Multi-axial fatigue numerical crack propagation in cruciform specimens

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Fatigue crack growth in a compressor stage of a turbofan engine by AIAS 2018 International Conference on Stress Analysis FEM-DBEM approach Fatigue crack growth inaa compressor stage of a turbofan engine by a b Venanzio Giannella *, Michele Perrella , Valery N. Shlyannikov XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal FEM-DBEM approach Dept. of Industrial Engineering, University of Salerno via Giovanni Paolo II, Fisciano, Italy Kazan Scientific Center of Russian Street, 2/31 - 420111 Kazan, Russia. Thermo-mechanical modeling a Lobachevsky high turbine blade of an a Academy of Sciences, a pressure b Venanzio Giannella *, MicheleofPerrella , Valery N. Shlyannikov airplane gas turbine engine Dept. of Industrial Engineering, University of Salerno via Giovanni Paolo II, Fisciano, Italy a

b

a

Abstract

Kazan Scientific Center of Russian Academy of Sciences, Lobachevsky Street, 2/31 - 420111 Kazan, Russia.

b

P. Brandãoa, V. Infanteb, A.M. Deusc*

In this work, the fatigue crack-growth process in a rotating disk of an aircraft gas turbine engine has been simulated. The considered a Department of Mechanical Instituto Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, crack nucleated in the attachmentEngineering, between a blade andSuperior the diskTécnico, of a compressor stage, both made of a two-phase titanium alloy. Portugal Abstract The bfatigue crack-growth process of such crack has been simulated by means of two codes, ABAQUS and BEASY, based on Finite IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Element Method (FEM) and Dual Boundary Element Method (DBEM) respectively. In particular, a variant of the submodelling Portugal In this work, the fatigue process in ahas rotating disk of aircraft gas engine has been simulated. The considered c technique, based on the crack-growth superposition principle, been used foran coupling the turbine two codes in order exploit simultaneously their CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av.toRovisco Pais, 1, 1049-001 Lisboa, crack nucleated in the attachment between a blade and the disk of a compressor stage, both made of a two-phase titanium peculiar strength points. The FEM code has been used to compute the global stress field whereas the DBEM code has beenalloy. used to Portugal The fatigue process ofuseful such crack has been simulated by means of two codes, ABAQUS andand BEASY, based on Strain Finite calculate thecrack-growth fracture parameters, to predict the crack-growth evolution. The J-integral method the Minimum Element MethodCriterion (FEM) and Dual have Boundary Method (DBEM) respectively. In particular, a variant of the submodelling Energy Density (MSED) been Element used for calculating K values and predicting crack kinking respectively. technique, based the superposition principle, has been used forthe coupling the two by codes in order to exploit simultaneously In Abstract this work, the on FEM-DBEM crack path is compared with both path obtained a full-scale experimental test and the their path peculiar strength points. FEM code has in been compute the global field whereas thewith DBEM code has been to predicted via a full FEMThe approach: having an used initialtostage considered the stress only centrifugal load, no allowance e.g. used for the calculate the fracture parameters, useful to predict the crack-growth evolution. The J-integral method and the Minimum Strain fluid pressure the blades and for the blade behaviour, some discrepancies are demanding found between numerical and During their on operation, modern aircraft enginedynamic components are subjected to increasingly operating conditions, Energy Density Criterion (MSED) have been used for calculating K values and predicting crack kinking respectively. experimental results. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent In this work, the FEM-DBEM crack path is compared with bothapproach the path obtained by a full-scale experimental test and the path The computational advantages of the submodelling are highlighted, in addition a preliminary fatigue degradation, one of which is creep. Aproposed model using the finite element method (FEM) was developed, intoorder to be able to predict predicted via a full FEM approach: having in an initial stage considered the only centrifugal load, with no allowance e.g. for the assessment for of theHPT considered (further analyses under development). the creepprovided behaviour blades.compressor Flight datadisk records (FDR) for are a specific aircraft, provided by a commercial aviation fluid pressure on the blades and for the blade dynamic behaviour, some discrepancies are found between numerical and company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model experimental results. forAuthors. the FEM analysis,byaElsevier HPT blade © needed 2018 The Published B.V. scrap was scanned, and its chemical composition and material properties were The computational advantages of the proposed submodelling approach are highlighted, in addition to a preliminary fatigue obtained. Theaccess data that wasunder gathered wasBY-NC-ND fed into thelicense FEM model and different simulations were run, first with a simplified 3D This is an open article the CC (http://creativecommons.org/licenses/by-nc-nd/3.0/) assessment provided for the considered compressor disk (further analyses are under development). rectangularunder blockresponsibility shape, in order establish the model, and then with the real 3DConference mesh obtained from Analysis. the blade scrap. The Peer-review of to thebetter Scientific Committee of AIAS 2018 International on Stress overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a © Published by Elsevier B.V. B.V. © 2018 2018The TheAuthors. Authors. Published byofElsevier model useful in the goalthe predicting turbine blade life, given a set of FDR data. This is an can openbe access article under CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the Scientific Committee of AIAS 2018 International Conference on Stress Analysis. Peer-review under responsibility theElsevier Scientific Committee of AIAS 2018 International Conference on Stress Analysis. © 2016 The Authors. Publishedofby 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.: +39-089-96-4111. E-mail address: [email protected] 2452-3216 © 2018 The Authors. Published by Elsevier B.V. * Corresponding author. Tel.: +39-089-96-4111. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) E-mail address: [email protected] Peer-review under responsibility of the Scientific Committee of AIAS 2018 International Conference on Stress Analysis. 2452-3216 © 2018 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/3.0/) Peer-review under responsibility of the218419991. Scientific Committee of AIAS 2018 International Conference on Stress Analysis. * Corresponding author. Tel.: +351 E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016.

2452-3216  2018 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/3.0/) Peer-review under responsibility of the Scientific Committee of AIAS 2018 International Conference on Stress Analysis. 10.1016/j.prostr.2018.11.077

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Keywords: FEM-DBEM; fatigue crack-growth; turbine vane; load spectrum.

1. Introduction Fatigue life prediction for turbofan engine disks, traditionally, has involved two distinct problems: a numerical stress analyses involving the critical regions and a crack modelling to perform the simulation of advancing damages, pre-existent in the component or in-service initiated. In order to predict the residual fatigue life of mechanical components successful applications of Fracture Mechanics (FM) methodologies must be achieved (e.g. in Maligno et al., 2015; Citarella et al., 2013, 2014b, 2018). An accurate characterization of crack paths and Crack Growth Rates (CGRs) require knowledge of flight cycle loading, accurate stress assessment, reliable material properties definition and precise Stress Intensity Factors (SIFs) evaluations. Experience has shown that advances in FM proceed through mutual interaction between numerical analyses and experimental observations (Citarella et al., 2005, 2014a, 2014c, 2015a; Calì et al., 2003). In this context, component design becomes very demanding due to high temperatures, complex mechanical loads, corrosive environment and long expected lifetimes. It is therefore of interest to accurately evaluate the impact of detected defects on these components (e.g. in Citarella et al., 2014d, 2016b) to avoid the catastrophic consequences of an in-service structural failure. Several fatigue failures of rotating compressor disks of civil aircraft engines were recorded during service. Such failures were the result of fatigue crack initiations and their propagation up to reaching a critical size. In all of the failures, the crack propagation started from part-through defects, initiated in between the compressor disk and the blade attachments (both made of a two-phase titanium alloy). These quarter-ellipse corner cracks developed in the slot fillets under the blades and near the disk outer surface. This paper provides a numerical investigation of a crack-growth process, following previously performed full-scale experimental tests of the same rotating disk, as available in literature (Shlyannikov et al., 2001). In particular, a peculiar variant of the submodelling methodology is implemented, based on a FEM model of the overall uncracked component, useful to calculate the global stress to subsequently use as input for the fracture assessment by a DBEM submodel (the crack is introduced at this stage). In order to reduce the computational efforts, the DBEM analyses are performed on a submodel including a restricted volume embedding the crack. FEM stresses, once converted into tractions, are applied on the crack face elements of the DBEM submodel, representing the only needed boundary conditions to perform the stress analysis and calculate SIFs, kink angles and CGRs. Wilson (1979) has briefly presented the theoretical background of such an approach in the past and, more recently, Giannella et al. (2017a, 2017b) have applied the approach to simulate fatigue crack-growth in real structures, also for thermal-stress problems with allowance for complex load spectra. The J-integral and the Minimum Strain Energy Density (MSED) criterion (Sih, 1974) have been used for calculating SIFs and kink angles respectively. The Paris’ law has been used to calculate CGRs. The crack path predicted by the FEM-DBEM approach has been compared with the path calculated by means of a FEM-FEM global-local approach and with an experimental path taken from literature. The adopted FEM based fracture simulation tool is FRANC3D (Wawrzynek et al., 2009). The adopted FEM and DBEM codes are ABAQUS (Dassault, 2011) and BEASY (BEASY, 2016) respectively. Nomenclature C da/dN E G J K

Paris’ law coefficient Crack-Growth Rate (CGR) Young’s modulus Shear modulus J-integral Stress Intensity Factor (SIF)

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Kth Kc Keff n W

Γ𝜌𝜌 υ ρ

3

Threshold value for K Critical value for K Effective value for K Paris’ law exponent Strain energy density Closed contour around crack tip Poisson’ ratio Mass density

2. FEM analyses 2.1. FEM modelling The global FEM model is representative of one quarter of a compressor disk, comprising nine blades (Fig. 1a). All the parts are made of a two-phase titanium alloy, with the main mechanical and fatigue properties listed in Tab. 1. The loading boundary condition consists only of the centrifugal force, with a rotational speed of 925 rad/s, applied to the whole domain. The kinematic boundary conditions consist of normal constraints, applied on the two orthogonal disk cut surfaces, in addition to axial constraints again applied on the two disk cut surfaces and on some blade surfaces (to prevent their rigid body motion; Fig. 1d). These boundary conditions allow the model to deform freely in the radial direction due to the centrifugal forces. A second level of submodelling has been performed (Fig. 1b) in order to gain accuracy (by further mesh refinement) in the volume surrounding the interface between blade roots and disk slots; such interface (Fig. 1c) is modelled with nonlinear contact conditions with allowance for friction (f=0.3).

Figure 1. (a) FEM model, (b) FEM submodel and (c) close-up on the crack insertion point (red dot) in the slot fillet; (d) boundary conditions applied on the two disk cut surfaces and on the Z+ surfaces of the blades.

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Table 1. Main mechanical and fatigue properties for the considered material. Young’s modulus E [GPa]

Poisson ratio υ [-]

Density ρ [kg/m3]

C [MPa^(1-n)/mm^(n/2)]

n [-]

100.1

0.3

4500

3E-14

3.6

2.2. FEM results Plots of the global-local radial displacements and von Mises stresses are reported in Fig. 2. In particular, Fig. 2d clearly highlights the area with highest stresses where a crack is likely to nucleate and start its growth due to the cyclic loading. Fatigue crack growth has been detected during a full-scale experimental test (Shlyannikov et al., 2001): the modelled crack is shown in Fig. 3a whereas in Fig. 3b some further cracked scenarios are shown.

Figure 2. Global (a) radial displacements [mm] and (b) von Mises stresses [MPa], (c) local von Mises stresses [MPa] with close-up (d) in the surrounding of crack insertion area.

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Figure 3. (a) Failure induced by the fatigue crack-growth; (b) typical disk failure scenarios (Shlyannikov et al., 2001).

3. DBEM analyses 3.1. Introduction A DBEM submodel (Fig. 4), enclosing the domain in which the crack is likely to propagate, is extracted by the FEM submodel (Fig. 2b). It must be sufficiently larger than the volume affected by the experimental crack-growth (Fig. 3a) and, when still uncracked, comprises 3099 linear elements. A quarter-circular part-trough crack, with radius equal to 0.15 mm, has been inserted in the DBEM submodel (Fig. 4), in the most critical point, as pointed out by the FEM global-local stress analyses and consistently with the experimental outcomes. The initial crack inclination (nearly 45°) has been defined consistently with the experimental inclination. The area surrounding the crack insertion point as well as the crack faces have been step-by-step remeshed along the propagation, using quadratic (9 nodes) quadrilateral elements or quadratic triangular (6 nodes) elements. During the remeshing process, several rings of internal points were positioned along the crack front in order to build the Jpaths needed to allow the J-integral evaluation. One element far from the crack has been constrained in the three directions in order to remove the rigid body motion (the load applied on the crack faces is self-equilibrated). SIFs are calculated with the J-integral approach (Rigby et al. 1993, 1998; Dell’Erba et al. 2000) whereas the crack path assessment is based on the Minimum Strain Energy Density (MSED) criterion (Sih, 1074). Different approaches have been proposed in the past to perform fracture assessments by using in combination FEM and DBEM: Fixed Displacement (FD), Fixed Load (FL) and Loaded Crack (LC). A complete benchmark among the three approaches has been presented by Giannella et al. (2017a) on a complex industrial application in which also load spectrum effects have been considered. A further benchmark between FD and LC approaches on a crack propagating in aeroengine turbine stage has been presented by Citarella et al. (2016b). In summary, the LC approach turned out to exhibit both computational and accuracy advantages vs. FD and FL and, therefore it was selected to tackle the fracture problem presented in this paper.

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Figure 4. (a) DBEM submodel with the initial crack; (b) close-up of the crack insertion area; (c) internal view of the crack with highlight of tractions applied on the crack face elements and J-paths along the crack front (X representing the normalised abscissa along crack front).

3.2. FEM-DBEM LC approach The FEM-DBEM LC approach is based on the application of the superposition principle to fracture mechanics problems and is here described (Fig. 5) with reference to a thermal-stress crack problem (see also Wilson, 1979):  starting from an original uncracked domain (A), a crack can be opened (B) and loaded with tractions corresponding to those calculated over the dashed line of the virtual crack in (A);  the new configuration (B), equivalent to the previous one (A), can be transformed by using the superposition principle, splitting the boundary conditions as provided in (C) and (D) (Eq. (2)). (C) represents the original problem to solve, whereas (D), after the tractions sign inversion turns into the equivalent problem (E) that will be effectively worked out; namely, SIFs for case (C) are equal (Eq. (2)) to those calculated for the simpler problem (E). In conclusion, using the boundary conditions retrieved from the considered thermal-stress problem (Fig. 5) a pure stress analysis for a crack problem can be solved, in which the crack faces undergo tractions equal in magnitude but opposite in sign to those calculated over the dashed line in Fig. 5 (position A). Ka = Kb = 0 = Kc + Kd

Kc = − Kd = Ke

(1) (2)

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Fig. 5: LC approach description.

3.3. J-integral formulation An application of the J-integral to 3D crack problems can be found in Rigby et al. (1993, 1998) and its generalization to 3D thermoelastic crack problems can be found in Dell’Erba et al. (2000). The J-integral for 3D problems can be defined as: 𝜕𝜕𝑢𝑢

𝐽𝐽 = ∫Γ (Wn1 − 𝜎𝜎ij 𝑖𝑖 𝑛𝑛j ) dΓ = 𝜕𝜕𝑥𝑥 𝜌𝜌

= ∫𝐶𝐶+𝜔𝜔 (Wn1 − 𝜎𝜎ij

1

𝜕𝜕𝑢𝑢𝑖𝑖

𝜕𝜕𝑥𝑥1

𝑛𝑛j ) dΓ − ∫Ω(𝐶𝐶)

𝜕𝜕

𝜕𝜕𝑥𝑥3

(𝜎𝜎𝑖𝑖3

𝜕𝜕𝑢𝑢𝑖𝑖

𝜕𝜕𝑥𝑥1

) dΩ

(3)

where Γ𝜌𝜌 is a contour identical to C𝜌𝜌 but proceeding in an anti-clockwise direction (Fig. 6). The integral J is defined in the plane x3 = 0 for any position on the crack front. Considering a traction free crack, the contour integral over the crack faces ω is zero, whereas, when considering loaded crack faces, as for the LC approach here presented, the

contribution of ∫𝜔𝜔 −𝜎𝜎ij

𝜕𝜕𝑢𝑢𝑖𝑖

𝜕𝜕𝑥𝑥1

𝑛𝑛j dω has to be added.

Figure 6. Closed path around the crack tip (Wilson, 1979).

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For mixed-mode 3D problems, the J-integral is related to the three basic fracture modes through the components J I , J II and J III : J = J I + J II + J III

(4)

K eff = √(K I + |K III |)2 + 2K II 2

(5)

𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 = C∆𝐾𝐾 𝑛𝑛

(6)

In Rigby et al. (1998), a decomposition method is presented, where the integrals J I , J II and J III in elastic problems are directly calculated from J. The method for deriving the three separate K values from J can be found in Rigby et al. (1998). Once the three values for K 𝐼𝐼 , K 𝐼𝐼𝐼𝐼 and K 𝐼𝐼𝐼𝐼𝐼𝐼 are known, an effective K 𝑒𝑒𝑒𝑒𝑒𝑒 value (BEASY 2016) is calculated In final, the K 𝑒𝑒𝑒𝑒𝑒𝑒 is inserted in Eq. 6 to compute the da/dN. where C and n are listed in Tab. 1. 4. FEM analyses A FEM submodel (Fig. 1b), implemented by FRANC3D code, was used to simulate the crack-growth. To this aim, it was necessary to define a further domain to use as a remeshing zone where to insert and propagate the crack (Fig. 7). In particular, the code generates and insert the crack in the remeshing domain (Fig. 7) adopting tie constraints at the connecting surfaces, and eventually applying submodelling boundary conditions (in terms of displacements) on the submodel cut surfaces. Fig. 8 shows the FEM submodel when the initial crack is inserted. Subsequently, the FEM crack-growth procedure proceeds step-by-step: for each growth step, the code calculates the new crack front, inserts the new crack surface in the remeshing domain, connects such remeshed domain to the FEM submodel and solves it considering the boundary conditions coming from the FEM global model. All the models comprises purely linear tetrahedral elements. SIFs along the crack front were calculated by means of the J-integral technique whereas the crack path was assessed by means of the Maximum Tangential Stress (MTS) criterion (Erdogan et al., 1963).

* Figure 7. FEM-FEM strategy for the crack-growth simulation.

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Figure 8. FEM cracked submodel with highlight of the initial crack.

Figure 9. SIFs calculated along the front of the initial crack at J-path positions.

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5. Results 5.1. FEM-DBEM Using the LC approach, accurate SIFs can be computed even with coarse meshes as shown in Giannella et al. (2017b) or Citarella et al. (2016b). Moreover, no modelling of the blades is required when using such an approach. The DBEM model containing the initial crack comprises 2341 linear and quadratic elements. SIFs vs. normalized abscissa, calculated along the along initial crack front (Fig. 4c), are shown in Fig. 9. The step-by-step crack propagation simulation was set up with 25 steps, each comprehensive of the following steps:  crack adder (the crack was inserted into the DBEM uncracked model);  crack loading (the crack faces were loaded with FEM stress results);  solve (the DBEM model was solved and SIFs were computed);  crack life (the da/dN was computed from Eq. 6);  crack extend (the new crack front was predicted). For each step of crack advance, loads on the new crack surface generated during the propagation were automatically added, as provided by the global FEM analysis of the uncracked model. The average crack advance per step has been set equal to 0.15 mm for the initial 5 steps and gradually increased up to 1.5 mm for the last 5 steps. Figure 10 shows the DBEM model containing the propagated crack; such model comprises 4101 linear and quadratic elements. 5.2. FEM The FEM propagated crack in Figure 11 clearly shows a crack path almost overlapped to the one calculated with DBEM code. Nevertheless, both crack paths calculated numerically do not replicate the experimental crack paths shown in Fig. 3. Such differences could be attributed to non modelled effects like blade vibrations, fluid pressure, etc. but the modelling is under development and in the near future will also add such loading conditions

Figure 10. (a) DBEM submodel with the propagated crack; (b) close-up on the propagated crack; (c) through-the-thickness view of the crack; loads applied on the crack face elements in orange; J-path to calculate J-integral along crack front in purple.

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Figure 11. FEM submodel with the propagated crack.

6. Conclusions A coupled FEM-DBEM procedure, based on the superposition principle applied to fracture mechanics, has been adopted to simulate a fracture process on a real aircraft gas turbine engine. Such an approach is fully automated and allows to predict SIFs and then the whole crack-growth and path with high accuracy. The initial crack has been inserted in the most critical point, as acquired from the experimental outcomes, of the compressor stage of interest. A good agreement between the numerical crack paths calculated with FEM-DBEM and FEM-FEM approaches has been shown, although the numerical paths did not match the experimental paths, probably because of the lack of allowance for relevant loading conditions coming from e.g. pressure on blades, residual stresses, transients, blade vibrations, etc. Some of the aforementioned effects will be introduced in the near future to improve the correlation between numerical and experimental outcomes. References Maligno, A.R., Citarella, R., Silberschmidt, V.V., Soutis, C., 2015. Assessment of Structural Integrity of Subsea Wellhead System: Analytical and Numerical Study. Frattura ed Integrità Strutturale 31 97-119. Citarella, R., Lepore, M., Maligno, A., Shlyannikov, V., 2015a. FEM simulation of a crack propagation in a round bar under combined tension and torsion fatigue loading. Frattura ed Integrità Strutturale 31 138-147. Citarella, R., Lepore, M., Shlyannikov, V., Yarullin, R., 2014a Fatigue surface crack growth in cylindrical specimen under combined loading. Engineering Fracture Mechanics 131, 439–453. Citarella, R., Perrella, M., 2005. Multiple surface crack propagation: numerical simulations and experimental tests. Fatigue Fract Eng Mater Struct 28, 135–48. Calì, C., Citarella, R., Perrella M., 2003. Three-dimensional crack growth: numerical evaluations and experimental tests, in: ”European Structural Integrity Society”. In: Biaxial/Multiaxial Fatigue and Fracture, Edited by Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli, 31(3), 3504. Citarella, R., Giannella, V., Lepore, M., Dhondt, G., 2018. Dual boundary element method and finite element method for mixed‐mode crack propagation simulations in a cracked hollow shaft. Fatigue & Fracture of Engineering Materials & Structures 41(1), 84-98. Citarella, R., Cricrì, G., 2014b. Three-Dimensional BEM and FEM Submodelling in a Cracked FML Full Scale Aeronautic Panel. Appl Compos Mater 21 (3) 557-577. Citarella, R., Lepore, M., Fellinger, J., Bykov, V., Schauer, F., 2013. Coupled FEM-DBEM method to assess crack growth in magnet system of Wendelstein 7-X. Fracture and Structural Integrity 26, 92-103. Citarella, R., Cricrì, G., Lepore, M., Perrella, M., 2014c. Assessment of Crack Growth from a Cold Worked Hole by Coupled FEM-DBEM Approach. Key Engineering Materials 577-578, 669-672 Trans Tech Publications, Switzerland.

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