Multiple crack propagation with Dual Boundary ...

Key Engineering Materials Vol. 560 (2013) pp 129-155 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.560.129

Multiple crack propagation with Dual Boundary Element Method in stiffened and reinforced full scale aeronautic panels R. Citarella*, G. Cricrì*, E. Armentani# *

Dept. of Industrial Engineering, University of Salerno, Via Ponte Don Melillo, 1, Fisciano (SA), Italy, e-mail: [email protected], tel. +39089964111 # Department of Chemical, Materials and Production Engineering, University of Naples Federico II, Italy

KEYWORDS: DBEM, full scale aeronautic panel, 2D/3D crack growth, MSD, doubler-skin assembly, damage tolerance

Abstract In this work, the performance of a new methodology, based on the Dual Boundary Element Method (DBEM) and applied to reinforced cracked aeronautic panels, is assessed. Such procedure is mainly based on two-dimensional stress analyses, whereas the three-dimensional modelling, always implemented in conjunction with the sub-modelling approach, is limited to those situations in which the so-called “secondary bending” effects cannot be neglected. The connection between the different layers (patches and main panel) is realised by rivets: a peculiar original arrangement of the rivet configuration in the two-dimensional DBEM model allows to take into account the real inplane panel stiffness and the transversal rivet stiffness, even with a two dimensional approach. Different in plane loading configurations are considered, depending on the presence of a biaxial or uniaxial remote load. The nonlinear hole/rivet contact, is simulated by gap elements when needed. The most stressed skin holes are highlighted, and the effect of through the thickness cracks, initiated from the aforementioned holes, is analysed in terms of stress redistribution, SIF evaluation and crack propagation. The two-dimensional approximation for such kind of problems is generally not detrimental to the accuracy level, due the low thickness of involved panels, and is particularly efficient for studying varying reinforcement configurations, where reduced run times and a lean preprocessing phase are prerequisites. The accuracy of the proposed approach is assessed by comparison with Finite Element Method (FEM) results and experimental tests available in literature. This approach aims at providing a general purpose prediction tool useful to improve the understanding of the fatigue resistance of aeronautic panels. 1.

Introduction

The ability to determine the fatigue life for a damaged structure has become increasingly important with the advent of the damage tolerance criteria mandated by FAA (Federal Aviation Administration) regulations for ageing transport aircraft; consequently, the repair or reinforcing techniques have been compared on the basis of their fatigue behaviour performances [1]. Damage tolerance is part of the fail-safe methodology and stands for the concept of damage initiation and growth as it pertains to the fracture analysis methodology. According to airworthiness regulations such as FAR 25, a structural failure which cannot be detected must not reduce the residual strength of the structure below ultimate load (UL). In the damage tolerance (DT) domain a reduction of residual strength up to limit load (LL) is allowed, if the damage is detectable. Since the damage tolerance philosophy states that the extent of damage and detectability determines the required load level to be sustained, there is a strong, multidisciplinary relationship between structural design, inspection and maintenance philosophy and non-destructive testing (NDT), that has to be covered by advanced methods and tools.

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 193.205.162.117, University of Salerno, Fisciano (SA), Italy-29/04/13,11:51:53)

130

Advances in Crack Growth Modeling

In order to develop an effective methodology for repair or reinforcement riveted panels, it is important to be able to accurately determine the complex stress field created by the added patches as well as, in case of crack initiation, the resulting reduction in the stress intensity factors (SIFs). Numerical simulations are useful to identify the most fatigue critical locations where to check the effect of possible cracks by simulating the single or multiple crack propagation under a given spectrum load. Many examples of BEM (Boundary Element Method) models applied to multiple crack propagation analyses can be found in the recent literature. Plate and shell analysis is a widely used approach in typical engineering practice [2-4] as a fully three-dimensional approach is typically too computationally demanding (even if, as will be shown in this paper, sometimes also a simple twodimensional approach can be quite effective). In [5], problems involving multiple crack–hole interaction and multiple crack interaction have been analysed; moreover, coalescence and localisation phenomena were also considered in the framework of plane applications of fatigue calculations; the authors used the Paris Law in evaluating the crack increments and the maximum circumferential stress criterion for calculating the local crack direction. Moreover, in [6] the DBEM model was applied by the authors coupled with a stochastic evaluation of the material strength parameters. They analysed a plate with an array of holes, from which cracks can start, submitted to traction fatigue load. Even more spread than the BEM , the finite element method (FEM) has often been used to analyse single or multiple crack growth problems. The finite element framework is particularly attractive for it provides the ease and flexibility of modelling complex structural components with arbitrary boundary conditions, nonlinear material behaviour, and anisotropic material properties. In spite of the successes achieved with finite elements in computational fracture mechanics, mesh generation in three-dimensions for crack growth simulations is still a formidable task. This is because in order to capture the evolution of the crack in fatigue growth simulations, remeshing along with local refinements around the crack front are required to obtain accurate solutions for fracture parameters (e.g. stress intensity factor or J-integral). Nevertheless, multiple crack growth plane problems can be successfully faced with this classic technique as was done, for instance, in [7]. In their work, the authors dealt with simulation of arbitrary shape crack growth in multiple cracked brittle bodies using an automated adaptive remeshing procedure. They also performed a stability analysis to determine active cracks from a set of competitive cracks and compared various crack growth criteria together with the respective crack trajectory prediction. However, standing the difficulty to apply and to keep under control any automated remeshing algorithm, extended finite element method (XFEM) and meshless methods are widely and successfully applied to crack propagation analyses in the last years. These techniques allow the entire crack to be represented independent of the mesh, thus remeshing is not required to model crack growth [8, 9]. Among many works, in [8] the authors apply the XFEM method to the problem of fatigue crack growth of multiple cracks in a quasi-brittle material in a plane stress model. They used linear elastic fracture mechanics and the Paris Law to drive the crack growth increment; the crack growth direction was determined based on the maximum hoop stress criterion. While BEM can also treat this class of problems, the extended finite element method has the advantage that it can readily be applied to inhomogeneous problems. It can also readily be applied to nonlinear problems, in contrast to BEM. Finally, coupled FEM-BEM methods are applied in some cases, as described in [10]:the authors dealt with the multiple crack propagation in plain concrete using the Finite Element – Scaled Boundary Finite Element coupled method (FEM–SBFEM). The scaled boundary finite element method (SBFEM) is a semi- analytical, numerical procedure developed by Song and Wolfin in 1996 [11]. It combines the advantages of both the FEM and BEM, and is very efficient in solving problems of unbounded media and problems with discontinuities and singularities. The method was recently coupled with the FEM by using cohesive interface elements to model cracks within the context of nonlinear fracture mechanics, resulting in the so-called FEM–SBFEM coupled method. In modelling mixed-mode multiple crack propagation in concrete, this method was found to exhibit

Key Engineering Materials Vol. 560

131

several advantages, the major of which are: (1) starting from analytical stress solutions, SIFs can be accurately calculated without usage of fine element meshes or special singularity elements; (2) the remeshing is much simpler compared to those used in classical FEM; (3) problems can be solved with substantially fewer degrees of freedom (DOF). On the other hand, this method is not easy to implement into industrial codes and its efficiency is to be demonstrated for complicated geometries. In this work, different configurations of mechanically fastened doublers for a damaged or reinforced aircraft skin are analysed in order to compare their fatigue performance. In particular an innovative DBEM (Dual Boundary Element Method) modelling approach was designed, using a commercial code (BEASY) together with in house made routines, capable of explicitly modelling the different test article layers with their rivet connections, even in a two-dimensional approach. The methodology is applied to the following test articles:  a riveted patch repair, applied on a cracked panel which undergoes a multiaxial spectrum loading;  a special multi-layer and multi-material specimen with a Multi Site Damage (MSD) scenario, obtained by cutting a rectangular notched area from the surrounding of the upper left corner of a wide body aircraft door (A380);  a riveted repair flat aeronautic panel, realised and tested in the context of the European research project “IARCAS” (VI framework); this panel is assembled in such a way to simulate the in service repairs, with doublers riveted over corresponding cut-out;  a flat stiffened full scale panel, designed as a fibre metal laminate (FML) in the context of the European research project “DIALFAST”, where it was tested under both static and fatigue biaxial loads. The results of the proposed DBEM procedure are validated by comparison with Finite Element Method (FEM) and experimental tests.

2. Theoretical aspects of crack growth with DBEM 2.1 Displacement and traction equations The present work is concerned with the numerical implementation of DBEM [12-13] for two- and three-dimensional analyses of crack-growth problems examined through the linear elastic fracture mechanics (LEFM). The DBEM, a particular variant of BEM, has successfully been applied to the incremental analysis of crack-extension problems. DBEM approach incorporates two independent boundary integral equations: the traction equation applied when the collocation point is on one of the two coincident crack surfaces and the displacement equation applied on the other crack surface and on all the remaining (non singular) boundaries. In the absence of body force, the displacement boundary equation (Eqn. 1) can be written as [12]: ( ) ( )

(

⨍

) ( )

(

∫

) ( )

(1)

where i,j denote Cartesian components, Tij (x’,x) and Uij(x’,x) represent the Kelvin traction and displacement fundamental solutions at a boundary point x, respectively. The symbol ⨍ stands for the Cauchy principal value integral and ( ) * ij for a smooth boundary. Assuming continuity of both strains and tractions at x’ on a smooth boundary, the stress components ij are given by (Eqn. 2): ( )

⨍

(

) ( )

⨎

(

)

( )

(2)

where Skij(x’,x) and Dkij(x’,x) contain derivatives of Tij (x’,x) and Uij(x’,x), respectively [12]. The symbol ⨎ stands for the Hadamard finite part of an integral.

132

Advances in Crack Growth Modeling

The traction components tj are given by the traction equation (Eqn. 3): ( )

( )⨍

(

) ( )

( )⨎

(

)

( )

(3)

where ni (x’) denotes the component of outward unit normal to the boundary at x’. Equations (1) and (3) constitute the basis of DBEM. The dual boundary integral equations and the crack modelling strategy is presented in [12] and in [13-14] for two- and for three-dimensional problems respectively.

2.2

The J–integral method

The J-integral technique, based on a path-independent integral and adopted for the stress intensity factor evaluation is presented in [12] and in [13-14] for two- and for three-dimensional problems respectively. Alternative approaches for SIF’s evaluations are presented in [15]. Consider a Cartesian reference system, with the origin at the tip of a traction-free crack and the x axis aligned with the crack. The J-integral is defined in eqn (4)

J   (Wn1 - t ju j,1 ) dS S

(4)

where S is an arbitrary circular contour surrounding the crack tip; W is the strain energy density, given by σij*εij/2, where σij and εij are the stress and strain tensor components, respectively; tj are traction components, given by σij*ni, where ni are the components of the unit outward normal to the contour path. Under linear elastic hypothesis the relationship between the J-integral and the SIFs (KI=opening mode I and KII=shear mode II) is given by eqn (5), where E’=E for plane stress conditions and E’=E/(1-ν2) for plane strain conditions (E is the elasticity Young modulus): J=(KI2+ KII2)/E’

(5)

In order to decouple the SIFs, the J integral is represented by the sum of two integrals as follows: J=JI+JII

(6)

but it is necessary to decompose the displacement and stress fields into their symmetric and antisymmetric components, and use the symmetric components to calculate J I and the antisymmetric ones for JII. Finally, the following relationships hold: JI= KI2/E’ and JII= KII2/E’

(7)

The implementation of this procedure into the boundary element method is straightforward. A circular contour path around the crack tip is defined with a set of internal points located at symmetrical positions with respect to the crack axis. The two contour points on the crack faces are the first point and the last point of the path, respectively. In a circular path, at these points it is always verified that n1=-1 and n2=0 and thus, for a traction-free crack, we have t2=0. The integration along the contour path can be accomplished either with the trapezoidal or Simpson’s rule or by Gaussian quadrature. For the sake of simplicity, only circular paths centred at the crack-tip and containing a pair of crack nodes were considered.

Key Engineering Materials Vol. 560

2.3

133

The incremental direction.

The Maximum Principal Stress Criterion and the Minimum Strain Energy Density criterion [16] are applied to predict the tangent direction of the continuous crack path for respectively two- and threedimensional problems. An incremental crack-extension analysis is performed to determine the crack path, with no need for an extensive remeshing during three dimensional crack propagation because the crack extension is modelled by just adding new boundary elements along the previous crack front, with the only mesh surrounding the break through points being updated. The explicit expression of strain energy density around the crack front can be written as:

dW S     1 dV r  cos 

(8)

where S() is given by

S    a11  K I2  2a12  K I  K II  a22  K II2  a33  K III2

(9)

and

1  3  4  cos    1  cos   16 1 a12   sin  cos   1  4  8 1 a22   41     1  cos    3 cos   1  1  cos   16 1 a12  4 a11 

in which stands for the shear modulus of elasticity and  is the Poisson ratio. S/cos represents the amplitude of intensity of the strain energy density field and it varies with the angle  and . It is apparent that the minimum of S/cos always occurs in the normal plane of the crack front curve, namely =0. S is known as strain energy density factor and plays a similar role to the stress intensity factor. The theory for three-dimensional cracks, is based on three hypotheses:  the direction of the crack growth at any point along the crack front is toward the region with the minimum value of strain energy density factor S as compared with other regions on the same spherical surface surrounding the point.  crack extension occurs when the strain energy density factor in the region determined by hypothesis S = Smin, reaches a critical value, say Scr.  the length, ro, of the initial crack extension is assumed to be proportional to Smin such that Smin/ro remains constant along the new crack front. It can be seen that the minimum strain energy density criterion can be used both in two and three dimensions. Note that the direction evaluated by the criterion in three dimensional cases is insensitive to KIII. The crack growth direction angle o is obtained by minimising the strain energy density factor S() of Eq.(9) with respect to . The minimum strain density factor S(o) is denoted by Smin. Locating the minimum of S() with respect to  is carried out numerically using the bisection method by solving:

134

Advances in Crack Growth Modeling

dS   0 d

   

The equivalent stress intensity factor, Keq, can be calculated from the minimum strain density factor as: K eq 

2.4

4  S min 2  1

(10)

The incremental size.

Determination of the incremental size poses two problems: the first is the determination of the amount of each increment in terms of a reference size, the second is the relationship between the maximum incremental size and other incremental sizes along the crack front. It is stated in the hypothesis (3) of the strain energy density criterion that the length ro of the initial crack extension is assumed to be proportional to Smin such that Smin/ro remains constant along the crack front. Since the strain energy factor Smin is proportional to the square power of the equivalent stress intensity factor Keq, the incremental size at the crack front point under consideration is given by:  K eq   a  a max   max K  eq  

2

(11)

where max{Keq} is the maximum SIF equivalent evaluated at a set of discrete points along the front, and amax is the incremental size at the point corresponding to the max{Keq} which is chosen beforehand as being the maximum distance from the crack front to the opposite side of the element containing the crack front. The above expression is used for standard crack growth. 2.5 Fatigue Growth Calculation. During fatigue crack growth, the relation between the incremental size and the number of load cycles may be represented by a number of crack growth laws, such as PARIS, FORMAN, RHODES or NASGRO. Alternatively, a tabulated form can be used to supply. During the fatigue analysis, there are options on the method of computing the dN values:  the SIF’s are constant over the step;  the SIF’s at the previous step and the current step are used to compute the dN value over the last crack growth step; this requires two analysis to be performed before the first dN value is computed (backward correction);  the previous two results are used to predict a guesswork for the dN over the next step; this may be inaccurate as the SIF’s may change significantly over the step (forward prediction).

3. 3.1

Problem description and results Problem 1: DBEM analysis on a non-linear multiple crack propagation in a doubler-skin assembly [17] The performance of a riveted patch repair (realised to remove prior damage), applied on a cracked panel (Fig. 1a), is simulated by using a DBEM procedure. A two-dimensional stress analysis on a single-sided repaired configuration is performed, neglecting the effects of secondary bending.

Key Engineering Materials Vol. 560

135

x  x

y

2

1

R

3

Wd

eg

Doubler 4

Ld

Rivets

P Ws

Wc

R

Ls Ws Ld=Wd Lc=Wc R P eg

265 80 70 16 10 20 10

mm mm mm mm mm mm mm

Skin

Lc Ls

Fig. 1a: Skin-Doubler assembly: only one quarter is shown because of the symmetry. (Problem 1) The connection between the two layers (patch and panel) is realised by 32 elastic rivets, with through-cracks (with a length equal to 1 mm) initiated on the most loaded holes (Figs. 1b-c), where the experience suggests the highest recurrence of crack initiations: as a matter of fact the optimization attempts are mostly concentrated on the corner hole N. 6, as showed by the outcomes of the European project on repairs “IARCAS”.

1

3

6

2

4

7

5

8

INITIATION POINTs

Fig. 1b: Skin maximum principal stress (MPa) evaluated by DBEM (left) and most stressed crack initiation points on hole N. 6 (right). (Problem 1)

136

Advances in Crack Growth Modeling

4

5

2

3

1

Fig. 1c: DBEM model deformed shape and Von Mises skin stresses [MPa] under biaxial traction load (the numbering here is related to cracks). (Problem 1) In alternative, an analytic approach to assess crack initiation can be provided, e.g. by an isotropic elastic-damaging model, implemented by use of integral equations to deal with localization phenomena that precede the crack initiation [18].The nonlinear hole/rivet contact is simulated by gap elements with a rivet-hole interference i=0 (in real applications there is a slight interference of about 0.25% of the pin radius, but this has no effect on the SIF values for the through cracks analysed in the following). In particular, the two plates (belonging to two different zones of the BEM model), attached by the rivet connections, are both explicitly modelled herein, and the connection is modelled by two circles representative of the two halves of the rivet projected on the medium plane. Such circles (belonging to two further zones of the BEM model) are connected to each other by internal springs (Fig. 2), with a stiffness (Kx=Ky=2.9E+4 N/mm) corresponding to the shear rivet stiffness, and are engaged with the two corresponding skin and doubler holes by the interposition of gap elements.

+

=

Fig. 2: Rivet half engaged with skin (left) and with doubler (centre) to form the assembled rivet (a fictitious hole is modelled in the pin center in order to be able to connect the two halves of the pin itself by internal springs). (Problem 1-4) The assembly of the final linear BEM system is obtained imposing continuity and equilibrium conditions at the interface between the different considered zones and specifically at the pin-hole interface, that is the only part of the boundary in which there is an indirect (through the pin) interaction between the overlapped plates. Different loading configurations are considered: a biaxial remote fatigue traction load with x=y=100 MPa (Figs. 1b-c) and a biaxial fatigue load (x=50 MPa,y=100 MPa) combined with a static shear load (=40 MPa) as shown in Fig. 3. In the latter case the multiaxial loading is nonproportional, so that the stress principal directions are varying along the combined load cycle; in such case it is important to point out that the crack path is calculated based on the SIF values calculated when the fatigue remote loads attain their maximum values (the two remote traction loads are synchronous). In all cases the stress ratio is R=min/max=0.1.

Key Engineering Materials Vol. 560

137

1

3 2 1

Fig. 3: DBEM maximum principal stresses (MPa) under a biaxial fatigue load combined with a static shear load. The fatigue critical point (up) and the multiple crack path (down) are highlighted. (Problem 1)

The most stressed skin holes are highlighted and the effect of through cracks from such holes is analysed in terms of SIFs vs. crack length and crack growth rates: in particular the behaviour of crack N. 1 is illustrated in Figs. 4-5, under a combination of static shear load and biaxial fatigue traction load (load case N. 1);

SIF Value

Stress intensity Factor vs. crack length

700 600 500 400 300 200 100 0 -50 0

Load Case index = 1: SIF1 Load Case index = 1: SIF2

1

2

3

4

5

crack length

6

7

8

Fig. 4: SIF values (MPa*mm1/2) vs. crack length (mm) for crack N. 1. (Problem 1)

4.5E-04 4.0E-04

da/dN Value

3.5E-04 3.0E-04 2.5E-04 2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

Crack Length

Fig. 5: Crack growth rates (mm/cycle) vs. crack length for crack N.1 under combined biaxial fatigue load and static shear (load case N. 1). (Problem 1)

138

Advances in Crack Growth Modeling

The fatigue properties of the 2024 T3 material are taken from the NASGRO 3 database (1 Clad; plt & sht; L-T [M2EA11AB1]).The spectrum effect related to the presence of an overload intermingled in the baseline sequence is taken into account by using a retardation model (Generalised Willenborg model) superimposed to the NASGRO 3 [19] law (Eqn. 12: in particular, one cycle with maxOL=2maxBL and minOL=minBL (OL=overload, BL=baseline) was initially applied before starting with the baseline fatigue load, and its effects are clearly visible, with an initial crack growth rate that is very low and, at the second crack propagation step, suddenly increases because of the vanishing retardation effect (Fig. 5).  Kth  1   da  1  f    K   C   K   q dn  1  R    Kt max  1   K c   p

n

(12)

Under mixed mode conditions the calculation of an equivalent K is mandatory: in this case the Yaoming Mi formula [20] was adopted. In [17], a comparison between the SIFs values and an assessment of computational and preprocessing efforts, as provided by DBEM and FEM, are available. It is worth to point out that more effective approaches (alternative to the usage of the empirical aforementioned retardation models) to take into account the spectrum effects, caused by induced residual stresses, are available in the literature [21-24]. 3.2 Problem 2: Non linear MSD crack growth for a riveted aeronautic reinforcement [25-27] A special specimen was created cutting a rectangular notched area from the surrounding of the upper left corner of a wide body aircraft (A380) door (Fig. 6a). Test article

Fig. 6a. Upper corner of an aircraft door (the depicted points are indicative of rivet connections). (Problem 2)

Key Engineering Materials Vol. 560

139

Test article A

A

SECT. A-A

Fig. 6b. Test article section with highlight A of strain gauge positions. (Problem 2) This part of the aircraft skin is made of three different layers made of different materials (one titanium Ti-6Al-4V and two aluminium 2524-T351 layers) and with variable thickness (Fig. 6b). Then a fatigue traction load is applied by a special servo-hydraulic machine (Fig. 6c) and two cracks, whose initiation position is predetermined by preliminary notches, appear in the skin and in the aluminium reinforcement (after a given number of cycles) in the same position, reproducing a realistic crack scenario (Fig. 7). Such through cracks are monitored during their propagation along the specimen width, providing a realistic initial scenario for the DBEM model (Fig. 8) and experimental propagation data, useful for the correlation with the simulated crack path and growth rates.

Figure 6c. Loading frame with clamped specimen undergoing fatigue load. (Problem 2)

140

Advances in Crack Growth Modeling

Fig. 7. Final cracked scenario: only the skin crack is visible but there is also an underlying crack in the aluminium reinforcement patch. (Problem 2)

Explicitly modelled rivets

Extruded part (blue line)

Fig. 8. Two-dimensional DBEM model with highlight of rivets (yellow and red symbols) . (Problem 2)

Key Engineering Materials Vol. 560

141

The DBEM model shown in Fig. 8 is used as such for the second part of crack propagation, whereas for the initial part the central area (delimited by the blue line) is extruded and modelled as threedimensional. In the two-dimensional model, the reinforcement part that is far from the cracked area is modelled as a single panel (only one BEM zone) with a thickness that is equal to the sum of the thicknesses of joined aluminium panels, whereas the central part, closed to the cracked area, is modelled by separate layers (different zones with specific mechanical properties) joined by titanium rivets with a diameter equal to 9.5 mm. Such rivets interact with the joined plates by gap elements (red symbols in Fig. 8) in the area adjacent to the cracks and by continuity conditions (yellow symbols in Fig. 8) in the surrounding area. The initial crack length in the skin and in the aluminium doubler are respectively equal to 3.25 mm and 3 mm. The crack growth rates are provided by the closure corrected Paris formula: da/dN= C (Keff )m=C(U(R))m(K)m = C’ (K)m

(13)

where  Keff=U(R)* K=(0.54+0.46*R+2.37*R2)* K

(14)

and C’=C* U(R)m

(15)

The fatigue constants for the Al 2524 T351 are: C’=1.26e-13 and m=3.5, when K is expressed in MPa*mm1/2 and da/dN in mm/cycle. The results of the two-dimensional simulation (Figs. 9a-b) exhibit a satisfactory correlation with the experimental crack path (Fig. 7) and growth rates (Fig. 9b).

Skin crack

Titanium doubler

doubler crack

Fig. 9a. Cracked area by DBEM approach (the titanium doubler remains intact). (Problem 2)

Advances in Crack Growth Modeling

crack length

142

170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

Experimental Crack Size Numerical Crack Size

0

5000

10000

15000

20000

cycles

Fig. 9b. Numerical (2d) and experimental crack length vs. cycles. (Problem 2) The comparison in Fig. 9b starts with a skin crack length equal to 9.5 mm because non negligible numerical-experimental discrepancies appear in the initial stage of crack propagation, and consequently, for such “short” crack range, a DBEM three-dimensional simulation [28-33] is attempted in order to properly allow for secondary bending effects. In particular, in order to improve the correlation in the initial stage of crack propagation, the cracked part of the global twodimensional model (Fig. 8) is extracted and “extruded” in order to generate a three dimensional submodel (Fig. 10), whose boundary conditions are imposed displacements, calculated along a virtual line corresponding to the submodel boundary (blue line in Fig. 8) by a two dimensional solution.

Fig.10: Initial three-dimensional cracked configuration with boundary conditions and Von Mises stresses [MPa]: external and internal view. (Problem 2) Nonlinear contact conditions are applied between the plate surfaces in the area surrounding the cracks, in order to precisely model the plate interactions in the area of major interest (Fig. 11).

Fig. 11. DBEM deformed plot of the reinforcement after the first step of crack propagation. (Problem 2)

Key Engineering Materials Vol. 560

143

The improvements on crack growth rates, in the initial part of the crack propagation (Fig. 12), justify the increased computational effort of such three dimensional approach. The experimental skin crack line is shifted in order to highlight that the lack of numerical-experimental correlation is limited to the initial crack propagation phase: as a matter of fact, when the crack becomes higher than 6 mm the inclinations of experimental and numerical (from three-dimensional calculus) curves are similar. Num 3D SKIN (z=3 mm)

Num 3D DOUBLER z=12 mm

Num. 2D Skin

Num. 2D Doubler

Num 3D SKIN (z=0)

EXPERIMENTAL SKIN CRACK

Crack Length (a)

EXPERIMENTAL SKIN CRACK (SHIFTED)

10,0 9,5 9,0 8,5 8,0 7,5 7,0 6,5 6,0 5,5 5,0 4,5 4,0 3,5 3,0 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 Number of Cycles (N)

Fig. 12. Crack length vs. number of cycles in the initial stage of crack propagation: comparison between two/three-dimensional numerical results and experimental data. (Problem 2)

3.3

Problem 3: FML Full Scale Aeronautic Panel Under Multiaxial Fatigue: Experimental Test and DBEM Simulation [34] This section concerns the numerical and experimental characterization of the static and fatigue strength of a flat stiffened panel, designed as a fibre metal laminate (FML) and made of Aluminium alloy and Fibre Glass FRP (Table 1). The panel, whose dimensions are 2181 x 2181 mm (excluding the aluminium gripping plates), consists of three bays joined together by butt-straps and z-shape stringer coupling; windows cut-outs are included in the structure. The stringer pitch and the frame pitch are equal to, respectively, 172.3 mm and 533 mm. The panel is made of two parts: an upper and a lower panel, joined by a lap joint at the stringer N.6. The frames are applied on both panel sides to minimize the secondary bending effects. The rivet compliance is calculated with the Swift formula: C

 1 5 1    0.8    D  E3 t  E t  2 E2   1 1

(16)

ti=thickness of connected panels, E1,2=Young modulus of the panels, E3=rivet Young modulus, D=rivet diameter. The panel is full scale (Fig. 13) and is tested under both static and fatigue biaxial loads (Fig. 14), applied by means of a multi-axial fatigue machine [35-36].

144

Advances in Crack Growth Modeling Table 1 Skin and stringer lay-up (A) and adopted material (B). (B) (right). Table 1: Skin(A) and stringer lay-up (left) and adopted material PLY

MATL

P1 P2 P3 P1 P3 P2 P1

LAMINA F/G F/G LAMINA F/G F/G LAMINA

SKIN ORIENT. N/A 0° 90° N/A 90° 0° N/A

STRINGER ORIE. N/A 0° 0° N/A 0° 0° N/A

THK [mm]

Component

Material

0.3 0.125 0.125 0.3 0.125 0.125 0.3

Lamina Skin FG Prepreg

Alloy 7475 – T761 FG FM 94-22% S2 GLASS – 187-460 Alloy 2024 – T3 CLAD Alloy 2024 – T3 CLAD Alloy 7075 – T651 Alloy 6056 – T4

Frame Shear cleats Window frame Plates

Figure 13. A330-300 fuselage barrel tested within the European project “DIALFAST”. (Problem 3)

Figure 14. Tested panel loaded by a multi-axial fatigue machine. (Problem 3) The static and fatigue tests are simulated by the Dual Boundary Element Method (DBEM) in a two-dimensional approach (only allowance for membrane stresses). The strain gauge (Fig. 15) outcomes from the static test, are compared with corresponding numerical results, getting a satisfactory correlation as shown in [34].

Key Engineering Materials Vol. 560

145

Figure 15. Strain gage and rosette configuration on the internal panel side (A). (Problem 3)  After the static test, an initial notch is created in the panel (Fig. 16) and a biaxial fatigue load is applied (Fig. 17), causing a crack initiation and propagation in the metal layers. The bi-axial fatigue test is characterised by Pxmax=125 kN and Pymax=250 kN, (frequency equal to 1 Hz) applied in both directions with a load ratio R = 0.04.

Figure 16. Notch and crack gages. (Problem 3)

Figure 17: DBEM deformed plot of the notched panel. (Problem 3)

146

Advances in Crack Growth Modeling

An original approach based on Dual Boundary Element Method (DBEM) is proposed to simulate the fatigue crack growth in the FML metal layer [34]. The initial experimental cracked scenario is based on two cracks on the two sides of the notch (Figure 4): with a length equal to 2 mm on the left side (C.G.2.) and 4 mm on the right side (C.G.1.). The boundary element mesh of the notched panel (no crack initiation yet) together with loading conditions are shown in Fig. 17, whereas the Maximum Principal Stresses are shown in Figs. 18a-b.

Figure 18a. DBEM deformed plot of notched panel with Maximum Principal stresses (MPa). (Problem 3)

Figure 18b. Maximum principal stresses (MPa) on a close up of the notched area: with subtraction of the delaminated area (up); composite delaminated part (centre); aluminium delaminated part (down). (Problem 3) Even if the panel is made of alternate bonded composite and metal layers, only one panel is modelled in the numerical analysis (Fig. 17), with global thickness and mechanical properties calculated with the “rule of mixture”. In particular: a) the skin (FML3 3/2 0.3) is approximated as isotropic because the fibre direction in the composite layers is alternatively at 0 and 90 degrees and the related mechanical properties (Young modulus E=52788 MPa and Poisson coefficient =0.31) are experimentally evaluated (using a small specimen with the same composition);

Key Engineering Materials Vol. 560

147

b) the stringer are still made of FML but this time the fibre orientation is always along the stringer direction (1) and consequently the exhibited mechanical behaviour is orthotropic (E1=61455 MPa, E2=42913 MPa, G12=20148 MPa, =0.31). In the BEM model such stringers are merged with the underlying skin, in such a way to define a single “zone” whose mechanical orthotropic properties are obtained with the “rule of mixture” (E1=59465 MPa, E2=45179 MPa, G12=20148 MPa, =0.31), whereas the zone thickness is obtained by imposing the same overall cross-sectional area. The aforementioned approach is applied to all the peculiar assemblies between metal and composite layers and is enabled by the code capability to solve for orthotropic (uncracked) domains. The available formulation is not capable to simulate crack propagation in orthotropic medium but this is not a limitation in our problem because the propagation is just simulated in the FML metal layer. The different FML layers are explicitly modelled only in the cracked area (Fig. 18b), as different zones sewed along the boundary by rivet rows (with the same system showed in par. 3.1). In Fig. 19 a contour plot with Maximum Principal stresses (MPa) is shown with reference to the delaminated area after 14960 cycles of propagation.

Figure 19: Maximum Principal stresses (MPa) in the delaminated area with highlight of stresses in the cracked aluminium layer; the high stress values in the underlying composite layer are partially visible behind the crack. (Problem 3) The geometry and dimensions of the delaminated area were not available from experiments so they were assessed by a ”mixed” approach based on qualitative information from literature [37-38] and on a calibration process aimed at minimisation of numerical-experimental crack growth rate discrepancies. Namely, such delaminated area was changed up to reaching a satisfactory correlation between numerical and experimental crack growth rates (Fig. 20). Due to the limited overall crack advance it is possible to assume that the fibres in the composite layers do not undergo rupture up to the final crack scenario considered. Such hypothesis is confirmed by the always decreasing experimental crack growth rates in the considered range of propagation (Fig. 20). A further approximation comes from considering the extension of the delaminated area fixed during the crack propagation: this can be acceptable due to the limited overall crack advances considered. A sensitivity analysis, to evaluate the impact of a changing delaminated area extension on the crack growth rates is available in [39].

148

Advances in Crack Growth Modeling

The crack growth law adopted for propagation in the aluminium layer is the NASGRO 4.0 (Eqn. 12) (whose calibration parameters are those related to AL 7475 T761). In Fig. 21 SIF’s are shown in correspondence of different crack sizes: the KI values are decreasing as a consequence of the bridging effects and the mode II is negligible.

Figure 20: Crack size (mm) vs. number of cycles for the two cracks. (Problem 3)

Figure 21. SIF’s vs. crack size for the two cracks. This approach aims at providing a general purpose prediction tool for a better understanding of the fatigue resistance of FML panels, providing a deeper insight into the roles of the fibre stiffness and of the delamination extension on the stress intensity factors [40]. The experimental test, realized in the context of a European research project (DIALFAST), is consistently reproduced by the proposed procedure (with the limitations previously mentioned on the delamination area assumption). 3.4 Problem 4: MSD crack propagation by DBEM on a repaired aeronautic panel [41] This section focusses on the use of the DBEM procedure to investigate the damage tolerance performance of a riveted repair flat aeronautic panel (Figs. 22a-b), realised and tested in the context of the European project “IARCAS” (VI framework). Such panel is assembled in such a way to simulate the in service repairs, with doublers riveted over corresponding cut-out. Skin and doubler have the material properties of Al 2024 T3 kin and doubler have the material properties of Al 2024 T3ct “IARCAS and Al 2017-NAS 1097. The assembly component dimensions and rivet mechanical properties are reported in Table 2. The panels, repair patches and rivets are modelled in a two-dimensional analysis with no allowance for out-of-plane bending (Fig. 23), with edge-cracks initiated from some skin rivet holes and growing due to fatigue load (Fig. 24a-c). The DBEM analyses are fully linear because, due to the relevant length of the initial considered cracks, the non-linearity coming from the pin-hole contact can be neglected.

Key Engineering Materials Vol. 560

149

Fig. 22a. Panel in testing configuration: front view (left) and backward view (right), with highlight of the analysed repairs (REP3, REP4, REP5). (Problem 4)

Fig. 22b: Drawing of the different riveted repairs. Table 2. Assembly component dimensions and rivet mechanical properties. Repair REP 3 REP 4 REP 5

Skin thickness (mm) 1.65 1.65 1.65

Doubler thickness (mm) 1.8 1.8 1.8

Stiffener section (mm2) 99 99 99

 rivets (mm) 4.0 4.0 4.0

Rivet transversal stiffness (Kx, Ky) in N/mm 2.9E+4 2.9E+4 2.9E+4

150

Advances in Crack Growth Modeling

Fig. 23. DBEM model of the repaired panel with highlight of analysed repairs, mesh and boundary conditions. (Problem 4)

Crack 1

Crack 2

Fig. 24a. Repaired skin by REP 3, with two cracks initiated from the corner hole. (Problem 4)

Crack 3

Crack 4

Fig. 24b. Repaired skin by REP 4 (left), with close-up of the damaged area (right), where two cracks initiated from the corner hole, are visible. (Problem 4)

Fig. 24c. Repaired skin by REP 5, with two cracks initiated from the skin corner hole. An arrow highlight the hole undergoing a position optimisation. (Problem 4)

Key Engineering Materials Vol. 560

151

In the DBEM model the layers representative of each component are overlapped but distinct (Fig. 23), with clearly no allowance for the existing offset and corresponding secondary bending effects; in other words the medium planes of each layer are geometrically coincident. In the numerical model, the skin and the repairs in the cracked area, are modelled by different patches, and the rivet connection between them is again (see par. 3.1) modelled by two (or more) circles, representative of the two (or more) parts in which the rivet is being divided, respectively engaged with the corresponding skin and doubler holes (Fig. 2). The same approach is adopted to connect the stiffeners to the skin. The repairs in the non-cracked areas are modelled as they were bonded to the skin, considering a local increase in the skin thickness corresponding to the repair thickness; only in the critical area affected by the crack propagation the rivet connection between doublers, skin and stiffeners is explicitly modelled, by different zones with different thicknesses and material properties. In particular, only the rivets corresponding to repairs N. 3 to 5 and to that part of the stiffener located between REP 3 and REP 4 are explicitly modelled. An example of possible results of such analysis is provided in Fig. 25, showing the crack scenario and related Von Mises stresses after 110000 fatigue cycles with a remote fatigue load applied equal to Pmax=28405 N with a stress ratio R=0.1; in particular the stiffened plate deformed plot with Von Mises stresses is shown, with highlight of the skin, the underlying reinforcement patches and the stiffeners. The contour plot shows not only the skin stresses but also the stringers and repairs stresses, as partially visible through the open crack. The propagation analysis is based on the NASGRO3 formula (eqn. 12), where the fatigue material properties, are related to the Al 2024 T3 (all the analysed cracks will affect the only skin). The numerical and experimental crack advances are compared in Figs. 26-28.

Fig. 25. Stiffened plate deformed plot (scale factor 15) with Von Mises stresses (MPa); the plots on the right hand side represent details of some critical zones of the plot on the left hand side as pointed out by arrows. (Problem 4)

152

Advances in Crack Growth Modeling

C R AC K S O N R E P AIR 3 80 70

C R AC K L E NG T H

60 50 40 crack 2 crack 1 s tringer s tringer crack 2 crack 1 -

30 20 10 0 97900

99400

experimental experimental

numerical numerical

100900

102400

103900

105400

106900

108400

CY CLES

Fig. 26. Numerical-experimental correlation for the growing cracks (length in mm): the crack N. 2, in the final part, propagates under a still resistant stiffener. (Problem 4) C R AC K S O N R E P AIR 4 110 100 s tringer s tringer crack 4 crack 3 crack 3 crack 4 -

90 80 C R AC K L E NG T H

70 60

experimental experimental numerical numerical

50 40 30 20 10 0 97900

99400

100900 102400 103900 105400 106900 108400 109900 111400 112900 CY CLES

Fig. 27. Numerical-experimental correlation for the growing cracks (length in mm): the crack N. 3, in the final part, propagates under a still resistant stiffener. (Problem 4) C R AC K S O N R E P AIR 5 55

C R AC K L E NG T H

50 45

crack 5 - experimental

crack 6 - experimental

40

crack 5 - numerical

crack 6 - numerical

35

s tringer

s tringer

30 25 20 15 10 5 0 97900

99400

100900

102400

103900

105400

106900

108400

109900

111400

CY CLES

Fig. 28. Numerical-experimental correlation for the growing cracks (length in mm). (Problem 4) The two-dimensional approximation for this problem is not detrimental to the accuracy of numerical-experimental correlation, so it become useful to study varying repair configurations, where reduced run times and a lean pre-processing phase are prerequisites.

Key Engineering Materials Vol. 560

4.

153

Conclusions

This paper shows the wide range of possibilities offered for this kind of problems, by the described DBEM approach, in terms of easy pre- and post-processing, reduced run-times and satisfactory accuracy. The aircraft design approach is nowadays mainly based on global-local analysis [42], generally adopted in order to analyse large scale components up to full aircraft configurations. In general, a global finite element model describes the overall structural behaviour, however local effects cannot be accurately captured in this model; therefore an automatic creation of local models is necessary including automatic remeshing, if required. In this context of global-local analyses, the proposed applications show how efficient is the DBEM and, in particular, the specific strategy adopted for modelling several joined overlapped plates in a two dimensional approach, when implementing local analyses for the aforementioned fracture problems, providing the aircraft companies with an efficient tool aimed at maintenance and inspection interval assessment. A further development direction is given by the implementation of nondeterministic analysis and optimization features. Uncertainty quantification has the potential to move us from the general factor-of-safety (FS) approach to a probability-of-failure (POF) approach. This will provide more flexibility in the design while increasing the effective safety of the system [43]. Consistently with such premises, the described DBEM procedure enable an automatic generation of many different cracked configurations, thanks to an in house made routine that, in combination with the main DBEM code (BEASY), allows the automatic generation of the rivet connections in such a complex two-dimensional model. Then a complete multi-site crack growth analysis can be performed in a few minute calculus with a powerful 64 bit parallel machine, enabling, in the future, the introduction of such deterministic procedure in a stochastic code, to get the randomisation of the crack propagation phenomena with associated probability of failure assessment. Future developments can be devoted to making automatic the model update when a crack link-up with a neighbouring hole, with the addition of an approximate criterion to assess the remaining ligament failure. References [1] Zhang X, Boscolo M, Figueroa-Gordon D, Allegri G, Irving PE (2009) Fail-Safe Design of Integral Metallic Aircraft Structures Reinforced by Bonded Crack Retarders. Engineering Fracture Mechanics 76:114-133. [2] Wen PH, Aliabadi MH, Young A (2000) Plane stress and plate bending coupling in BEM analysis of shallow shells. Int J Numer Meth Engng 48:1107–25. [3] Dirgantara T, Aliabadi MH (2000) Crack growth analysis of plates loaded by bending and tension using dual boundary element method. Int J Fract 105:27–47. [4] Wen PH, Aliabadi MH, Young A (2004) Crack growth analysis for multi-layered airframe structures by boundary element method. Engineering Fracture Mechanics 71: 619–631. [5] Leonel E. D., Venturini W. S. (2011) Multiple random crack propagation using a boundary element formulation. Engineering Fracture Mechanics 78: 1077-1090. [6] Romlay F.R.M., Ouyang H, Ariffin A.K., Mohamed N.A.N. (2010) Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method. Engineering Analysis with Boundary Elements 34: 297–305. [7] Azadi H., Khoei A. R. (2011) Numerical simulation of multiple crack growth in brittle materials with adaptive remeshing. Int. J. Numer. Meth. Engng; 85:1017–1048. [8] Goangseup Z., Jeong-Hoon S., Budyn E., Sang-Ho L., Belytschko T. (2004) A method for growing multiple cracks without remeshing and its application to fatigue crack growth. Modelling Simul. Mater. Sci. Eng. 12; 901–915.

154

Advances in Crack Growth Modeling

[9] Chopp D.L., Sukumar N. (2003) Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method; International Journal of Engineering Science 41; 845–869. [10] Ooi E.T., Yang Z.J. (2009) Modelling multiple cohesive crack propagation using a finite element–scaled boundary finite element coupled method; Engineering Analysis with Boundary Elements 33; 915–929. [11] Wolf J.P., Song C.M. (1996) Finite-element modelling of unbounded media. Chichester: Wiley. [12] Portela A, Aliabadi MH & Rooke DP (1991) The dual boundary element method: effective implementation for crack problems. Int. Journ. Num. Meth. Engng. 33:1269-1287. [13] Mi Y, Aliabadi MH (1994) Three-dimensional crack growth simulation using BEM. Computers & Structures 52:871-878. [14] Mi Y (1996) Three-dimensional analysis of crack growth - Topics in Engineering 28, Computational Mechanics Publ., Southampton, U.K. [15] Salvadori A., Gray L. (2007) Analytical integrations and SIFs computation in 2D fracture mechanics – Int. J. Numer. Meth. Engng 70:445–495. [16] Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 86:519-527. [17] Armentani E, Citarella R (2006) DBEM and FEM analysis on non-linear multiple crack propagation in an aeronautic doubler-skin assembly. International Journal of Fatigue 28:598– 608. [18] Mallardo V (2009) Integral equations and nonlocal damage theory: a numerical implementation using the BDEM. International Journal of Fracture 157 (1-2): 13-32. [19] Forman RG, Shivakumar V, Newman JC (1993) Fatigue Crack Growth Computer Program “NASA/FLAGRO” Version 3.0, National Aeronautics and Space Administration Lyndon B. Johson Space Center, Houston, Texas. [20] Tanaka K (1974) Fatigue crack propagation from a crack inclined to the cyclic tensile axis, Engineering Fracture Mechanics 6:493-500. [21] Citarella R, Cricrì G (2009) A two-parameter model for crack growth simulation by combined FEM-DBEM approach, Advances in Engineering Software 40 (5), 363–377. [22] Häusler SM, Baiz PM, Tavares SMO, Brot A, Horst P, Aliabadi MH, de Castro PMST and Peleg-Wolfin Y (2011) Crack Growth Simulation in Integrally Stiffened Structures Including Residual Stress Effects from Manufacturing. Part I: Model Overview, SDHM 7(3), 163-190, [23] Häusler SM, Baiz PM, Tavares SMO, Brot A, Horst P, Aliabadi MH, de Castro PMST and Peleg-Wolfin Y (2011) Crack Growth Simulation in Integrally Stiffened Structures Including Residual Stress Effects from Manufacturing. Part II: Model Overview, SDHM 7(3), 191-209. [24] Carlone P., Citarella R., Lepore M. and Palazzo G.S., Numerical Crack Growth Analysis in AA2024-T3 Friction Stir Welded Butt Joints, Proceedings of The Eighth International Conference on Engineering Computational Technology, 4-7 September 2012, Dubrovnik– Croatia. [25] Citarella R, Silvestri M, Apicella A (2006) DBEM crack growth simulation and experimental results for a multi-layer and multi-material aeronautic panel. Key Engineering Materials 324325:1123-1126. [26] Citarella R, Lepore M, Apicella A, Calì C (2007) DBEM Crack Growth Simulation for a Riveted Aeronautic Reinforcement under Non-linear Contact Conditions. Key Engineering Materials 348-349:593-596. [27] Citarella R (2009) Non Linear MSD crack growth by DBEM for a riveted aeronautic reinforcement. Advances in Engineering Software 40 (4):253–259. doi:10.1016/j.advengsoft.2008.04.007. [28] Citarella R, Cricrì G (2010) Comparison of DBEM and FEM Crack Path Predictions in a notched Shaft under Torsion. Engineering Fracture Mechanics 77:1730-1749.

Key Engineering Materials Vol. 560

155

[29] Citarella R, Buchholz F-G (2008) Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading. Engineering Fracture Mechanics 75:489–509. [30] Citarella R, Buchholz F-G (2007) Comparison of DBEM and FEM Crack Path Predictions with Experimental Findings for a SEN-Specimen under Anti-Plane Shear Loading. Key Engineering Materials 348-349: 129-132. [31] Citarella R, Soprano A (2006) Some SIF's evaluations by Dual BEM for 3D cracked plates. Journal of Achievements in Materials and Manufacturing Engineering 19(2): 64-72. [32] Citarella R, Perrella M (2005) Multiple surface crack propagation: numerical simulations and experimental tests. Fatigue and Fracture of Engineering Material and Structures 28:135-148. [33] Calì C, Citarella R, Perrella M (2003) Three-dimensional crack growth: numerical evaluations and experimental tests, Biaxial/Multiaxial Fatigue and Fracture, ESIS Publication 31: 341-360, Ed. Elsevier. [34] Armentani E, Citarella R, Sepe R (2011) FML Full Scale Aeronautic Panel Under Multiaxial Fatigue: Experimental Test and DBEM Simulation. EFM Special Issue on 'Multiaxial Fracture' 78 (8):1717-1728. [35] Armentani E, Caputo F, Esposito R, Godono G (2001) A new three loading axes machine for static and fatigue tests, Proceedings of the Sixth International Conference on Biaxial/Multiaxial Fatigue & Fracture, Lisboa, Portugal, June 25-28, Vol. 1, pp. 323-330. [36] Apicella A, Armentani E, Esposito R (2006) Fatigue test on a full scale panel. Key Engineering Materials 324-325: 719-722. [37] Alderliesten RC (2005) Fatigue Crack Propagation and Delamination Growth in Glare. PhD Thesis, Delft University of Technology. [38] Riccio A. (2005) Effects of geometrical and material features on damage onset and propagation in single-lap bolted composite joints under tensile load: Part II - Numerical studies, Journal of Composite Materials 39(23): 2091-2113. [39] Citarella R., Ascione V., Lepore M., Calì C., Fatigue Crack propagation by DBEM in a FML aeronautic full scale panel, Proceedings of the Seventh International Conference on Engineering Computational Technology ECT 2010, 14-17 September, Valencia, Spain, 2010, ISBN 978-1-905088-41-6. [40] Armentani E, Caputo F, Esposito R, Godono G (2004) Evaluation of energy release rate for delamination defects at the skin/stringer interface of a stiffened composite panel. Engineering Fracture Mechanics 71(4-6): 885-895. [41] Citarella R (2011) MSD Crack propagation on a repaired aeronautic panel by DBEM. Advances in Engineering Software 42(10): 887-901. [42] Calomfirescu M, Daoud F, Pühlhofer T (2010) A new look into structural design philosophies for aerostructures with advanced optimization methods and tools, IV European Conference on Computational Mechanics, Palais des Congrès, Paris, France, May 16-21. [43] Citarella R, Apicella A (2006) Advanced Design Concepts and Maintenance by Integrated Risk Evaluation for Aerostructures. Structural Durability and Health Monitoring 2(3):183-196.