2023) Preface Željko Božić a, *, Siegfried Schmauder b , Katarina Monkova c , George Pantazopoulos d , Sergio Baragetti e , Francesco Iacoviello f a University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lu č i ć a 5, 10000 Zagreb, Croatia b University of Stuttgart, Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), Pfaffenwaldring 32, Stuttgart, Germany c Technical University of Kosice, Faculty of Manufacturing Technologies, Sturova 31, 080 01 Presov, Slovakia d ELKEME - Hellenic Research Centre for Metals S.A., Athens – Lamia National Road , 32011 Oinofyta Viotias, Greece e University of Bergamo, Department of Management, Information and Production Engineering, Viale Marconi 5, Dalmine 24044, Italy f Università di Cassino e del Lazio Meridionale, via G. DI Biasio 43, 03043, Cassino (FR), Italy © 2024 The Authors. Published by ELSEVIER B.V. Keywords: Preface; fatigue; fracture; structural integrity 1. Preface The 7 th International Conference on Structural Integrity and Durability, ICSID 2023, was organized by the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, and the Croatian National Group of the European Structural Integrity Society (ESIS), in Dubrovnik in Croatia from September 19 to 22, 2023. ICSID 2023 was organized as the Hybrid Conference i.e, a combination of on-site and online participation to provide convenience for participants. The Conference was held in the Centre for Advanced Academic Studies (CAAS) of the University of Zagreb, in the city of Dubrovnik. The magnificent old building of CAAS is situated in the center of Dubrovnik on the Croatian Adriatic Coast, in the vicinity of the most prominent historical places of the Old Town (https://icsid2023.fsb.hr/). Online presentations were held using a commercial meeting platform. In total 52 on-site and online presentations were given, including five plenary lectures. 7th International Conference on Structural Integrity and Durability (ICSID 2023) Preface Željko Božić a, *, Siegfried Schmauder b , Katarina Monkova c , George Pantazopoulos d , Sergio Baragetti e , Francesco Iacoviello f a University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lu č i ć a 5, 10000 Zagreb, Croatia b University of Stuttgart, Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), Pfaffenwaldring 32, Stuttgart, Germany

* Corresponding author. Tel.: +385 1 6168 536; fax: +385 1 6156 940. E-mail address: zeljko.bozic@fsb.unizg.hr * Corresponding author. Tel.: +385 1 6168 536; fax: +385 1 6156 940. E-mail address: zeljko.bozic@fsb.unizg.hr

2452-3216 © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers 2452-3216 © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers

Željko Božić et al. / Procedia Structural Integrity 58 (2024) 1– 2

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Ž. Boži ć et al. / Structural Integrity Procedia 00 (2019) 000–000

The objective of the ICSID 2023 Conference was to bring together scientists, researchers, and engineers from around the world to discuss how to analyze, predict and assess the fatigue and fracture of structural materials and components. The Conference provided a forum for discussion of contemporary and future trends in experimental, analytical, and numerical fracture mechanics, fatigue, failure analysis, structural integrity assessment, and other important issues in the field. A wide range of topics was covered such as: Failure investigation and analysis; Structural integrity assessment; Analytical Models; Advanced testing and evaluation techniques; Applications to components and structures; Non destructive evaluation (NDE); Fatigue and fracture simulation and testing at all length scales; Fracture and failure criteria; Multiscale materials modeling; Mixed-mode and multiaxial fatigue and fracture; Durability and life extension of structures and components; Structural integrity of 3D-printed structures; Models, criteria and methods in fracture mechanics; Finite element methods and their applications; Effect of residual stresses; Fatigue and fracture of weldments, welded components, joints and adhesives; Corrosion, environmentally enhanced degradation and cracking, corrosion fatigue; Fracture and damage of cementitious materials; Fatigue and fracture of polymers, elastomers, composites and biomaterials; and others. Prior to the ICSID 2023 Conference a two-day Summer School with the topic “Fatigue and fracture modelling and analysis” was organized for graduate students, researchers, and engineers from industry. Those participants who came to Dubrovnik, besides the excellent technical program, had the opportunity to enjoy their stay in Dubrovnik, one of the most famous Mediterranean cities, world celebrated symbol of historical heritage and beauty, which has found its place in the UNESCO World Heritage List. As the Guest Editors of this Conference Proceedings, we wish to thank all authors for their contributions. Guest Editors of the Procedia Structural Integrity ICSID 2023 Conference Proceedings: Željko Božić, University of Zagreb, Croatia Siegfried Schmauder, University of Stuttgart, Germany Katarina Monkova, Technical University in Kosice, Slovakia George Pantazopoulos, ELKEME - Hellenic Research Centre for Metals S.A., Athens, Greece Sergio Baragetti, University of Bergamo, Italy Francesco Iacoviello, Università di Cassino e del Lazio Meridionale – DICeM, Italy

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2023) 000–000

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Procedia Structural Integrity 58 (2024) 137–143

© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Abstract This paper deals with the problem of analyzing the energy dissipation in a non-linear viscoelastic beam structural component made of a functionally graded material. The beam is in skew bending. The viscoelastic model used for dealing with the beam mechanical behaviour is set-up by combining of linear as well as non-linear springs and dashpots. The viscoelastic model is under strains which change in time on sinusoidal law. The material properties of the viscoelastic model are distributed smoothly along the thickness of the beam structure. The energy dissipation in the beam is due to dashpots which are parts of the viscoelastic model. Therefore, the strain energy dissipation density is derived by analyzing the stresses and strains in the dashpots. Then the strain energy dissipation density is integrated in the volume of the beam in order to obtain solution of the dissipated energy. The solution is applied for investigating the change of the dissipated energy due to variation of the properties of the model and the loading conditions of the beam. © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Keywords: Skew bending; dissipation; viscoelastic beam; inhomogeneity; material non-linearity 1. Introduction In contrast to conventional structural materials like metals and fibre reinforced composites, functionally graded engineering materials are special composites which have a graded microstructure resulting in continuous material inhomogeneity (Chen and Lin (2008), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Rizov (2020)). Thus, the properties of structural components made of functionally graded materials are smooth functions of one or more 7th International Conference on Structural Integrity and Durability (ICSID 2023) Analysis of energy dissipation in non-linear viscoelastic beam in skew bending Victor Rizov* Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., 1046 – Sofia, Bulgaria

* Corresponding author. Tel.: + (359-2) 963 52 45 / 664; fax: + (359-2) 86 56 863 E-mail address: v_rizov_fhe@uacg.bg

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

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coordinates (Chikh (2019), Han et al. (2001), Kou et al. (2012), Rizov and Altenbach (2020)). The functionally graded materials are preferred in many applications in different areas of modern engineering (Kieback et al. (2003), Mahamood and Akinlabi (2017), Yan et al. (2020), Hao et al. (2002)). This is because of their superior properties in comparison to the conventional materials. The wide impingement of functionally graded materials in the engineering practice in the recent decades is closely related to analyzing of various aspects of the mechanical behaviour and performance of structural components under different loadings. One of the important problems is the energy dissipation in structural components made of functionally graded materials with non-linear viscoelastic mechanical behaviour. Energy dissipation has bearing on issues like integrity, reliability and life of engineering structures. The present theoretical paper deals with analyzing of the energy dissipation in a beam structural component made of a functionally graded material. There are two novelties in the present paper: (i) the beam exhibits non-linear viscoelastic behaviour and (ii) the beam is under time-dependent skew bending. It should be underlined here that previous energy dissipation studies deal usually with linear viscoelastic beam structures subjected to bending around the horizontal axis of the cross-section (Narisawa (1987), Rizov (2021)). In the present paper, the non-linear viscoelastic behaviour of the beam is treated by a model representing a combination of linear as well as non-linear springs and dashpots. A solution of the energy dissipation problem is derived. The influence of the distribution of material properties, the beam size and the parameters of the loading on the dissipated energy is clarified. 2. Analytical model Fig. 1 shows the static schema of a beam structural component of rectangular cross-section.

Fig. 1. Static schema of beam loaded in skew bending. The beam is loaded in skew bending so that the beam free end angle of rotation,  , presented as a vector in Fig. 1 is inclined under angle,  , with respect to the horizontal centric axis, y , of the beam. The variation of  with time,

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

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t , is represented by formula (1), i.e.   t    sin 0  ,

(1)

where 0  and  are parameters.

Fig. 2. Schema of viscoelastic model.

The non-linear viscoelastic mechanical behaviour of the beam under consideration is taken to be represented by the theoretical viscoelastic model shown in Fig. 2. The model combines two linear springs with moduli of elasticity, 1 E an 2 E , one linear dashpot with coefficient of viscosity,  , as well as non-linear spring and dashpot marked with ( ) nls and ( ) nld , respectively (Fig. 2). The stress-strain relation of the non-linear spring is given by







(2)

nls

B D 



where nls  is the stress,  is the strain, B and D are material properties. The constitutive law of the non-linear dashpot is taken to be represented by the following expression:

  P Q 





(3)

nld





where nld  is the stress, P and Q are material properties. The beam under consideration is functionally graded along its thickness, i.e. in  z direction. Thus, the continuous distribution of the properties of the viscoelastic model along the beam thickness is governed by exponential laws

h 2



1 

g E E e  1 1

(4)

h 2



2 

g E E e  2 2

(5)

h 2



g e 3   



(6)

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

140

h 2



4 

g B B e 

(7)

h 2



5 

g D D e 

(8)

h 2



6 

g P P e 

(9)

h 2



7 

g Q Q e 

(10)

where

z h    .

2 h

(11)

In formulas (4) – (11),    are parameters, h is the beam thickness. The viscoelastic model in Fig. 2 is under strain,  , variation of which with respect to time is expressed by   t    sin 0  , (12) where 0  is a parameter. The stress,  , in the viscoelastic model is determined by 7 1 2 , ,..., Since the beam analyzed here has a high length to thickness ratio, the distribution of strains in the beam cross section is written in the form z y z y C        , (14) where C  is the strain in the centre, y  and z  are the beam curvatures in xy and xz planes, respectively. We use the following approach for deriving C  , y  and z  . First, by applying the integrals of Maxwell-Mohr for expressing the free beam end angle of rotation and projecting it on y and z , we get nld nls    2 E     . (13)

cos

 





(15)

sin

 

z 



(16)

We use equilibrium equation (18) for obtaining C  .

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

141

A    ( )

(17)

where N is the axial force (here,

0  N ), A is the cross-section area.

The unit dissipation energy, 0 u , is given by



 dt

  0

 

     ,

(18)

nld

 

  and   are the stress and strain in the linear dashpot, respectively. For the dissipated energy, U , in the beam we get

where

U u dV V 0 ( )   ,

(19)

where V is the volume of the beam (the integral in (19) is solved by the MatLab software). 3. Numerical results The numerical results are derived with purpose to investigate the change of the dissipated energy due to variation of the properties of the model, the loading conditions and the beam geometry.

0.2

0.5

0.8

3  

1  (curve 1 – at

Fig. 3. Dissipated energy versus parameter,

, curve 2 – at

and curve 3 – at

/200  

0.400  l m,

0.006  b m,

0.010  h m,



When deriving the numerical results, we assume that

and 0.004   . The results are graphically presented below (refer to Fig. 3, Fig. 4 and Fig. 5). First, we investigate how the dissipated energy in the beam is influenced by parameters, 1  and 3  . The variation of the dissipated energy is shown in Fig. 3. The rise of parameter, 1  , induces a gradual growth of the dissipated energy (Fig. 3). The influence of parameter, 3  , over the dissipated energy is similar to this of 1  as indicated by Fig. 3. The curves shown in Fig. 4 illustrate the influence which parameter, 4  , has over the dissipated energy for three

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

142 6

l h / ratios. Fig. 4 indicates a reduction of the dissipated energy with rise of 4  . It can also be seen in Fig. 4 that the dissipated energy increases rapidly when l h / ratio grows.

Fig. 4. Dissipated energy versus parameter, 50  l h ). The variations which the dissipated energy undergoes when the angle  and the parameter 0  change are also studied. 4  (curve 1 – at / 30  l h , curve 2 – at / 40  l h and curve 3 – at /

/300  

/200  

Fig. 5. Dissipated energy versus angle,  (curve 1 – at



, curve 2 – at

and curve 3 – at

/100  



These variations are visualized in Fig. 5 where the dissipated energy is plotted versus  at three values of 0  . One can see in Fig. 5 that the dissipated energy grows as result of increase of  and 0  . 4. Conclusions The energy dissipation in a non-linear viscoelastic functionally graded beam structural component loaded in skew bending is studied theoretically. It is detected that:  the rise of parameter 1  induces a gradual growth of the dissipated energy;  the influence which the parameter 3  has over the dissipated energy is similar to this of 1  ;

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

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 the dissipated energy reduces with rise of 4  ;  the dissipated energy increases rapidly when l h / ratio grows;  increase of  and 0  results in growth of the dissipated energy. References

Chen, Y., Lin, X., 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Computational Materials Science 44, 581-581. Chikh, A., 2019. Investigations in static response and free vibration of a functionally graded beam resting on elastic foundations. Frattura ed Integrità Strutturale 14, 115-126. Han, X., Liu, G.R., Lam, K.Y., 2001. Transient waves in plates of functionally graded materials, International Journal for Numerical Methods in Engineering 52, 851-865. Hao, Y.X., Chen, L.H., Zhang, W., Lei, J.G., 2002. Nonlinear oscillations and chaos of functionally graded materials plate. Journal of Sound and vibration 312, 862-892. Kieback, B., Neubrand, A., Riedel, H., 2003. Processing techniques for functionally graded materials. Materials Science and Engineering: A 362, 81-106. Kou, X.Y., Parks, G.T., Tan, S.T., 2012. Optimal design of functionally graded materials, using a procedural model and particle swarm optimization. Computer Aided Design 44, 300-310. Mahamood, R.M., Akinlabi, E.T., 2017. Functionally Graded Materials. Springer. Narisawa, I., 1987. Strength of Polymer Materials. Chemistry. Rizov, V.I, Altenbach, H., 2020. Longitudinal fracture analysis of inhomogeneous beams with continuously varying sizes of the cross-section along the beam length, Frattura ed Integrità Strutturale 53, 38-50. Rizov, V.I., 2020. Analysis of Two Lengthwise Cracks in a Viscoelastic Inhomogeneous Beam Structure. Engineering Transactions 68, 397–415. Rizov, V.I., 2021. Energy Dissipation in Viscoelastic Multilayered Inhomogeneous Beam Structures: An Analytical Study. Materials Science Forum 1046, 39-44. Saidi, H., Sahla, M., 2019. Vibration analysis of functionally graded plates with porosity composed of a mixture of Aluminum (Al) and Alumina (Al2O3) embedded in an elastic medium. Frattura ed Integrità Strutturale 13, 286-299. Saiyathibrahim, A., Subramaniyan, R., Dhanapl, P., 2016. Centrefugally cast functionally graded materials – review. International Conference on Systems, Science, Control, Communications, Engineering and Technology, 68-73. Li, Y., Feng, Z., Hao, L., Huang, L., Xin, C., Wang, Y., Bilotti, E., Essa, K., Zhang, H.m Li, Z., Yan, F., Peijs, T., 2020. A review on functionally graded materials and structures via additive manufacturing: from multi-scale design to versatile functional properties. Advanced Materials Technologies 5, 1900981.

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Procedia Structural Integrity 58 (2024) 115–121

© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Abstract Fracture analysis of functionally graded plates is a challenging problem in materials engineering due to the complex material behavior and the presence of cracks. In this study, we propose an efficient open-source MOOSE implementation of phase field fracture analysis of functionally graded. Automatically oriented exponential finite elements are used to discretize the model, which allows for efficient and accurate computation of the fracture behavior. The functionally graded material properties are modeled using a power law function, which captures the variation of material properties along the thickness direction of the plate. The phase field model is coupled with a finite element solver to simulate the mechanical response of the plate under loading. The proposed implementation is validated using several benchmark problems available in literature. The results show that the proposed MOOSE implementation provides accurate predictions of the crack propagation and fracture behavior of functionally graded plates. © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Keywords: Crack; Phase-field; MOOSE; Functionally graded materials; Fracture 1. Introduction Functionally graded materials (FGMs) have gained significant attention in recent years for their versatile applications in aerospace, energy, and biomedical engineering, thanks to their unique composition and microstructure variations across their volume. While FGMs exhibit exceptional mechanical and thermal properties, the challenge of understanding and predicting fracture behavior within these intricate structures remains a critical concern. 7th International Conference on Structural Integrity and Durability (ICSID 2023) An open-source moose implementation of phase-field modeling of fracture in functionally graded materials P.C. Sidharth a , B.N. Rao a, * a Indian Institute of Technology Madras, Chennai, Tamilnadu, 600036, India Abstract Fracture analysis of functionally graded plates is a challenging problem in materials engineering due to the complex material behavior and the presence of cracks. In this study, we propose an efficient open-source MOOSE implementation of phase field fracture analysis of functionally graded. Automatically oriented exponential finite elements are used to discretize the model, which allows for efficient and accurate computation of the fracture behavior. The functionally graded material properties are modeled using a power law function, which captures the variation of material properties along the thickness direction of the plate. The phase field model is coupled with a finite element solver to simulate the mechanical response of the plate under loading. The proposed implementation is validated using several benchmark problems available in literature. The results show that the proposed MOOSE implementation provides accurate predictions of the crack propagation and fracture behavior of functionally graded plates. © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Keywords: Crack; Phase-field; MOOSE; Functionally graded materials; Fracture 1. Introduction Functionally graded materials (FGMs) have gained significant attention in recent years for their versatile applications in aerospace, energy, and biomedical engineering, thanks to their unique composition and microstructure variations across their volume. While FGMs exhibit exceptional mechanical and thermal properties, the challenge of understanding and predicting fracture behavior within these intricate structures remains a critical concern. 7th International Conference on Structural Integrity and Durability (ICSID 2023) An open-source moose implementation of phase-field modeling of fracture in functionally graded materials P.C. Sidharth a , B.N. Rao a, * a Indian Institute of Technology Madras, Chennai, Tamilnadu, 600036, India

* Corresponding author. Tel.: +91-044-22574285 E-mail address: bnrao@iitm.ac.in * Corresponding author. Tel.: +91-044-22574285 E-mail address: bnrao@iitm.ac.in

P.C. Sidharth et al. / Procedia Structural Integrity 58 (2024) 115–121 P.C. Sidharth and B.N. Rao/ Structural Integrity Procedia 00 (2019) 000–000

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Fracture is a fundamental consideration in the performance and reliability of engineering materials, and FGMs are no exception. However, traditional fracture mechanics approaches, suitable for homogeneous materials, encounter limitations when dealing with the spatially varying material properties inherent in FGMs. Consequently, there is a growing need for innovative computational techniques capable of effectively capturing the complex interplay between microstructure, material gradients, and fracture processes within these materials. In this regard, the present paper explores the utilization of exponential finite element shape functions as a key element in phase-field modeling for the simulation of fracture in functionally graded materials. This approach offers a robust and versatile framework to address intricate fracture phenomena while seamlessly accommodating the heterogeneous nature of FGMs. By representing the fracture process as a diffuse interface within a continuous field variable and leveraging exponential finite element shape functions, we can gain valuable insights into critical aspects such as crack initiation, propagation, and branching within FGMs. Through an in-depth examination of the fundamental principles behind phase-field modeling with exponential finite element shape functions, we emphasize the benefits and unique capabilities of this approach in the context of FGMs. Drawing upon recent research and practical case studies, we demonstrate how this methodology contributes to a more profound understanding of fracture mechanisms in FGMs and empowers the optimization of these advanced materials across various engineering applications. By leveraging exponential finite element shape functions within the phase-field modeling framework, our paper seeks to bridge the gap between computational mechanics and the intricate material properties of FGMs. This novel approach not only promises to advance the understanding of fracture behavior in functionally graded materials but also facilitates their design and engineering for enhanced structural integrity and reliability. 1.1. Functionally graded materials In our current study, we have focused on two-dimensional functionally graded material (FGM) plates, where the properties vary along the length of the plate. However, it is important to note that in typical applications like thermal barriers and protective coatings, the property gradient occurs along the thickness direction of the plate, which we have not considered in this study. This aspect is reserved for future research and exploration. For our analysis, we have employed the Voigt rule of mixtures to homogenize the graded properties of the FGM plate at material points. Consequently, the spatial distribution of these varying material properties can be expressed in terms of the spatial coordinates as follows: (1) In this context, the symbol "P" represents a specific material property whose variation is mathematically described by equation 1. This property can include parameters such as Young's modulus (E), Poisson's ratio (ν), or critical energy release rate (G c ). The term "V x/y " refers to the material volume fraction of one of the two constituents that make up the functionally graded (FG) material. The subscripts "x" and "y" are indicative of the chosen direction of gradation. To further clarify, we define "p" as the volume fraction exponent or the material power law index, while "l x " and "l y " represent the dimensions of the plate in the x and y directions, respectively. The expressions for volume fractions along the x and y coordinates can be expressed as follows: / ( ) 1 ( 2 1) x y P x P P P V   

  

  

  

  

1 2

(2)

 

  

1.2. Phase field modeling of fracture Within phase-field models, the depiction of cracks relies on the utilization of a phase-field parameter. This parameter, denoted by a variable spanning from 0 to 1, serves as a representation of the transition between uncracked and cracked material phases. Consequently, the system's total energy can be conceptualized as a summation of the strain energy contained within the material's bulk and the energy dissipation stemming from the

P.C. Sidharth et al. / Procedia Structural Integrity 58 (2024) 115–121 P.C. Sidharth and B.N. Rao / Structural Integrity Procedia 00 (2019) 000–000 3 existence of the crack. In this context, the symbol  denotes the phase-field order parameter, while Gc signifies the critical fracture energy release rate specific to the material. The function  characterizes the crack surface and is influenced by both the phase-field parameter  and its spatial fluctuations. The balance equations of the phase-field model are derived through the application of the Gauss divergence theorem, as follows. 117

 

  

 2 1      

   H  ε

   σ b

c G l 





(3)

In this particular model, a scalar auxiliary field denoted as  , ranging from 0 to 1, serves as the damage indicator. In this framework, a  value of zero signifies undamaged or intact material, whereas a  value of one corresponds to completely damaged material. To establish boundary conditions, Dirichlet-type conditions are utilized to prescribe displacements in the undamaged region, while Neumann-type conditions are employed to prescribe tractions in the damaged region. In this study, the spectral decomposition of strain energy into tensile and compressive parts proposed by Miehe et al. (2010) is used, along with isotropic constitutive equations, following the hybrid implementation of Ambati et al. (2015). 2. Methodology To solve phase-field fracture problems using the finite element method, the discretization strategy is to divide the domain into 4-noded quadrilateral elements. Weak forms for equations are derived, leading to the creation of residual and stiffness matrices. The displacement field is represented using standard linear finite element (LFE) shape functions, while exponential finite element (EFE) shape functions are used to approximate the phase-field parameter. The resulting system of nonlinear equations is solved iteratively through the Newton-Raphson method. The coupled equations of the phase-field model are then implemented and solved within MOOSE, which stands for Multiphysics Object-Oriented Simulation Environment. The Multiphysics Object-Oriented Simulation Environment, developed by Idaho National Laboratories (INL) in the USA, is a cutting-edge open-source, parallel finite element framework. This powerful simulation environment is equipped with the Jacobian-Free Newton-Krylov (JFNK) solver, which enhances computational efficiency and accuracy. The software allows a wide variety of physics modules within this framework, making it versatile for a multitude of applications. For specialized needs, MOOSE sports dedicated modules for tackling complex phenomena, including fracture analysis, phase field modeling, and XFEM simulations. This approach follows a staggered scheme where displacement and phase degrees of freedom are alternately calculated in subsequent load increments. This methodology allows for the efficient resolution of phase-field fracture problems. 2.1. Exponential finite element shape functions The motivation behind the construction of exponential finite element (EFE) shape functions (Kuhn et al. 2010) arises from the exponential nature of the analytical solution. To accurately capture this exponential behavior, the authors suggest the use of EFE shape functions instead of the conventional standard linear finite element (LFE) shape functions. A visualization of EFE shape functions is shown in Fig. 1.

(1 ') 

 

  

  

  

  

exp



(4)

( ', ) 1  

( ', )  

 



2       

exp

 

The exponential finite element (EFE) shape functions described in equation (4) are defined by a scalar parameter denoted as  . This parameter signifies the ratio of the element edge length, h , to the length scale parameter, l .

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Therefore,  , represented as / h l   , serves as an indicator of the mesh density within the fracture zone. These shape functions possess the unique characteristic of adaptability to both the mesh size and the regularization length. a b

Fig. 1. (a) 1D analytical solution of a stationary crack; (b) Plot of EFE shape functions for two 1D elements having common node at x=0.

This adaptability arises from their dependence on both the element size, h, and the regularization parameter, l . For maintaining accuracy, it is imperative to correctly align the exponential finite element (EFE) shape functions with respect to the nodal values of the phase-field variable within all elements. In contrast, linear finite element (LFE) shape functions are symmetric and do not rely on the phase-field variable  . However, this limitation can be overcome by aligning the exponential finite element (EFE) shape functions with the nodal values of  , leading to

enhanced accuracy in simulations. 2.2. Finite element implementation

MOOSE is specifically designed to be programmed in Linux environments, offering a robust and stable platform for computational tasks. The input file is structured to house essential finite element mesh data, including nodal coordinates and element connectivity information. The heart of the executable is composed of C++ objects and classes, allowing for modular and object-oriented programming, which enhances the software's flexibility and maintainability. Output files are obtained in Exodus and CSV formats, making it adaptable to various post processing and data analysis needs. The results are visually represented and analyzed using Paraview. Additionally, the software supports massive parallelization with scalability up to 30,000 cores. Built-in physics modules and natural multi-physics support are employed to simplify complex simulations with one kernel object per PDE. Additionally, it offers time and mesh adaptivity for dynamic adjustments during runtime. 3. Numerical examples and discussion This study aims to assess the accuracy of exponential finite element (EFE) shape functions in modeling fracture responses in functionally graded materials (FGMs), given their variable material properties. Phase-field simulations are conducted using EFE and linear finite element (LFE) shape functions, with a focus on smaller length scale parameters. Results are compared, and a refined mesh with increased elements is used. Different length scale values correspond to distinct load responses, and EFE shape functions require additional quadrature points for integration. This study considers four examples, including tension and shear loading cases. 3.1. Cracking of alumina/zirconia FG plate under tension Fracture analysis in functionally graded alumina/zirconia plates is a benchmark case for Functionally Graded Materials (FGMs). This FGM comprises varying volume fractions of alumina (Al2O3) and zirconia (ZrO2), and the material properties for these constituents are detailed in Table 1. The plate contains a mode-I edge crack under external uniaxial tension, assuming plane strain conditions. We investigate four configurations with variations along

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the x-axis and the crack positioned at either the ZrO2 edge (X1) or Al2O3 edge (X2), and similar configurations along the y-axis (Y1 and Y2). Load-deflection responses are assessed using both linear finite element (LFE) and exponential finite element (EFE) shape functions. In cases where material properties vary along the y-axis, the elastic and fracture characteristics remain consistent in the direction of the crack. However, this introduces mode mixity at the crack tip due to the material gradient. The load responses consistently demonstrate that higher values of the parameter "p" lead to increased stiffness and earlier fracture, which is in agreement with previous research. This behavior is anticipated because the material along the crack path has a higher "E" value but provides less resistance to fracture. In all four scenarios, the utilization of exponential finite element (EFE) shape functions consistently predicts higher peak load responses compared to linear finite element (LFE) predictions, while also achieving matching peak displacements.

Table 1. Material properties of alumina and zirconia

Al 2 O 3

ZrO 2 210 0.31

Young’s modulus, E (GPa)

380 0.26

Poisson’s ratio ν

5.2

9.6

Fracture toughness, K IC ( MPa m )

Fig. 2. Alumina/zirconia FG plate; Boundary conditions and crack propagation path for the specimen loaded in tension. The simulation uses adaptive mesh refinement scheme in MOOSE, where critical elements are refined automatically.

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3.2. Cracking of alumina/zirconia FG plate under shear In this section, we comprehensively analyze shear-induced fracture behavior in an alumina/zirconia functionally graded (FG) plate. Our focus is to understand the mechanisms of secondary fracture. We extend our previous discussion on plate geometry and properties. Fracture predictions are made using both linear finite element (LFE) and exponential finite element (EFE) shape functions on a 14,000-element coarse mesh. EFE shape functions are particularly important due to the complex curvilinear crack patterns. Results from EFE simulations are compared to those from LFE simulations in terms of crack path, peak load, corresponding displacement, and computational time. The crack paths forecasted by EFE shape functions closely mirror those predicted by LFE shape functions. Remarkably, the crack path predictions from both LFE and EFE shape functions on the coarse mesh sample show only minimal deviations compared to finer mesh predictions. Additionally, load-displacement responses anticipated with EFE on the coarse mesh display enhanced accuracy compared to LFE predictions under similar conditions, closely aligning with converged solutions from finer mesh simulations. Crucially, the computational time required for EFE is only marginally higher than that for LFE, resulting in minimal force response errors.

Fig. 3. Force response versus displacement plots of alumina-zirconia functionally graded plate; (a) FGMX, (b) FGMY, (c) FGMYR, and (d) FGMXY.

Fig. 4. Alumina/zirconia FG plate; Boundary conditions and crack propagation path for the specimen loaded in shear. The simulation uses adaptive mesh refinement scheme. The crack path matches the one available in literature.

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4. Conclusions In conclusion, this study has leveraged the MOOSE (Multiphysics Object-Oriented Simulation Environment) to predict crack propagation in functionally graded materials utilizing the phase field model. The implementation of exponential finite element shape functions for discretizing the domain has proven to be an effective and accurate approach for modeling these complex materials. The incorporation of MOOSE's inbuilt adaptive mesh refinement technique has been instrumental in achieving precision in the prediction of force and displacement responses across a range of loading conditions, including tension and shear. Notably, the results obtained through this comprehensive simulation framework align closely with existing data in the literature, reaffirming the reliability and robustness of the MOOSE-based approach. This success in replicating established findings underscores the software's capability to handle multifaceted multiphysics problems and demonstrates its value as a vital tool in the field of materials science and engineering. As we continue to explore the boundaries of computational materials research, MOOSE stands as a trusted and proficient resource for advancing our understanding of functionally graded materials and their mechanical behavior, offering a significant contribution to the scientific community and facilitating future breakthroughs in the field. References Ambati, M., Gerasimov, T., De Lorenzis, L., 2015. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Computational Mechanics 55(2), 383–405. Hirshikesh, Natarajan, S., Annabattula, R. K., Martínez-Pañeda, E., 2019. Phase field modelling of crack propagation in functionally graded materials. Composites Part B: Engineering 169, 239–248. Kuhn, C., Müller, R., 2010. A continuum phase field model for fracture. Engineering Fracture Mechanics 77, 3625–3634. Li, Z., Shen, Y., Han, F., Yang, Z., 2021. A phase field method for plane-stress fracture problems with tension-compression asymmetry. Engineering Fracture Mechanics 257, 107995. Miehe, C., Welschinger, F., Hofacker, M., 2010. Thermodynamically consistent phase-field models of fracture: Variational principles and multi field FE implementations. International Journal for Numerical Methods in Engineering 83, 1273–1311.

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2023) 000–000

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ScienceDirect

Procedia Structural Integrity 58 (2024) 122–129

© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Abstract The behaviour of a component under real/random loading conditions can be studied by converting the random load into block load. Therefore, knowledge of the initiation and propagation of a crack under block loading conditions is very important for the safety and reliability of machine components / structures. Electromechanical impedance (EMI) is a real-time damage evaluation technique in which simultaneous use of the high-frequency mechanical excitations and responses of the structure are made by employing a piezoelectric transducer. In the present study, an experimental methodology to monitor fatigue crack initiation and propagation under block loading conditions for AA6061 aluminium alloy has been studied with the help of the EMI technique. The piezoelectric sensor bonded on the Compact Tension (CT) specimen acts as an actuator and sensor both. The conductance signature provided a sufficient information about the damage and crack growth under high low and low high block loading conditions. The conventional crack growth rate versus range of stress intensity factor plot matches well with the crack growth rate versus shift in resonant frequency. © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the ICSID 2023 Organizers Keywords: Fatigue crack growth; Electro Mechanical Impedance; Resonance Frequency; Conductance 7th International Conference on Structural Integrity and Durability (ICSID 2023) Application of EMI technique in crack propagation under the block loading conditions for AA6061 alloy K N Pandey a, *, Gurkirat Singh a a Department of Mechanical Engineering, MNNITA,Prayagraj 211004, India

* Corresponding author. Tel.: +91-0532-2271511; fax: +91-0532- 2545341. E-mail address: knpandey@mnnit.ac.in

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