PSI - Issue 2_A

Volume 2 • 201 6 A

ISSN 2452-3216

ELSEVIER

21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Guest Editors: Francesco I acoviello L uca Susmel

D onato Firrao Giuse pp e Ferro

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www.elsevier.com/locate/procedia XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandão a , V. Infante b , A.M. Deus c * a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy Editorial Francesco Iacoviello a *, Luca Susmel b , Donato Firrao c , Giuseppe Ferro c a Università di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043 Cassino (FR) Italy b University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK c Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. For the second time in the history of ESIS, the biennial European Conference of Fracture (ECF) was organized in Italy. In 1990 the ECF was held in Turin, in the northern Italy. Twenty-six years later, the picturesque Sicilian city of Catania hosted the 21 st European Conference of Fracture (ECF21). Sicilian ECF21 was characterized by some unique features. Among them, this conference was the first one with the proceedings being published in Procedia Structural Integrity , Elsevier’s open access on-line repository covering Fracture, Fatigue and Structural Integrity. Furthermore, during the conference, the drafts of the proceedings papers as well as relevant organizing information were shared with the participants by equipping them with tablets. This allowed us to minimize the usage of paper, turning ECF21 into the very first “green” conference ever run by ESIS. We do hope that these two key aspects together with some other unique organizational peculiarities gave to the participants a lot of good memories to be brought home, such good memories together with the friendly hospitality of the Sicilians, the blue of the Sicilian sea and the aroma of the Sicilian evenings being with the ECF participants for a long time. Selecting the plen ry peak s was not a simple task, since the fracture community offers a great choice of outstanding researc ers. T identify our esteemed plenary speakers, we used th following criteria: the plenary speakers had to be different from those attending the previous editions of the ECF, a uniform “geographical” distribution (trying to avoid geographical “polarization”) and, last, but not least, a close connection with the Italian Group of Fracture (www.gruppofrattura.it) – i.e., the Italian representative of ESIS that organized ECF21. This 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy Editorial Francesco Iacoviello a *, Luca Susmel b , Donato Firrao c , Giuseppe Ferro c a Università di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043 Cassino (FR) Italy b University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK c Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. For the second time in the history of ESIS, the biennial European Conference of Fracture (ECF) was organized in Italy. In 1990 the ECF was held in Turin, in the northern Italy. Twenty-six years later, the picturesque Sicilian city of Catania hosted the 21 st European Conf re ce of Fracture (ECF21). Sicilian ECF21 was characterized by some unique features. Among them, this conference was the first one with the proceedings being published in Procedia Structural Integrity , Elsevier’s open access on-line repository covering Fracture, Fatigue and Structural Integrity. Furthermore, during the conference, the drafts of the proceedings papers as well as relevant organizing information were shared with the participants by equipping them with tablets. This allowed us to minimize the usage of paper, turning ECF21 into the very first “green” conference ever run by ESIS. We do hope that these two key aspects together with some other unique organizational peculiarities gave to the participants a lot of good memories to be brought home, such good memories together with the friendly hospitality of the Sicilians, the blue of the Sicilian sea and the aroma of the Sicilian evenings being with the ECF participants for a long time. Selecting the plenary speakers was not a simple t sk, sinc the f acture community offers a great choice f outstan ing researchers. To identify our esteemed plenary speakers, we used the followi g crit ria: the pl nary speakers had to be different from those attending the previous editions of the ECF, a uniform “geographical” distribution (trying to avoid geographical “polarization”) and, last, but not least, a close connection with the Italian Group of Fracture (www.gruppofrattura.it) – i.e., the Italian representative of ESIS that organized ECF21. This 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy Editorial Francesco Iacoviello a *, Luca Susmel b , Donato Firrao c , Giuseppe Ferro c a Università di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043 Cassino (FR) Italy b University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK c Politecnico di Torino, Corso Duca degli A ruzzi 24, 10129, Torino, Italy © 2016 Th Authors. Published by Els vier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. For th second time in the history of ESIS, the biennial European Conferenc of Fracture (ECF) was rganiz d in Italy. In 1990 th ECF was held in Turin, in the northern Italy. Tw nty-six years later, the pictur sque Sicilian city of Catania hosted the 21 st European Conference of Fracture (ECF21). Sicilian ECF21 was c aracterized by some unique features. Among them, this conference was th first one with the pr ceedings being published in Procedia Structural Integrity , Elsevier’s pen access on-line repository covering Fracture, Fatigue and Structural Integrity. Furthermore, during the conference, the drafts of the proceedings papers as w ll as relevant organizing information were shared with t participants by equipping t em with tablets. Thi allowed us to minimize the usage of paper, turning ECF21 into the very rst “green” conf rence ever run by ESIS. We do hope that these two key aspects together with ome other unique organizational peculiarities gave to the participa ts a lot of good memories to be brought home, such good memories together with the friendly hospitalit of the Sicilians, the blue of the Sicilian sea and the aroma of the Sicilian evenings being with the ECF participants for a long time. Selecting the plenary speakers was not a simple task, since the fracture community offers a great choice of outstanding researchers. To identify our esteemed plenary speakers, we used the following criteria: the plenary speakers had to be different from those attending the previous editions of the ECF, a uniform “geographical” distribution (trying to avo d geographi al “polarization”) and, last, but not least, a close connection with the Italian Group of Fracture (www.gruppofrattura.it) – i.e., the Italian representative of ESIS that organized ECF21. This Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/lic nses/by-nc-nd/4.0/). Peer-review under respon ibility of the Scientifi Committee of ECF21. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016.

Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. * Corresponding author. Tel.: +39.07762993681; fax: +39.07762993781. E-mail address: iacoviello@unicas.it

* Corresponding author. Tel.: +351 218419991. E-mail address: amd@tecnico.ulisboa.pt 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. 2452-3216 © 2016 The Auth rs. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. * Corresponding author. Tel.: +39.07762993681; fax: +39.07762993781. E-mail address: iacoviello@unicas.it * Corresponding author. Tel.: +39.07762993681; fax: +39.07762993781. E-mail address: iacoviello@unicas.it

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.001

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resulted in outstanding plenary lectures, with the first article published in this issue of Procedia Structural Integrity summarizing the key aspects covered by some of these talks. The conference was organized by running a number mini-symposia, with each mini-symposium being leaded by invited chairpersons. The number of mini-symposia being organized by different colleagues was impressive, i.e.: - ESIS TC4: Advanced Fracture Mechanics Testing of Polymers, Adhesives and Composites (Andrea Pavan, with many TC4 components: L. Banks-Sills, B. R. K. Blackman, R. Frassine, A. J. Kinloch, J.-A. Pascoe, M. Rink, L. Warnet, J. G. Williams); - ESIS TC5- Dynamic Fracture (Y. Petrov, V. Silberschmidt); - ESIS TC10: Environmentally Assisted Cracking & Hydrogen Embrittlement (J. Toribio); - Constitutive modeling of cyclic plasticity and damage of materials (S. Krishna); - Constraint Effects in Fracture (V. Shlyannikov, T.Meshii); - Creep and fracture at elevated temperature (L. Esposito, C. Davies); - Deformation and fracture at high strain rate (N. Bonora, E. Brown, N. Bourne, G. Mirone); - Dissimilar metal welds: degradation phenomena and inspection (P. Trampus, L. Tòth, S. Szavai); - Fatigue crack initiation and propagation in gigacycle regime (T. Palin-Luc, M. Zimmermann); - Fracture in biomechanics: theoretical and experimental approaches to the resistance of biomaterials and medical devices (G. La Rosa); - Fracture Nanomechanics (T. Kitamura, Z. Zhang, J. He); - Functional fatigue, fracture and failure analysis of shape memory alloys and devices (A. Tuissi, C. Maletta); Considering that numerous reviewers were also involved in the reviewing process, and considering that several colleagues served as session chairmen during the conference, the number of volunteers that was involved in organization of ECF21 was really impressive! In addition, further sessions were organized to allow those contributions that were not involved in specific minsymposia to be presented. These sessions were organized by the members of the IGF Ex-Co, i.e., V. Di Cocco, G. Ferro, D. Firrao, F. Iacoviello, C. Maletta, A. Pirondi, G. Risitano, A. Spagnoli and L. Susmel. Finally, we would like to express our heartfelt thank to Angelo Finelli, the treasurer of both the IGF and ECF21: his long-time and continuous work as treasurer allowed all the events run by the IGF in the last decades as well as the organization of ECF21 to be very successful! As a result of this team work, it was possible to organize ECF21 having about 700 presentations delivered by colleagues coming from 47 different countries. This issue of Procedia Structural Integrity containing 471 papers is the tangible proof of the scientific success of ECF21. Professor Francesco Iacoviello Università di Cassino, Cassino, Italy Professor Luca Susmel University of Sheffield, Sheffield, UK Professor Donato Firrao Politecnico di Torino, Turin, Italy Professor Giuseppe Ferro Politecnico di Torino, Turin, Italy - Local Approaches to Cleavage and Ductile Fracture (C. Ruggieri, F. Minami); - Material Challenges for Extreme Oil and Gas Environments (M. Elboujdaini); - Mixed mode fracture at different scales (L. Marsavina, F. Berto, M. Ayatolahy); - Nonlinear, statistical and scaling aspects of damage-failure transition (O. Naimark, V. Loic, S. Santucci); - Recent advances in the framework of Finite Fracture Mechanics (D. Leguillon, A. Sapora, V. Mantic); - Structural integrity assessment of off-shore structures (A. Carlucci, D. Gentile); - Thermomechanical fatigue testing and life assessment models (A. M. Meizoso); - Thermal Methods for fatigue assessment (V. Crupi)

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XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandão a , V. Infante b , A.M. Deus c * a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 2D-lattice modelling of crack propagation induced by fluid injection in heterogeneous quasi-brittle materials David Gre´goire ∗ , Vincent Lefort, Olivier Nouailletas, Gilles Pijaudier-Cabot Universite´ Pau & Pays Adour, LFCR (UMR5150), Campus Montaury, 64600 Anglet, France Abstract Characterizing the path of a hydraulic fracture in a heterogeneous medium is one of the challenges of current research on hydraulic fracturing. We present here a 2D lattice hydro-mechanical model for this purpose. Natural joints are represented introducing elements with a plastic-damage behaviour. The action of fluid pressure on skeleton is represented using Biot’s theory. The interactions of cracks on fluid flow are represented considering a Poisueille’s flow between two parallel plates. The model is simplified by neglecting the e ff ect of deformation in the equation governing fluid flow. Numerical coupling is achieved with a staggered coupling scheme. We consider first the propagation of fractur restricted to th homogeneous case. The numerical model is compared to analytical solutions. It is found that the model is consistent with LEFM in he pure mechanic l case, and with analytical solutions from the literature in the case where the leak o ff is d minant. In very tight formations, deviations are obs rved, as expected, cause f the assumption in the flow m del. Finally, the influence of a natural joint of finite length cro sed by the fracture is shown. Tw cases re considered, the case of a j int perpendicular to the crack and the case of an inclined joint. In the first case, the crack passes through the joint, which is damaged due to the intrusion of the fluid. In the second case, the crack follows the joint and propagatio starts again from the tip. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 2D-lattice modelling of crack propagation induced by fluid injection in heterogeneous quasi-brittle materials David Gre´goire ∗ , Vincent Lefort, Olivier Nouailletas, Gilles Pijaudier-Cabot Universite´ Pau & Pays Adour, LFCR (UMR5150), Campus Montaury, 64600 Anglet, France Abstract Characterizing the path of a hydraulic fracture in a heterogeneous medium is one of the challenges of current research on hydraulic fracturing. We present here a 2D lattice hydro-mechanical model for this purpose. Natural joints are represented i troducing elements with a plastic-damage behaviour. The action of fluid pressure on skeleton is represented using Biot’s theory. The interactions of cracks on fluid flow are represented considering a Poisueille’s flow between two parallel plates. The model is simplified by neglecting the e ff ect of deformation in the equation governing fluid flow. Numerical coupling is achieved with a staggered coupling scheme. We consider first the propagation of fracture restricted to the homogeneous case. The numerical model is compared to analytical solutions. It is found that the model is consistent with LEFM in the pure mechanical case, and with analytical solutions from the literature in the case where the leak o ff is dominant. In very tight formations, deviations are observed, as expected, because of the assumption in the flow model. Finally, the influence of a natural joint of finite length crossed by the fracture is shown. Two cases are considered, the case of a joint perpendicular to the crack and the case of an inclined joint. In the first case, the crack passes through the joint, which is damaged due to the intrusion of the fluid. In the second case, the crack follows the joint and propagation starts again from the tip. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: Fracture Process Zone; Hydro-mechanical Coupling; Lattice Analysis; Jointed rock Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). er-review under esponsibility of the Scientific Committee of ECF21. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: Fracture Process Zone; H dro-mechanical Coupling; Lattice Analysis; Jointed rock

Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

1. Introduction 1. Introduction

Crack propagation under fluid injection is a coupled and complex problem with various applications, from magma transport in the lithosphere to oil and gas reservoir stimulation from the 40’s (Economides and Nolte, 2000). Di ff erent coupled e ff ects have been studied in the literature such as the interaction between di ff erent stimulated cracks Crack propagation under fluid injection is a coupled and complex problem with various applications, from magma transport in the lithosphere to oil and gas reservoir stimulation from the 40’s (Economides and Nolte, 2000). Di ff erent coupled e ff ects have been studied in the literature such as the interaction between di ff erent stimulated cracks

* Corresponding author. Tel.: +351 218419991. E-mail address: amd@tecnico.ulisboa.pt ∗ Corresponding author. Tel.: + 33-5-5957-4479 ; fax: + 33-5-5957-4439. E-mail address: david.gregoire@univ-pau.fr ∗ Corresponding author. Tel.: + 33-5-5957-4479 ; fax: + 33-5-5957-4439. E-mail address: david.gregoire@univ-pau.fr

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.337 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review und r responsibility of the Scientifi Committee of ECF21. 2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

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(Lecampion and Desroches, 2015), the influence of the spatial variation of the rock mechanical properties on the crack extensions (King, 2010), the proppant capacity to fill the crack space and its influence on the crack openings (Cipolla et al., 2010) or the influence of dynamic sollicitations on the variation of the rock global permeability (Chen et al., 2012). For homogeneous materials, di ff erent analytical solutions have been proposed for bi-wing crack configurations. For the KGD 1 configuration, see e.g. Khristianovic and Zheltov (1955) or Geertsma and De Klerk (1969). For the PKN 2 configuration, see e.g. Perkins and L. R. Kern (1961), Nordgren (1972) or Adachi and Detournay (2008). For a comparison of the two, see e.g. Geertsma and Haafkens (1979). These analytical solutions predict the width and extend of hydraulically-induced fractures by taking into account the fluid transfer within the matrix through a Carter’s leak-o ff coe ffi cient (Howard and Fast, 1957) but to solve the problem analytically, di ff erent asymptotic regimes are distinguished. For instance the fluid flow may be dominated by the leak-o ff or the fluid may preferentially stored within the propagating crack. The mechanical energy may be preferentially dissipated through the matrix fracture (toughness-dominated) or through frictional shear forces within the fluid (viscosity-dominated) (e.g. see Bunger et al., 2005, for a study of a toughness-dominated hydraulic fracture with leak-o ff ). For intermediate cases, the global model cannot be solved analytically and numerical modeling is needed to characterise width and extend of hydraulically-induced fractures and there interactions with the natural network of pre-existing joints. The mechanical and hydraulic behaviour of a rock formation is highly dominated by the natural network of pre-existing joints. This is particularly the case for source rocks, which have been intensively fractured hydraulically for oil and gas extraction (see e.g. Engelder et al., 2009, for a description of middle and upper devonian gas shales of the Appalachian basin, Gale et al., 2007, for a description of natural fractures in a Barnett shale or Warpinski and Teufel, 1987, for the influence of geologic discontinuities on hydraulic fracture propagation). Natural joints may have been cemented by geological fluid flows and the global permeability of the system will highly depend on the capacity of the hydraulic fracture to reactive these natural joints. This research study aims at developing a lattice-type numerical model allowing the simulation of crack propagation under fluid injection in a quasi-brittle heterogeneous medium. A lattice-based modeling description has been chosen because it has been shown in previous studies that this mesoscale approach is capable not only to provide consistent global responses (e.g. Force v.s. CMOD responses, see Grassl et al., 2012, or Gre´goire et al., 2015) but also to capture the local failure process realistically (see Gre´goire et al., 2015, or Lefort et al., 2015). Moreover, this numerical model is based on a dual Vorono / Delaunay description, which is very e ffi cient to represent dual mechanical / hydraulic couplings (Grassl et al., 2015). Therefore, this numerical tool will be used here to get a better understanding of initiation and propagation conditions of cracks in rock materials presenting natural joints where the coupling between mechanical damage and fluid transfer properties are at stake. This paper is organized as follows. After having briefly recalled in Section 2 the lattice model used in this paper, we proceed in Section 3 to the comparisons to analytical solutions. It is found that the model is consistent with LEFM in the pure mechanical case, and with analytical solutions from the literature in the case where the leak o ff is dominant. In very tight formations, deviations are observed, as expected, because of the assumption in the flow model. Section 4 presents the influence of a natural joint of finite length crossed by the fracture is shown. Two cases are considered, the case of a joint perpendicular to the crack and the case of an inclined joint. In the first case, the crack passes through the joint, which is damaged due to the intrusion of the fluid. In the second case, the crack follows the joint and propagation starts again from the tip.

1 Kristonovich-Geertsma-de Klerk 2 Perkins-Kern-Nordgren

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2. Lattice modeling

2.1. Mechanical description

2.1.1. Matrix description A 2D plane-stress lattice model is used to characterize the initiation and propagation conditions of cracks in rock materials presenting natural joints. This lattice model is based on the numerical framework proposed by Grassl and Jirasek (Grassl and Jira´sek, 2010). It has been shown in previous studies that this mesoscale approach is capable not only to provide consistent global responses (e.g. Force v.s. CMOD responses) (Grassl et al., 2012; Gre´goire et al., 2015) but also to capture the local failure process realistically (Gre´goire et al., 2015) for quasi-brittle materials such as concrete or rocks. The numerical procedure is briefly presented in this section. The reader may refer to references (Grassl and Jira´sek, 2010; Grassl et al., 2012; Gre´goire et al., 2015; Lefort et al., 2015) for further details. The matrix is supposed to be homogeneous at the scale of the study and the lattice is made of beam elements, which idealize the material structure. The matrix structure is meshed by randomly locating nodes in the domain, such that a minimum distance is enforced. The lattice elements result then from a Delaunay triangulation (solid lines in figure 1a) whereby the middle cross-sections of the lattice elements are the edges of the polygons of the dual Voronoi tesselation (dashed lines in figure 1a).

cross section

lattice element

(a)

(b)

(c)

Fig. 1: (a) Set of lattice elements (solid lines) with middle cross-sections (dashed lines) obtained from the Voronoi tessellation of the domain. (c) and (d) Lattice element in the global coordinate system (Reproduced from Grassl et al. (2012)).

Each node has three degrees of freedom: two translations ( u , v ) and one rotation ( φ ) as depicted in figure 1c. In the global coordinate system, the degrees of freedom of nodes 1 and 2, noted u e = ( u 1 , v 1 , φ 1 , u 2 , v 2 , φ 2 ) T , are linked to the displacement jumps in the local coordinate system of point C , u c = ( u c , v c , φ c ) T . See (Grassl and Jira´sek, 2010; Grassl et al., 2012; Gre´goire et al., 2015; Lefort et al., 2015) for details.

An isotropic damage model is used to describe the mechanical response of lattice element within the matrix.

The elastic constants and the model parameters in the damage models are calibrated from an inverse analysis technique.

2.2. Natural joint description

Natural joints are explicitly described within the model with beam elements with a new elasto-plastic damage constitutive law (figure 2). The originality of the model lies in the coupling between mechanical damage under normal strain and plasticity under tangential strain. Mechanical damage induces a decrease of the material cohesion whereas the plastic strains, in both normal and tangential directions, participate to the damage evolution. This new constitutive

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Lattice points on both sides of the natural joints Joint explicitly meshed within the lattice description Polyhedral interfaces Mechanical beam elements representing the joint constitutive behaviour Ghost elements with zero-section

Fig. 2: Lattice description of a 30 o inclined natural joint of finite length.

Experimental Elasto−plastic damage model Classical Mohr−Coulomb

Vertical displacement

80

70

Plaster joint Mortar

60

50

200mm

40

45

30

Force [kN]

20

Clamped face

10

0

100mm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vertical Displacement [mm]

Fig. 3: Comparison between experimental and numerical results for an indirect shear test.

law is able to reproduce indirect shear experimental tests performed on mortar specimens presenting a plaster joint where a classical Mohr-Coulomb criterion fails (see Figure 3) .

2.3. Hydraulic description

The hydro-mechanical coupling is introduced through a poromechanical framework based on the intrinsic and dual hydro-mechanical description of the lattice model, which is based on a hydraulic Voronoi tessellation and a mechanical Delauney triangulation (figure 4). The total stress links the mechanical stress and the pore pressure through the Biot coe ffi cient of the medium whereas the local permeability, which drives the hydraulic pressure gradient, depends on the local crack openings. The following assumptions are made: the matrix porosity is connected and its volume depends on the fluid pressure; the fluid saturates the porous medium and is incompressible; only laminar flows are considered

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P 19

P 40

Π 37

×

×

Π 32

P 16

h

Π 34

×

P 21

×

P 23

P 17

×

P 23

×

Π 30

Π 30

P 36

P 28

×

×

Π 36

Π 36

l

P

×

P 5

5

×

P 8

×

×

P 24

Π 31

P 6

Π 30 : Mechanical node P 5 : Hydraulic node

Π 29

Hydraulic pipe Mechanical beam

Fig. 4: Dual mechanical / hydraulic lattice description.

and gravity e ff ects are neglected. The model is simplified by neglecting the e ff ect of deformation in the equation governing fluid flow. Numerical coupling is achieved with a staggered coupling scheme.

3. Comparisons to analytical solutions

3.1. LEFM comparison for an impermeable crack

Linear elastic fracture mechanics (LEFM) links the fluid pressure to the crack extend for quasi-static stable crack propagation. In this section and for LEFM comparison purposes the crack path is pre-meshed within the lattice model ing. The surrounding matrix is assumed perfectly elastic and impermeable. A fluid flow is imposed in a short prenotch and net pressure versus crack tip abscissa are compared in Figure 5. A good agreement is observed as soon as the crack extend is high enough to neglect the fracture process zone influence.

3.2. Leak-o ff representation and comparison with Carter’s model

When an hydraulically stimulated crack propagates within a permeable medium, its extend depends on the so called leak-o ff , the quantity of fluid which drives out within the matrix. Carter’s model represents the leak-o ff as an unidimensional di ff usive flow perpendicular to the crack lips. For comparison purposes, we propose a simple hydraulic problem where a fully saturated beam is submitted to a pressure gradient (figure 6). A good agreement is observed for this simple di ff usive flow problem.

3.3. Analytical comparison for a permeable crack

Bunger et al., 2005 presents an analytical solution for the study of a toughness-dominated hydraulic fracture with leak-o ff . The geometry presented in figure 5 is used with a permeable crack. Since the analytical solution presented by Bunger et al., 2005 is based on brittle fracture, the crack path is still pre-meshed within the lattice modeling. Figure 7 presents the comparison between the analytical solution and the lattice results in term of crack extend evolution with time and crack opening repartition.

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notch

elastic matrix

σ x

Q 0

60 m

pre-meshed crack

45 m

σ y

(a)

(b)

(c)

(d)

Fig. 5: (a) Geometry. (b)-(c) Mechanical and hydraulic boundary conditions. (d) Net pressure vs. crack tip abscissa – Comparison between LEFM and the lattice results.

y

20 m

x

100 m

: permeable : impermeable

(b)

(a)

(d)

(c)

Fig. 6: (a) Geometry. (b) Pressure repartition at t = 10 6 s . (c)-(d) Comparison between Carter’s model and the lattice results in term of pressure repartition at t = 10 6 s and fluid flux temporal evolution at x = 0 . 01 m position.

4. Influence of a natural joint on the hydraulic fracture crack path

Since the model has been validated by di ff erent analytical comparisons in section 3, we propose here to study the free propagation of an hydraulically-induced fracture and its interaction with a pre-existing joint. The crack path is

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(a)

(b)

Fig. 7: Comparison between the lattice results and an analytical solution for a toughness-dominated hydraulic fracture with leak-o ff (Bunger et al., 2005): (a) crack extend evolution with time; (b) crack opening / aperture repartition at t = 100 s .

(a)

(b)

Fig. 8: Influence of a 45 o inclined joint on the crack path of a hydraulically-induced fracture: (a) crack path after t = 100 s ; (b) pressure repartition after t = 100 s .

not pre-meshed and is a direct results of the lattice modeling (Figure 8.a) as well as the pressure repartition within the host matrix (Figure 8.b).

5. Concluding remarks

In this paper, a new hydro-mechanical coupled lattice-based model has been proposed for 2D-simulation of crack propagation induced by fluid injection in heterogeneous quasi-brittle materials. The hydro-mechanical coupling is introduced through a poromechanical framework based on the intrinsic and dual hydro-mechanical description of the

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lattice model. Di ff erent configurations have been studied to test the relevance of the lattice description. It has been shown that it is capable of representing the crack propagation in both impermeable and permeable media by taking into account the leak-o ff phenomena accurately. A new elasto-plastic damage constitutive law has been proposed to mimic natural joint behaviour. This new constitutive law is able to reproduce indirect shear experimental tests performed on mortar specimens presenting a plaster joint where a classical Mohr-Coulomb criterion fails. Within this framework, the influence of a 45 o inclined joint on the crack path of a hydraulically-induced fracture has been studied and pressure repartition within the host matrix has been presented.

Acknowledgements

Financial support from TOTAL Exploration & Production and Conseil Ge´ne´ral des Pyre´ne´es Atlantique (CG64) is gratefully acknowledged. The authors wish also to acknowledge the University of Bordeaux and Universite´ Pau & Pays Adour , for the use of clusters AVAKAS and PYRENE respectively.

References

Adachi, J., Detournay, E., 2008. Plane strain propagation of a hydraulic fracture in a permeable rock. Engineering Fracture Mechanics 75, 4666–4694. Bunger, A., Detournay, E., Garagash, D., 2005. Toughness-dominated hydraulic fracture with leak-o ff . International Journal of Fracture 134, 175–190. Chen, W., Maurel, O., Reess, T., De Ferron, A., La Borderie, C., Pijaudier-Cabot, G., Rey-Bethbeder, F., Jacques, A., 2012. Experimental study on an alternative oil stimulation technique for tight gas reservoirs based on dynamic shock waves generated by Pulsed Arc Electrohydraulic Discharges. Journal of Petroleum Science and Engineering 88-89, 67–74. Cipolla, C., Warpinski, N., Mayerhofer, M., Lolon, E., Vincent, M., 2010. The Relationship Between Fracture Complexity, Reservoir Properties, and Fracture-Treatment Design. SPE Production & Operations 25, 1–25. Economides, M., Nolte, K., 2000. Reservoir Stimulation. John Wiley & Sons Ltd, West Sussex, England. Engelder, T., Lash, G., Uzca´tegui, R., 2009. Joint sets that enhance production from Middle and Upper Devonian gas shales of the Appalachian Basin. AAPG Bulletin 93, 857–889. Gale, J., Reed, R., Holder, J., 2007. Natural fractures in the Barnett Shale and their importance for hydraulic fracture treatments. AAPG Bulletin 91, 603–622. Geertsma, J., De Klerk, F., 1969. A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures. Journal of Petroleum Technology 21, 1571–1581. Geertsma, J., Haafkens, R., 1979. A Comparison of the Theories for Predicting Width and Extent of Vertical Hydraulically Induced Fractures. Journal of Energy Resources Technology 101, 8. Grassl, P., Fahy, C., Gallipoli, D., Wheeler, S.J., 2015. On a 2D hydro-mechanical lattice approach for modelling hydraulic fracture. Journal of the Mechanics and Physics of Solids 75, 104–118. Grassl, P., Gre´goire, D., Rojas-Solano, L., Pijaudier-Cabot, G., 2012. Meso-scale modelling of the size e ff ect on the fracture process zone of concrete. International Journal of Solids and Structures 49, 1818–1827. Grassl, P., Jira´sek, M., 2010. Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension. International Journal of Solids and Structures 47, 957–968. Gre´goire, D., Verdon, L., Lefort, V., Grassl, P., Saliba, J., Regoin, J., Loukili, A., Pijaudier-Cabot, G., 2015. Mesoscale analysis of failure in quasi brittle materials: comparison between lattice model and acoustic emission data. International Journal for Numerical and Analytical Methods in Geomechanics 39, 1639–1664. Howard, G., Fast, C.R., 1957. Optimum Fluid Characteristics for Fracture Extension. Proceedings of the American Petroleum Institute , 261–270. Khristianovic, S., Zheltov, Y., 1955. Khristianovic, S.A., Zheltov, Y.P., in: 4th World Petroleum Congress, Carlo Colombo, Rome., Rome. pp. 579–586. King, G., 2010. Thirty years of gas-shale fracturing: What have we learned? JPT Journal of Petroleum Technology 62, 88–90. Lecampion, B., Desroches, J., 2015. Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore. Journal of the Mechanics and Physics of Solids 82, 235–258. Lefort, V., Pijaudier-Cabot, G., Gre´goire, D., 2015. Analysis by Ripleys function of the correlations involved during failure in quasi-brittle materials: Experimental and numerical investigations at the mesoscale. Engineering Fracture Mechanics 147, 449–467. Nordgren, R., 1972. Propagation of a Vertical Hydraulic Fracture. Society of Petroleum Engineers Journal 12, 306–314. Perkins, T.K., L. R. Kern, 1961. Widths of Hydraulic Fractures. Journal of Petroleum Technology 13, 937 – 949. Warpinski, N., Teufel, L., 1987. Influence of Geologic Discontinuities on Hydraulic Fracture Propagation. Journal of Petroleum Technology , 209–220.

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XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine P. Brandão a , V. Infante b , A.M. Deus c * a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal c CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal Abstract During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 3D J-integral for functionally graded and temperature dependent thermoelastic materials J. Hein, M. Kuna ∗ Institute of Mechanics and Fluid Dynamics, Technical University Freiberg, Lampadiusstraße 4, 09599 Freiberg, Germany Abstract In order to mitigate the thermal shock impact, many ceramics, efr ctories or heat protecting layers are made of functionally graded (FGM) or lay red stru tures. Fractur mechanical methods are needed to evaluat the resistance of such structures against failure under thermal shock. Despite a lot of theoretical work has been done for two-dimensional crack configurations in thermo elastic FGM, real defects and structural components are of three-dimensional (3D) nature. In most cases the real gradation of the thermoelastic material properties does not obey simple mathematical functions, but show a complex dependency on location due to manufacturing. Moreover, the elastic and thermodynamic properties depend on temperature itself. In the paper, we present the derivation of the 3D J -integral for arbitrary location and temperature dependent material behavior. The J -integral is implemented in FEM by means of the equivalent domain integral technique. The method is applied to a thermally loaded plate of FGM with a surface crack. The influence of the material gradation on the results is investigated in various examples. c 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: J-integral ; functionally graded material ; temperature dependent material ; 3D crack ; thermal shock N menclatur 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 3D J-integral for functionally graded and temperature dependent thermoelastic materials J. Hein, M. Kuna ∗ Institute of Mechanics and Fluid Dynamics, Technical University Freiberg, Lampadiusstraße 4, 09599 Freiberg, Germany Abstract In order to mitigate the thermal shock impact, many ceramics, refractories or heat protecting layers are made of functionally graded (FGM) or layered structures. Fracture mechanical methods are needed to evaluate the resistance of such structures against failure under thermal shock. Despite a lot of theoretical work has been done for two-dimensional crack configurations in thermo elastic FGM, real defects and structural components are of three-dimensional (3D) nature. In most cases the real gradation of the thermoelastic mate ial pr perties does not obey simple mathematical functions, but show a complex dependency on location due to manufacturing. Moreover, the elastic and thermodynamic properties depend on temperature itself. In the paper, we present the derivation of the 3D J -integral for arbitrary location and temperature dependent material behavior. The J -integral is implemented in FEM by means of the equivalent domain integral technique. The method is applied to a thermally loaded plate of FGM with a surface crack. The influence of the material gradation on the results is investigated in various examples. c 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: J-integral ; functionally graded material ; temperature dependent material ; 3D crack ; thermal sh ck Nomenclature A f e t F th t i r Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). P er-review under esponsibility of the Scientific Committee of ECF21.

thermal expansion coe ffi cient

α

© 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. α thermal expansion coe ffi cient δ i j Kronecker symbol i j strain tensor consisting of mechanical m σ i j stress tensor ∆ A virtually extended crack area along a crack segment ∆ s ∆ a crack growth ∆ l m virtual crack propagation vector a crack length δ i j Kronecker symbo i j strain tensor consi ting of mechanical m Φ synonym for a material property σ i j stress tensor ∆ A virtually extended crack area along a crack segment ∆ s ∆ a crack growth l m virtual crack propagation vector a crack length α K m s a v Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation. i j and thermal strain tensor th i j i j and th rmal strain tensor th i j th Φ synonym for a material property

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Peer review under responsibility of the Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.281 ∗ Corresponding author. Tel.: + 49-3731-39-2092 ; fax: + 49-373-39-3455 E-mail address: meinhard.kuna@imfd.tu-freiberg.de 2452-3216 c 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. ∗ Corresponding author. Tel.: + 49-3731-39-2092 ; fax: + 49-373-39-3455 E-mail address: meinhard.kuna@imfd.tu-freiberg.de 2452-3216 c 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. rrespond + E-mail address: meinhard.kuna@imfd.tu-freiberg.de 016 r-r ific Committe * Corresponding author. Tel.: +351 218419991. E-mail address: amd@tecnico.ulisboa.pt