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Assessing the effect of residual stresses on the fatigue behavior of integrally stiffened structures
Theoretical and Applied Fracture Mechanics 51 (2009) 95–101
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Assessing the effect of residual stresses on the fatigue behavior of integrally stiffened structures G. Labeas *, I. Diamantakos, Th. Kermanidis Laboratory of Technology and Strength of Materials, University of Patras, Panepistimioupolis Rion, 26500 Patras, Greece
a r t i c l e
i n f o
Article history: Available online 9 April 2009 Keywords: Residual stresses Stress intensity factor Crack propagation Integral structures
a b s t r a c t Residual stresses emerge quite often in real structures due to the various manufacturing processes such as, welding, forming, cutting, milling, etc. In such cases, development of cracks at regions influenced by manufacturing operations demand additional attention. In the present work a numerical methodology has been developed, based on three-dimensional Finite Element Analysis, for the calculation of Stress Intensity Factors at cracks in welded components. The residual stress fields, which are used in SIF calculations, have been computed by the numerical simulation of the thermo-mechanical process. A numerical algorithm based on interpolation principles is developed, in order to introduce the three-dimensional field in the computational model of the cracked structure. The SIF calculation methodology is initially validated for the case of a welded plate by comparison of numerical results with existing analytical solutions. A cracked stiffened panel is analysed afterwards and the calculated fatigue crack propagation results are compared to experimentally measured data. Finally, the numerical procedure is applied to study the effect of more complicated residual stress fields on SIF values developing at cracks located in stiffened panels. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Numerous engineering structures operate under the presence of residual stresses (RS) arising from various thermo-mechanical manufacturing processes, such as, welding, forming, cutting, milling, etc. [1,2]; RS development may lead to undesired geometrical distortions [3,4], as well as to a reduction of structural integrity [5,6]. Furthermore, the behavior of cracks located inside or near the RS affected zone is seriously influenced; crack initiation period, crack propagation rate or angle and residual strength may be significantly affected by the presence of RS in the vicinity of a cracked area. Stress Intensity Factor (SIF) concept is commonly used for assessing the behavior of cracked structures. Simple formulas for fatigue crack propagation, e.g. Paris law [7], Forman law [8], or more sophisticated laws and criteria, like the Strain Energy Density criterion [9] can be used for the calculation of fatigue crack propagation rates and residual strength. For the calculation of SIFs under RS fields, limited works have been published. Tada and Paris [10] and Terada co-worker [11,12] have used a customary method based on the superposition principle and Muskhelishvili’s stress functions for the calculation of SIFs at cracks situated perpendicular to the welding bead. In the above mentioned works, only mode* Corresponding author. Tel.: +30 2610 969498; fax: +30 2610 997190. E-mail address: [email protected] (G. Labeas). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.04.003
I cracks are considered, while in real applications cracks at various angles with respect to the weld line usually appear. In the present work the effect of residual stresses due to welding process on SIFs at cracks situated in welded panels is studied. A numerical methodology has been developed for the calculation of SIFs at cracks in welded components. Realistic RS fields resulting from Laser Beam Welding (LBW) have been obtained by detailed numerical simulation of the thermo-mechanical process. As the FE models used for the LBW simulation and cracked structure assessment are different, RS field transfer between the two different FE models is performed using a specially developed routine, which is based on the interpolation kit-tool of MATLAB software. The three-dimensional FE model used for the calculation of SIFs at cracked structures under the presence of RS is initially validated by the comparison of numerical results to existing analytical solutions for the case of a flat cracked plate, including an RS field described by an analytical relationship. Then, the validated numerical methodology is applied in the case of a stiffened panel produced by High Speed Machining (HSM) process that introduces no significant RS in the structure. Calculated fatigue crack propagation results are compared to experimental measurements and useful conclusions are drawn concerning the SIF values governing the propagation procedure. Finally, the methodology is applied on the case of a stiffened panel produced by LBW for the assessment of the effect of RS field on the calculated SIFs.
G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101
2. Integral structure problem description
450 150
3
2.5
30
Cross section 1000 700 540
Two cases of welded integral structures have been studied. The first refers to a flat centrally cracked plate including a typical welding residual stresses field; this problem has been used for the validation of the numerical methodology, as analytical SIF solutions exist for this problem. The second investigated problem is an integrally stiffened panel, typical of modern aeronautical fuselage construction. This stiffened panel has been produced by two different manufacturing techniques; HSM, which is considered to develop no significant RS in the welded panels and LBW that introduces a considerable RS field in the structure.
2
96
2.1. Cracked unstiffened plate and RS field considered 2a
The investigated problem #1 comprises a 300 mm square plate of 1 mm thickness, resulting from the welding of two identical flat segments. A centrally located crack of half length a is considered, as shown in Fig. 1. The crack is perpendicular to the vertical (weld-line) axis. The stress ry, which represents the RS field, is considered to be constant parallel to the weld line (y-direction) and through-the-thickness (z-direction), while its variation along xcoordinate is schematically presented in Fig. 1. The considered RS distribution, typical for LBW process, may be described by the following equation:
ry ðxÞ ¼ r0y
1 1þ
2 x c0
4
ð1Þ
x c0
where, r0y is a parameter defining the maximum value of the tensile residual stress and c0 is the distance from y-axis at which the residual stress field changes from positive to negative, i.e. from tension to compression. 2.2. Stiffened panel geometry and resulting RS field The geometry of the stiffened panel problem #2 is presented in Fig. 2. The panel is 450 mm wide, 1000 mm long and is made of 6056 T3 Al alloy. The skin thickness is 2 mm all over the plate, except in the area of the stiffeners, where 3 mm thick sockets exist. The stiffeners are of blade type with 2.5 mm web thickness and are welded on the panel at 150 mm pitch. After manufacturing the integral panel, an artificial through crack of 20 mm length is created at its centre, vertical to the stiffeners direction. Two different manufacturing processes have been applied for the production of the stiffened panels, namely HSM and LBW. Using HSM process, stiffened panels are produced from machining
Crack Clamping area
Fig. 2. Integrally stiffened panel geometry (problem #2).
thick plates. It can be considered that the application of HSM process leads to almost zero RS in the machined integral structure. The alternative LBW process applies two laser beams to weld stiffeners on the flat skin creating T-joints, as shown in the detail of Figs. 2 and 3. Nd YAG laser of 3.5 kW power, 4 m/min welding speed, 600 lm fibre diameter and 150–200 mm focal length have been used for the welding. The RS field, resulting by welding the stiffeners using LBW, are calculated by numerical thermo-mechanical simulation of LBW process, as described in [13]. The distribution of rx , which is the calculated RS component parallel to the weld line, is presented in Fig. 4, along the panel width and at the stiffened panel centre (crack area). The residual stress values at the skin top and bottom are presented in the main diagram of Fig. 4, while in the detail diagram inserted in Fig. 4, the stress variation at the stiffener foot area is presented. It may be observed that higher RS are developed at skin top surface compared to the skin bottom area. 3. Fracture mechanics analysis of cracked structures 3.1. Numerical analysis methodology and validation For the stress analysis and the calculation of SIFs, appropriate FE models of cracked structures have to be developed. Three-dimensional (3-D) FE modelling has been used in order to enable investigation of through-the-thickness stress effects and SIF variation. The most important region in a fracture analysis model is the region around the crack edge, which is referred as the crack front in a 3-D model. In order to be able to simulate stress singularity of the crack front, characteristic of Linear Elastic Fracture Mechanics analysis, the elements around the crack front should be quadratic singular elements, having the mid-side nodes placed at the quarter points.
Laser beam
Fig. 1. Unstiffened cracked plate geometry and considered RS field (problem #1).
Laser beam
Fig. 3. T-joint of stiffeners and skin.
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400 350
350
Top
300
Stiffeners
300
σx (MPa)
400
Bot
250
250
200
200
150 100
150
50
100
0 0.05 -50
50 0 -0.25 -50
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.06
0.20
0.07
0.08
0.09
0.10
0.25
Position from panel centre [mm] Fig. 4. Variation of the calculated RS field rx component along the panel width.
For the development of the computational model for the cracked plate problem #1, SOLID95 element type of ANSYS FE code has been used. It is a higher order 3-D 20-node solid element having three degrees of freedom per node, namely translations in the
Fig. 5. (a) Typical FE model of flat welded plate, (b) FE mesh detail around the crack tip.
nodal x, y, and z directions. As SOLID95 element exhibits quadratic displacement behavior, it can be used for the fracture mechanics analysis at the crack front. The FE model of the flat cracked buttwelded plate is presented in Fig. 5a, while the discretization in the area around the crack tip can be seen in the FE mesh detail of Fig. 5b. Consequently, the RS values are calculated at element centroids according to Eq. (1) and are introduced in the FE model as initial stresses. Typical values of RS parameters r0y = 200 MPa and c0 = 2.5 mm, which are usually measured at butt-welded panels, have been considered in this study. Linear elastic analysis of the structure is executed, from which stresses, strains and displacements in the plate are computed. For the SIF calculation, nodal displacements in the vicinity of the crack tip are used. Due to the fact that displacements are not constant through the plate thickness, because of the plane-stress/plane-strain effect, calculated SIFs also exhibit a similar variation, as can be seen in the diagram of Fig. 6, where calculated SIF values at various depths of the plate are depicted for a 5 mm half crack length. In order to validate the numerical methodology, the analytical relationships of [14], providing SIFs for mode-I cracks located in a one-dimensional RS field have been used. Three typical values
7
0.5
SIF (MPa m )
6.5 6 5.5 5 4.5 4 0
0.2
0.4
0.6
Distance from plate surface (mm) Fig. 6. Through-the-thickness variation of calculated SIF.
0.8
1
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12
Analytical [14] FEM (ΔK ( xmax)) FEM (ΔK ( aaver)) FEM (ΔK ( nmin))
KI (MPa m1/2)
10 8 6 4 2 0
0
0.002
0.004
0.006
0.008
0.01
0.012
Crack length (m) Fig. 7. Comparison of analytical and numerical SIF results for mode-I cracks under RS.
(Kmin) and average (Kaver) values. In Fig. 7, computed SIFs for different crack lengths are compared to those analytically derived, indicating a very good agreement, especially for the maximum and average values of the through-the-thickness SIF variations. 3.2. Analysis of stiffened HSM panels
Fig. 8. (a) FE model of an integrally stiffened cracked LBW panel and (b) ‘spiderweb’ type mesh pattern around the crack front.
of the numerically computed through-the-thickness SIF distribution have been considered, namely the maximum (Kmax), minimum
The FE model validated in the centrally cracked flat plate problem #1, is extended to the modelling of the cracked integrally stiffened panel problem #2. The corresponding FE model is presented in Fig. 8a. The same computational model is used for both panels produced either by HSM or by LBW. As the panels have two axes of symmetry, only one-quarter of the geometry is modelled and proper symmetry boundary conditions are applied. The FE model consists of about 2800 elements and 15,500 nodes, varying slightly depending on the size of the crack. Special care is taken in the FE discretization of the crack front area, where singular elements are used in concentric ring element patterns around the crack front, forming a ‘spider-web’ type mesh pattern, as shown in Fig. 8b.
80
HSM - R=0.1
SIF range [ MPa m 0.5]
70 60 50 40 30
Κμαξ ΔK max
20
Καϖ ΔK aver
Κμιν ΔK min
10 0 0
20
40
60
80
100
120
Half crack length a [mm] Fig. 9. Variation DKmax, DKmin and DKaver versus half crack length for RS free HSM panels.
140
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G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101
The crack front has been considered straight and vertical to the crack propagation direction. A remote tension fatigue load of rmax = 80 MPa and R ¼ rmax =rmin ratio equal to 0.1 is applied at the panel. For the calculation of SIF range DK, an ANSYS built-in algorithm is utilised, which computes SIFs from the calculated crack tip nodal displacements. Due to the 3-D modelling of the plate and the developing bending moments, a variation of the calculated SIF through-the-thickness of the plate is observed. The variation of the basic DK distribution is plotted against crack length in Fig. 9, in the form of maximum (DK max ), minimum (DK min ) and average (DK aver ) values. The three calculated SIF expressions DK max ,DK min and DK aver are used along with Paris law in order to assess which expression leads to fatigue crack propagation predictions that coincide better to the respective experimental results, obtained in the frame of DaToN Project [15]. Paris law is expressed in the form:
da ¼ C DK m dN
ð2Þ
In Eq. (2) the constants used are C = 2.32 1012 and m = 2.92. In Fig. 10, the calculated crack propagation using DK max , DK min and DK aver is compared to respective experimental measurements. It can be observed that maximum and average values of SIF range DK lead to the better predictions for fatigue crack propagation, as compared to DK min . 3.3. Analysis of integrally stiffened LBW panels As mentioned above, to assess the effect of RS field in crack propagation of integrally stiffened panels, residual stresses due to the LBW process, as calculated by the numerical thermo-mechanical simulation should be introduced in the fracture mechanics
120
HSM - R=0.1
Half crack length a [ mm]
100 80 DKmax ΔK max DKaver ΔK aver DKmin ΔK min
60 40
Experiment 1
20
Experiment 2 Experiment 3 Experiment 4 Experiment 5
0 0
20000
40000
60000
80000
100000 120000 140000 160000
Loading cycles N Fig. 10. Comparison of calculated crack propagation with experimental measurements.
80
DKaver ΔKaver
60
DKmin ΔKmin
0.5
SIF range [ MPa m ]
LBW - R=0.1
ΔKmax DKmax
70
HSM ΔK DKaver aver
50 40 30 20 10 0 0
20
40
60
80
100
120
140
Half crack length a [mm] Fig. 11. Comparison of DKmax, DKmin and DKaver versus half crack length a, between LBW panel and HSM panel.
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G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101
160
Half crack length a (mm)
140 120 100 80 60 40
HSM Panels LBW Panels
20 0 0
50000
100000
150000
200000
Cycles Fig. 12. Experimental fatigue crack propagation in HSM and LBW stiffened panels.
analysis. However, the thermo-mechanical computational model used for the simulation of welding process is significantly different from the fracture mechanics model because: (a) the mesh density and mesh pattern requirements for LBW simulation are different from those for fracture analysis; in the case of LBW simulation, high mesh density is required at the area of welding, while proper meshing of a fracture mechanics model requires high element concentration in the crack front area and (b) the results of one thermomechanical simulation should be used for fracture mechanics analysis of several crack configurations (e.g. different crack lengths, locations and orientations). The transfer of RS fields between the two different computational models is performed using a specially developed routine, which is based on the interpolation kit-tool of MATLAB software. The computed RS values by the LBW simulation model are interpolated at the element centroids of the respective elements of the fracture mechanics model. Calculated residual stresses are applied as initial stresses at the FE model, together with a remote tension fatigue load of rmax = 80 MPa and R = 0.1. In Fig. 11, the variation of calculated DKmax, DKmin and DKaver magnitudes is plotted against crack length for the case of LBW panel and compared to the DKaver computed for the HSM panel, which is assumed to be free of RS. It is clear that residual stresses due to LBW lead to a reduction of the calculated SIF values. This is in qualitative agreement to experimental results of LBW panels, which exhibit lower fatigue crack propagation rates as compared to the respective in HSM panels (see Fig. 12). This observation may be attributed to the fact that, as observed in Fig. 4, RS values are compressive at the area where crack fronts are located in most of the fatigue crack propagation period. Calculation of fatigue crack propagation in LBW panels could not be performed, as Forman law constants for the specific Al alloy (6056) were not available and Paris law is not applicable, as it neglects the effect of RS on the calculated fatigue crack propagation rate. 4. Conclusions A numerical methodology has been developed for the calculation of crack SIFs in presence of residual stresses and the assessment of the effect of RS field to crack propagation. The proposed
methodology is based on three-dimensional computational FE models, developed for the cases of butt-welded flat plates, as well as stiffened panels produced by HSM and LBW process. Throughthe-thickness SIF variation is observed, due to the transition from plane stress condition at the panel surface to plane strain condition at the panel internal area. The average value of the through-thethickness computed SIF variation seems to be appropriate for fatigue crack propagation predictions, as it has been shown to fit well the experimental results. Finally, for the crack configurations studied, the developed residual stress field due to LBW has led to a reduction of SIF values, as the critical panel areas are affected by compressive residual stresses. Acknowledgements Part of this work was performed in the frame of the European Research Programme ‘‘Innovative Fatigue and Damage Tolerance Methods for the Application of New Structural Concepts” (DaToN); the financial support of the European Union under contract AST4CT-2005-516053 is gratefully acknowledged. References [1] L.J. Yang, Z.M. Xiao, Elastic–plastic modelling of the residual stress caused by welding, J. Mater. Process. Technol. 48 (1995) 589–601. [2] K. Masubuchi, Analysis of Welded Structures, Pergamon Press, Oxford, 1980. [3] S.A. Tsirkas, P. Papanikos, K. Pericleous, N. Strusevich, F. Boitout, J.M. Bergheau, Evaluation of distortions in laser welded shipbuilding parts using local–global finite element approach, Sci. Technol. Weld. Joi. 8 (2003) 79–88. [4] S.A. Tsirkas, P. Papanikos, Th. Kermanidis, Numerical simulation of the laser welding process in butt-joint specimens, J. Mater. Process. Technol. 134 (1995) 59–69. [5] C. Dalle Donne, G. Biallas, T. Ghidini, G. Raimbeaux, Effect of weld imperfections and residual stresses on the fatigue crack propagation in friction stir welded joints, in: Proceedings of the Second International Conference on Friction Stir Welding, 26–28 June, 2000, TWI, Gothenburg, Sweden. [6] S.K. Cho, Y.S. Yang, K.J. Son, J.Y. Kim, Fatigue strength in laser welding of the lap joint, Finite Elem. Anal. Des. 49 (2004) 1059–1070. [7] P. Paris, F. Erdogan, A critical analysis of crack propagation laws, J. Basic Eng. Trans. ASME 85 (1963) 528–534. [8] R.G. Forman, V.R. Kearney, R.M. Engle, Numerical analysis of crack propagation in a cyclic-loaded structure, J. Basic Eng. Trans. ASME 89D (1967) 459. [9] G.C. Sih, Strain–energy–density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305–321.
G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101 [10] H. Tada, P.C. Paris, The stress intensity factor for a crack perpendicular to the welding bead, Int. J. Fract. 21 (1982) 279–284. [11] H. Terada, An analysis of the stress intensity factor of a crack perpendicular to the welding bead, Eng. Fract. Mech. 8 (1976) 441–444. [12] H. Terada, T. Nakajima, Analysis of stress intensity factor of a crack approaching welding bead, Int. J. Fract. 27 (1985) 83–90. [13] G.A. Moraitis, G.N. Labeas, Residual stress and distortions calculation of laser beam welding for aluminum lap joints, J. Mater. Process. Technol. 198 (2008) 260–269.
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[14] H. Tada, P.C. Paris, G. Irwin, The Stress Analysis of Cracks Handbook, The American Society of Mechanical Engineers, New York, NY, USA, 2000. [15] European Research Programme, Innovative Fatigue and Damage Tolerance Methods for the Application of New Structural Concepts (DaToN), Contract AST4-CT-2005-516053.
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Assessing the effect of residual stresses on the fatigue behavior of integrally stiffened structures G. Labeas *, I. Diamantakos, Th. Kermanidis Laboratory of Technology and Strength of Materials, University of Patras, Panepistimioupolis Rion, 26500 Patras, Greece
a r t i c l e
i n f o
Article history: Available online 9 April 2009 Keywords: Residual stresses Stress intensity factor Crack propagation Integral structures
a b s t r a c t Residual stresses emerge quite often in real structures due to the various manufacturing processes such as, welding, forming, cutting, milling, etc. In such cases, development of cracks at regions influenced by manufacturing operations demand additional attention. In the present work a numerical methodology has been developed, based on three-dimensional Finite Element Analysis, for the calculation of Stress Intensity Factors at cracks in welded components. The residual stress fields, which are used in SIF calculations, have been computed by the numerical simulation of the thermo-mechanical process. A numerical algorithm based on interpolation principles is developed, in order to introduce the three-dimensional field in the computational model of the cracked structure. The SIF calculation methodology is initially validated for the case of a welded plate by comparison of numerical results with existing analytical solutions. A cracked stiffened panel is analysed afterwards and the calculated fatigue crack propagation results are compared to experimentally measured data. Finally, the numerical procedure is applied to study the effect of more complicated residual stress fields on SIF values developing at cracks located in stiffened panels. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Numerous engineering structures operate under the presence of residual stresses (RS) arising from various thermo-mechanical manufacturing processes, such as, welding, forming, cutting, milling, etc. [1,2]; RS development may lead to undesired geometrical distortions [3,4], as well as to a reduction of structural integrity [5,6]. Furthermore, the behavior of cracks located inside or near the RS affected zone is seriously influenced; crack initiation period, crack propagation rate or angle and residual strength may be significantly affected by the presence of RS in the vicinity of a cracked area. Stress Intensity Factor (SIF) concept is commonly used for assessing the behavior of cracked structures. Simple formulas for fatigue crack propagation, e.g. Paris law [7], Forman law [8], or more sophisticated laws and criteria, like the Strain Energy Density criterion [9] can be used for the calculation of fatigue crack propagation rates and residual strength. For the calculation of SIFs under RS fields, limited works have been published. Tada and Paris [10] and Terada co-worker [11,12] have used a customary method based on the superposition principle and Muskhelishvili’s stress functions for the calculation of SIFs at cracks situated perpendicular to the welding bead. In the above mentioned works, only mode* Corresponding author. Tel.: +30 2610 969498; fax: +30 2610 997190. E-mail address: [email protected] (G. Labeas). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.04.003
I cracks are considered, while in real applications cracks at various angles with respect to the weld line usually appear. In the present work the effect of residual stresses due to welding process on SIFs at cracks situated in welded panels is studied. A numerical methodology has been developed for the calculation of SIFs at cracks in welded components. Realistic RS fields resulting from Laser Beam Welding (LBW) have been obtained by detailed numerical simulation of the thermo-mechanical process. As the FE models used for the LBW simulation and cracked structure assessment are different, RS field transfer between the two different FE models is performed using a specially developed routine, which is based on the interpolation kit-tool of MATLAB software. The three-dimensional FE model used for the calculation of SIFs at cracked structures under the presence of RS is initially validated by the comparison of numerical results to existing analytical solutions for the case of a flat cracked plate, including an RS field described by an analytical relationship. Then, the validated numerical methodology is applied in the case of a stiffened panel produced by High Speed Machining (HSM) process that introduces no significant RS in the structure. Calculated fatigue crack propagation results are compared to experimental measurements and useful conclusions are drawn concerning the SIF values governing the propagation procedure. Finally, the methodology is applied on the case of a stiffened panel produced by LBW for the assessment of the effect of RS field on the calculated SIFs.
G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101
2. Integral structure problem description
450 150
3
2.5
30
Cross section 1000 700 540
Two cases of welded integral structures have been studied. The first refers to a flat centrally cracked plate including a typical welding residual stresses field; this problem has been used for the validation of the numerical methodology, as analytical SIF solutions exist for this problem. The second investigated problem is an integrally stiffened panel, typical of modern aeronautical fuselage construction. This stiffened panel has been produced by two different manufacturing techniques; HSM, which is considered to develop no significant RS in the welded panels and LBW that introduces a considerable RS field in the structure.
2
96
2.1. Cracked unstiffened plate and RS field considered 2a
The investigated problem #1 comprises a 300 mm square plate of 1 mm thickness, resulting from the welding of two identical flat segments. A centrally located crack of half length a is considered, as shown in Fig. 1. The crack is perpendicular to the vertical (weld-line) axis. The stress ry, which represents the RS field, is considered to be constant parallel to the weld line (y-direction) and through-the-thickness (z-direction), while its variation along xcoordinate is schematically presented in Fig. 1. The considered RS distribution, typical for LBW process, may be described by the following equation:
ry ðxÞ ¼ r0y
1 1þ
2 x c0
4
ð1Þ
x c0
where, r0y is a parameter defining the maximum value of the tensile residual stress and c0 is the distance from y-axis at which the residual stress field changes from positive to negative, i.e. from tension to compression. 2.2. Stiffened panel geometry and resulting RS field The geometry of the stiffened panel problem #2 is presented in Fig. 2. The panel is 450 mm wide, 1000 mm long and is made of 6056 T3 Al alloy. The skin thickness is 2 mm all over the plate, except in the area of the stiffeners, where 3 mm thick sockets exist. The stiffeners are of blade type with 2.5 mm web thickness and are welded on the panel at 150 mm pitch. After manufacturing the integral panel, an artificial through crack of 20 mm length is created at its centre, vertical to the stiffeners direction. Two different manufacturing processes have been applied for the production of the stiffened panels, namely HSM and LBW. Using HSM process, stiffened panels are produced from machining
Crack Clamping area
Fig. 2. Integrally stiffened panel geometry (problem #2).
thick plates. It can be considered that the application of HSM process leads to almost zero RS in the machined integral structure. The alternative LBW process applies two laser beams to weld stiffeners on the flat skin creating T-joints, as shown in the detail of Figs. 2 and 3. Nd YAG laser of 3.5 kW power, 4 m/min welding speed, 600 lm fibre diameter and 150–200 mm focal length have been used for the welding. The RS field, resulting by welding the stiffeners using LBW, are calculated by numerical thermo-mechanical simulation of LBW process, as described in [13]. The distribution of rx , which is the calculated RS component parallel to the weld line, is presented in Fig. 4, along the panel width and at the stiffened panel centre (crack area). The residual stress values at the skin top and bottom are presented in the main diagram of Fig. 4, while in the detail diagram inserted in Fig. 4, the stress variation at the stiffener foot area is presented. It may be observed that higher RS are developed at skin top surface compared to the skin bottom area. 3. Fracture mechanics analysis of cracked structures 3.1. Numerical analysis methodology and validation For the stress analysis and the calculation of SIFs, appropriate FE models of cracked structures have to be developed. Three-dimensional (3-D) FE modelling has been used in order to enable investigation of through-the-thickness stress effects and SIF variation. The most important region in a fracture analysis model is the region around the crack edge, which is referred as the crack front in a 3-D model. In order to be able to simulate stress singularity of the crack front, characteristic of Linear Elastic Fracture Mechanics analysis, the elements around the crack front should be quadratic singular elements, having the mid-side nodes placed at the quarter points.
Laser beam
Fig. 1. Unstiffened cracked plate geometry and considered RS field (problem #1).
Laser beam
Fig. 3. T-joint of stiffeners and skin.
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400 350
350
Top
300
Stiffeners
300
σx (MPa)
400
Bot
250
250
200
200
150 100
150
50
100
0 0.05 -50
50 0 -0.25 -50
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.06
0.20
0.07
0.08
0.09
0.10
0.25
Position from panel centre [mm] Fig. 4. Variation of the calculated RS field rx component along the panel width.
For the development of the computational model for the cracked plate problem #1, SOLID95 element type of ANSYS FE code has been used. It is a higher order 3-D 20-node solid element having three degrees of freedom per node, namely translations in the
Fig. 5. (a) Typical FE model of flat welded plate, (b) FE mesh detail around the crack tip.
nodal x, y, and z directions. As SOLID95 element exhibits quadratic displacement behavior, it can be used for the fracture mechanics analysis at the crack front. The FE model of the flat cracked buttwelded plate is presented in Fig. 5a, while the discretization in the area around the crack tip can be seen in the FE mesh detail of Fig. 5b. Consequently, the RS values are calculated at element centroids according to Eq. (1) and are introduced in the FE model as initial stresses. Typical values of RS parameters r0y = 200 MPa and c0 = 2.5 mm, which are usually measured at butt-welded panels, have been considered in this study. Linear elastic analysis of the structure is executed, from which stresses, strains and displacements in the plate are computed. For the SIF calculation, nodal displacements in the vicinity of the crack tip are used. Due to the fact that displacements are not constant through the plate thickness, because of the plane-stress/plane-strain effect, calculated SIFs also exhibit a similar variation, as can be seen in the diagram of Fig. 6, where calculated SIF values at various depths of the plate are depicted for a 5 mm half crack length. In order to validate the numerical methodology, the analytical relationships of [14], providing SIFs for mode-I cracks located in a one-dimensional RS field have been used. Three typical values
7
0.5
SIF (MPa m )
6.5 6 5.5 5 4.5 4 0
0.2
0.4
0.6
Distance from plate surface (mm) Fig. 6. Through-the-thickness variation of calculated SIF.
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Analytical [14] FEM (ΔK ( xmax)) FEM (ΔK ( aaver)) FEM (ΔK ( nmin))
KI (MPa m1/2)
10 8 6 4 2 0
0
0.002
0.004
0.006
0.008
0.01
0.012
Crack length (m) Fig. 7. Comparison of analytical and numerical SIF results for mode-I cracks under RS.
(Kmin) and average (Kaver) values. In Fig. 7, computed SIFs for different crack lengths are compared to those analytically derived, indicating a very good agreement, especially for the maximum and average values of the through-the-thickness SIF variations. 3.2. Analysis of stiffened HSM panels
Fig. 8. (a) FE model of an integrally stiffened cracked LBW panel and (b) ‘spiderweb’ type mesh pattern around the crack front.
of the numerically computed through-the-thickness SIF distribution have been considered, namely the maximum (Kmax), minimum
The FE model validated in the centrally cracked flat plate problem #1, is extended to the modelling of the cracked integrally stiffened panel problem #2. The corresponding FE model is presented in Fig. 8a. The same computational model is used for both panels produced either by HSM or by LBW. As the panels have two axes of symmetry, only one-quarter of the geometry is modelled and proper symmetry boundary conditions are applied. The FE model consists of about 2800 elements and 15,500 nodes, varying slightly depending on the size of the crack. Special care is taken in the FE discretization of the crack front area, where singular elements are used in concentric ring element patterns around the crack front, forming a ‘spider-web’ type mesh pattern, as shown in Fig. 8b.
80
HSM - R=0.1
SIF range [ MPa m 0.5]
70 60 50 40 30
Κμαξ ΔK max
20
Καϖ ΔK aver
Κμιν ΔK min
10 0 0
20
40
60
80
100
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Half crack length a [mm] Fig. 9. Variation DKmax, DKmin and DKaver versus half crack length for RS free HSM panels.
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The crack front has been considered straight and vertical to the crack propagation direction. A remote tension fatigue load of rmax = 80 MPa and R ¼ rmax =rmin ratio equal to 0.1 is applied at the panel. For the calculation of SIF range DK, an ANSYS built-in algorithm is utilised, which computes SIFs from the calculated crack tip nodal displacements. Due to the 3-D modelling of the plate and the developing bending moments, a variation of the calculated SIF through-the-thickness of the plate is observed. The variation of the basic DK distribution is plotted against crack length in Fig. 9, in the form of maximum (DK max ), minimum (DK min ) and average (DK aver ) values. The three calculated SIF expressions DK max ,DK min and DK aver are used along with Paris law in order to assess which expression leads to fatigue crack propagation predictions that coincide better to the respective experimental results, obtained in the frame of DaToN Project [15]. Paris law is expressed in the form:
da ¼ C DK m dN
ð2Þ
In Eq. (2) the constants used are C = 2.32 1012 and m = 2.92. In Fig. 10, the calculated crack propagation using DK max , DK min and DK aver is compared to respective experimental measurements. It can be observed that maximum and average values of SIF range DK lead to the better predictions for fatigue crack propagation, as compared to DK min . 3.3. Analysis of integrally stiffened LBW panels As mentioned above, to assess the effect of RS field in crack propagation of integrally stiffened panels, residual stresses due to the LBW process, as calculated by the numerical thermo-mechanical simulation should be introduced in the fracture mechanics
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HSM - R=0.1
Half crack length a [ mm]
100 80 DKmax ΔK max DKaver ΔK aver DKmin ΔK min
60 40
Experiment 1
20
Experiment 2 Experiment 3 Experiment 4 Experiment 5
0 0
20000
40000
60000
80000
100000 120000 140000 160000
Loading cycles N Fig. 10. Comparison of calculated crack propagation with experimental measurements.
80
DKaver ΔKaver
60
DKmin ΔKmin
0.5
SIF range [ MPa m ]
LBW - R=0.1
ΔKmax DKmax
70
HSM ΔK DKaver aver
50 40 30 20 10 0 0
20
40
60
80
100
120
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Half crack length a [mm] Fig. 11. Comparison of DKmax, DKmin and DKaver versus half crack length a, between LBW panel and HSM panel.
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Half crack length a (mm)
140 120 100 80 60 40
HSM Panels LBW Panels
20 0 0
50000
100000
150000
200000
Cycles Fig. 12. Experimental fatigue crack propagation in HSM and LBW stiffened panels.
analysis. However, the thermo-mechanical computational model used for the simulation of welding process is significantly different from the fracture mechanics model because: (a) the mesh density and mesh pattern requirements for LBW simulation are different from those for fracture analysis; in the case of LBW simulation, high mesh density is required at the area of welding, while proper meshing of a fracture mechanics model requires high element concentration in the crack front area and (b) the results of one thermomechanical simulation should be used for fracture mechanics analysis of several crack configurations (e.g. different crack lengths, locations and orientations). The transfer of RS fields between the two different computational models is performed using a specially developed routine, which is based on the interpolation kit-tool of MATLAB software. The computed RS values by the LBW simulation model are interpolated at the element centroids of the respective elements of the fracture mechanics model. Calculated residual stresses are applied as initial stresses at the FE model, together with a remote tension fatigue load of rmax = 80 MPa and R = 0.1. In Fig. 11, the variation of calculated DKmax, DKmin and DKaver magnitudes is plotted against crack length for the case of LBW panel and compared to the DKaver computed for the HSM panel, which is assumed to be free of RS. It is clear that residual stresses due to LBW lead to a reduction of the calculated SIF values. This is in qualitative agreement to experimental results of LBW panels, which exhibit lower fatigue crack propagation rates as compared to the respective in HSM panels (see Fig. 12). This observation may be attributed to the fact that, as observed in Fig. 4, RS values are compressive at the area where crack fronts are located in most of the fatigue crack propagation period. Calculation of fatigue crack propagation in LBW panels could not be performed, as Forman law constants for the specific Al alloy (6056) were not available and Paris law is not applicable, as it neglects the effect of RS on the calculated fatigue crack propagation rate. 4. Conclusions A numerical methodology has been developed for the calculation of crack SIFs in presence of residual stresses and the assessment of the effect of RS field to crack propagation. The proposed
methodology is based on three-dimensional computational FE models, developed for the cases of butt-welded flat plates, as well as stiffened panels produced by HSM and LBW process. Throughthe-thickness SIF variation is observed, due to the transition from plane stress condition at the panel surface to plane strain condition at the panel internal area. The average value of the through-thethickness computed SIF variation seems to be appropriate for fatigue crack propagation predictions, as it has been shown to fit well the experimental results. Finally, for the crack configurations studied, the developed residual stress field due to LBW has led to a reduction of SIF values, as the critical panel areas are affected by compressive residual stresses. Acknowledgements Part of this work was performed in the frame of the European Research Programme ‘‘Innovative Fatigue and Damage Tolerance Methods for the Application of New Structural Concepts” (DaToN); the financial support of the European Union under contract AST4CT-2005-516053 is gratefully acknowledged. References [1] L.J. Yang, Z.M. Xiao, Elastic–plastic modelling of the residual stress caused by welding, J. Mater. Process. Technol. 48 (1995) 589–601. [2] K. Masubuchi, Analysis of Welded Structures, Pergamon Press, Oxford, 1980. [3] S.A. Tsirkas, P. Papanikos, K. Pericleous, N. Strusevich, F. Boitout, J.M. Bergheau, Evaluation of distortions in laser welded shipbuilding parts using local–global finite element approach, Sci. Technol. Weld. Joi. 8 (2003) 79–88. [4] S.A. Tsirkas, P. Papanikos, Th. Kermanidis, Numerical simulation of the laser welding process in butt-joint specimens, J. Mater. Process. Technol. 134 (1995) 59–69. [5] C. Dalle Donne, G. Biallas, T. Ghidini, G. Raimbeaux, Effect of weld imperfections and residual stresses on the fatigue crack propagation in friction stir welded joints, in: Proceedings of the Second International Conference on Friction Stir Welding, 26–28 June, 2000, TWI, Gothenburg, Sweden. [6] S.K. Cho, Y.S. Yang, K.J. Son, J.Y. Kim, Fatigue strength in laser welding of the lap joint, Finite Elem. Anal. Des. 49 (2004) 1059–1070. [7] P. Paris, F. Erdogan, A critical analysis of crack propagation laws, J. Basic Eng. Trans. ASME 85 (1963) 528–534. [8] R.G. Forman, V.R. Kearney, R.M. Engle, Numerical analysis of crack propagation in a cyclic-loaded structure, J. Basic Eng. Trans. ASME 89D (1967) 459. [9] G.C. Sih, Strain–energy–density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305–321.
G. Labeas et al. / Theoretical and Applied Fracture Mechanics 51 (2009) 95–101 [10] H. Tada, P.C. Paris, The stress intensity factor for a crack perpendicular to the welding bead, Int. J. Fract. 21 (1982) 279–284. [11] H. Terada, An analysis of the stress intensity factor of a crack perpendicular to the welding bead, Eng. Fract. Mech. 8 (1976) 441–444. [12] H. Terada, T. Nakajima, Analysis of stress intensity factor of a crack approaching welding bead, Int. J. Fract. 27 (1985) 83–90. [13] G.A. Moraitis, G.N. Labeas, Residual stress and distortions calculation of laser beam welding for aluminum lap joints, J. Mater. Process. Technol. 198 (2008) 260–269.
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[14] H. Tada, P.C. Paris, G. Irwin, The Stress Analysis of Cracks Handbook, The American Society of Mechanical Engineers, New York, NY, USA, 2000. [15] European Research Programme, Innovative Fatigue and Damage Tolerance Methods for the Application of New Structural Concepts (DaToN), Contract AST4-CT-2005-516053.