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Materials and Design 50 (2013) 360–369
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Materials and Design
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Residual stresses in Linear Friction Welding of aluminium alloys
0261-3069/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.matdes.2013.03.051
⇑ Corresponding author at: Singapore Institute of Manufacturing Technology,A⇑STAR, Singapore 638075, Singapore. Tel.: +65 6793 8595.
E-mail address: [email protected] (X. Song).
X. Song a,b,⇑, M. Xie b, F. Hofmann b, T.S. Jun b,c, T. Connolley d, C. Reinhard d, R.C. Atwood d, L. Connor d,M. Drakopoulos d, S. Harding e, A.M. Korsunsky b
a Singapore Institute of Manufacturing Technology, A�STAR, Singapore 638075, Singaporeb Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKc Welding & Joining Research Group, POSCO, Pohang 790-300, Republic of Koread Diamond Light Source, Beamline I12 JEEP, Didcot, Oxon OX11 0DE, UKe Combustion Systems – Engineering, WH-45, Rolls Royce Plc, Bristol BS34 7QE, UK
a r t i c l e i n f o
Article history:Received 28 December 2012Accepted 15 March 2013Available online 26 March 2013
Keywords:Linear Friction WeldingResidual stresses distribution functionFinite element simulationSynchrotron X-ray diffraction
a b s t r a c t
In the present study, the residual stresses distribution in the Linear Friction Welded (LFW) pieces werestudied using Finite Element (FE) simulation and high energy synchrotron X-ray diffraction. Fully-cou-pled implicit thermo-mechanical analysis simulation procedure was employed, with semi-automaticre-meshing using Python scripting to accommodate large deformation and output contour trackingmethods for extracting deformed configurations. Model validated by the multi-detector polychromaticsynchrotron X-ray experimental data were used to study the distribution of residual stresses distributionin the Linear Friction Welded (LFW) pieces to provide a general function to describe the residual stressesdistribution. A characteristic length scale was found in the plot.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Linear Friction Welding (LFW) is a solid state joining process inwhich the bonding of two flat-edged components is accomplishedby their relative reciprocating motion under axial (compressive)forces [1]. It has many advantages over conventional fusion weld-ing: it can provide high quality welds with improved structuralintegrity (strength and fatigue resistance). Further, it is suitablefor complex geometries and almost all metal types, includingdissimilar materials, thus offering the potential to achieve weightreduction, making overall designs more energy-efficient andeco-friendly [2]. The LFW joining technique has been successfullyapplied to the manufacture of bladed disks (blisks) used in theaerospace industry [3], and other potential applications in the fabri-cation of aero-engine components have also been mentioned [4,5].
As in the case of all thermo-mechanical and joining processes,the consequences of joining need to be carefully assessed, so thatthe effects on structural integrity can be evaluated and appropri-ately incorporated in design rules. Microstructural changes duringwelding (including LFW) are typically studied by means of electronmicroscopy, which provides information on grain morphology, sizeand orientation distribution (texture) [2,6]. It is believed thatthe microstructure of a LFW is usually characterised by a
thermo-mechanically affected zone (TMAZ) close to the weld line,a heat affected zone (HAZ) further from the weld line and the unaf-fected parent material microstructure further more [1]. The TMAZusually displays a region at the weld line which is recrystallizedand contains fine grains (the weld zone), and a region further awayfrom the weld line which is heavily deformed (but not recrystal-lized). The HAZ is affected by the transferred heat during welding,but is not plastically deformed [2,6].
A further significant consequence of joining is the residualstress state that ensues, and that has direct influence on perfor-mance [7]. Residual stresses are the type of stresses that remainafter the original causes of deformation (external forces, heat andtemperature gradients) have been removed [8]. Residual stressescan be thought to be caused by misfitting permanent (inelastic)strains and may arise or become modified at every stage of theengineering component life cycle, from original material produc-tion to component fabrication to service operation [9]. The signifi-cance of residual stresses is that they may greatly influence thematerial and component performance through various mecha-nisms: ductility exhaustion, LCF and HCF, creep rupture, etc.) [3].Therefore, modern engineering practice demands that they shouldbe taken into consideration in design, performance prediction, andassessment [10].
In the course of the LFW process, significant heat is evolved byfriction at the component interface, resulting in severe local plasticdeformation and also giving rise to potentially large residual stres-ses. In order to achieve validated evaluation of residual stresses
X. Song et al. / Materials and Design 50 (2013) 360–369 361
from LFW, a combined approach is used, consisting of finite ele-ment LFW process simulation and experimental measurement ofthe residual strains by synchrotron X-ray diffraction.
Finite element simulation of the Linear Friction Welding processhas been attempted by a number of researchers [11–16]. To aidclassification and analysis, the LFW process was abstracted intofour different stages [11,12]. A key simplification often used inthe early models was that reciprocal motion was ignored, and rep-resented merely by the inflow of heat into the component(s) acrossthe interface. However, it was later established that this simplifiedheat generation model does not account for some crucial aspects ofthe thermal–mechanical interaction, namely, the fact that fric-tional heat generation happens at the same time as temperature-dependent plastic flow and associated large deformation [11–13].To capture the full details of the problem, a fully coupled approachis proposed [14–16]. The advantage of the proposed approach isthat is allows parametric analysis of the thermo-mechanical inter-actions during LFW. Although most publications acknowledge thatLFW is predominantly a steady state process [16], explicit FE algo-rithms (rather than implicit algorithms) have mostly been em-ployed to carry out the simulation. This allows accommodationof the large deformation created by the reciprocal motion andextrusion of materials, and avoids numerical convergence prob-lems. However, explicit simulations often lead to longer computa-tional time, and may be associated with larger numerical errors.Furthermore, failure to match the experimentally observed flashshape has been reported [15]. In the present study, an implicit fullycoupled thermal–mechanical finite element simulation of the Lin-ear Friction Welding process was chosen to ensure more accurateprediction of both the residual stresses and deformed shape,including flash. Automatic re-meshing had to be employed to limitelement distortion.
Although finite element simulations provide full field strain andstress information, the significant number of complex parametersused in the model means that confidence in the results can onlybe ensured by validation against experimental evidence. Experi-mentally, residual stresses in LFW pieces can be evaluated usingeither destructive (or semi-destructive) methods or non-destruc-tive methods. Destructive methods such as hole-drilling [9], crackcompliance [17] and contour methods [18] are relatively cheap andportable. However, the residual stress information is limited to afew measurement points or a line, and typically only one or twostress components are found. Moreover, due to the inherentdestructive nature, these methods are not suitable for further in-line inspection or processing.
In order to overcome these limitations, various non-destructivemethods have been developed, including electromagnetic [19,20]and ultrasonic [21] techniques, etc. All of these methods sharesignificant sensitivity to material microstructure, limited measure-ment accuracy, and some limitations in terms of directional sensi-tivity [9]. The methods based on diffraction of penetratingradiation, on the other hand, allow truly non-destructive measure-ment of residual elastic strains (and the deduction of residualstresses), and possess good spatial and directional resolution. Thenew multi-cell energy dispersive detector installed on the I12 JEEPbeamline at the Diamond Light Source is specially designed forsynchrotron X-ray diffraction using a polychromatic setup[22,23]. The configuration combines the merits of high energy syn-chrotron X-rays (excellent penetration depth) with high whitebeam flux (hence fast data collection) with simultaneous multi-detector acquisition (allowing complete 2D strain state at a pointto be assessed after one sub-second exposure). This opens up thepossibility of acquiring complete strain maps of entire LFW pieceswithin periods of a few hours. This newly-developed setup wasused to measure residual stresses in LFW work pieces, with
additional consideration given to the effects of texture on stressstate and data quality.
It is worth noting that, despite the advantages the method pos-sesses, it is relatively expensive for industrial users, and access tothe synchrotron beamline is limited. Therefore, one of the objec-tives of the present study was to seek a functional description ofthe residual stress state, which allows derivation of the completestress–strain distribution from a limited number of data points.This is particularly useful, e.g. for the estimation of the highest ten-sile residual stress value and its position within the joint: insteadof searching for this point experimentally, an estimation can be de-rived by fitting a known function to a few measured values.
2. The finite element simulation
2.1. The LFW process parameters
In the present study, LFW of two pieces of similar material, alu-minium alloy AA2024, was considered. The case of LFW of similarmaterial (AA2024/AA2024) provides an important ‘‘baseline’’, andcan be used to generate the generic residual stress distributionfunction from this relatively simple case. In future, this paramet-ric/analytic approach may be extended to other cases (i.e. dissim-ilar material LFW).
Linear Friction Welding of AA2024/AA2024 was performed atTWI (The Welding Institute, Cambridge, UK). Processing parame-ters were collected in real time during welding. A schematic illus-tration of the setup is shown in Fig. 1. Details of the weldingprocess parameters are given in Table 1, and plotted in Fig. 2.
In Table 1, total upset refers to the total axial shortening of thetwo components after welding, while burn-off indicates the criticalinitial shortening at which the LFW control system begins to reducethe oscillation amplitude. In the simulation, one bar of the assemblywas maintained stationary, while the other was subjected to oscil-latory movement in the x-direction whilst experiencing a compres-sive force in the y-direction applied at the top end of the bar.
2.2. 2D LFW simulation: a novel framework
For the purposes of the current study, a 2D model was createdwith the exact in-plane dimensions of the specimens, i.e. the widthof 33.6 mm and length of 80 mm for each bar (Fig. 3). As illustratedin the figure, the bottom bar was encastred, whilst the top bar hadtime-varying displacement boundary conditions applied to repre-sent the oscillatory movement during the welding process. TwoReference Points (RPs) were created in the model, for the followingpurposes:
(1) To apply fixed or moving displacement boundary conditions,by linking the RPs through the ‘‘�Equations’’ input commandwith the specific edges of the model, highlighted in colour inFig. 3a.
(2) To act as a sensor through a user subroutine (UEL) for mea-suring the current weld upset. When a critical user-definedupset distance is reached or exceeded, the UEL calls the util-ity routine XIT to trigger re-meshing and to ensure that theelements are not excessively distorted;
(3) To play the role of an information channel between the inputfile and various subroutines at the beginning of each re-mesh analysis step, so that the total run time informationcan be made available for the user-defined subroutine UAMPthat defines the amplitudes of oscillation and pressure. Inthis way, the kinematic aspects of the bars being joined werefully monitored and controlled via the respective RPs.
Fig. 1. Illustration of the LFW arrangement, and the coordinate system used in the welding setup.
Table 1Linear Friction Welding process parameters for AA2024/AA2024.
Joint Force(kN)
Pressure(MPa)
Frequency(Hz)
Amplitude(mm)
Burn-off(mm)
Piece initial length(mm)
Weldment length(mm)
Total upset(mm)
AA2024/AA2024
77 154 50 2 1.5 80 150.49 9.51
0
0.5
1
1.5
2
2.5
0
10
20
30
40
50
60
70
80
90
0 0.5 1 1.5 2 2.5 3 3.5
Am
plitude (mm
)Fo
rce
(kN
), U
pset
(mm
)
Time (sec)
LFW of AA2024/AA2024
Fig. 2. Plots of time history of the applied force, amplitude and upset during LFW process of AA2024/AA2024.
362 X. Song et al. / Materials and Design 50 (2013) 360–369
The LFW process model developed in the present study uses anonlinear, quasi-static, thermo-mechanically coupled analyticalframework. Each analysis step in the simulation sequence repre-sents a single fully coupled temperature–displacement calculation.The exact duration of the step is not known a priori, but is con-trolled by the user element subroutine UEL.
Re-meshing capability was a key aspect of the implementationcrucial for successful completion of the simulation. During LFW,large deformation occurs in the near weld region, where localshearing, forging and flash formation take place. If a single staticmesh were used, no matter how fine, element distortion wouldaccumulate and soon render the calculation impossible. To limitelement distortion, re-meshing had to be triggered when certaincriteria were fulfilled. The procedure was automated through the
use of Python scripts. To capture the significant changes in thecomponent shape, the capabilities of ABAQUS/CAE were used toextract the outer contour of the bars, re-seed the surface, and cre-ate the new mesh in automatic mode. In each bar, the mesh wasdivided into two regions (see Fig. 3b). Smaller elements were usedin the region near the bond line, and also within the flash.
To describe the contact conditions, two types of contact interac-tions were defined: weld contact and self-contact. The weld con-tact was described by a pair of interactions that wassymmetrical, in the following sense. The first interaction definedthe bottom surface of the top bar as the master surface and thetop surface of the bottom bar as the slave surface. In the secondinteraction definition this master–slave relationship was reversed.This ‘‘balanced master–slave’’ arrangement ensures more accurate
(a) (b)
T
O
P
B
O
T
T
O
M
Fig. 3. (a) 2D model setup with boundary conditions and loads and (b) 2D mesh with fine and coarse mesh regions.
X. Song et al. / Materials and Design 50 (2013) 360–369 363
and stable description of the contact pressure at the weld interfaceand avoids ‘‘hourglass’’ effects. Furthermore, it was combined witha softened contact interaction description to promote the re-distri-bution of the contact pressure between nodes along and to bothsides of the interface.
The other type of contact interaction was introduced to addressthe possibility of self-contact that may cause problems during re-meshing. The Part2DGeomFrom2DMesh command was used to gen-erate the new, current configuration geometry. This is achieved byperforming curve-fit operations, and these in turn may lead to self-intersections of the boundary, with consequent invalid part topol-ogy and meshing failure. To overcome this problem, a softenedcontact model was used that introduced a normal pressure evenfor a small separation distance (0.01 mm). The separation distancewas kept as small as practical to avoid introducing non-physicalassumptions into the contact behaviour.
2.3. Material properties
In LFW of similar material, AA2024-T351 material propertieswere employed. The inelastic deformation response was describedby constitutive laws that incorporated temperature and strain ratedependence of yield stress. The strain rate dependence was definedby the Johnson–Cook law where the prevailing exponential coeffi-cient used was C = 0.0083 [24]. The temperature dependence of theother relevant physical and mechanical properties was found fromthe ASM Metals Handbook [25]. It is worth noting that the temper-ature dependence of the yield stress exerts crucial influence on theLFW process in terms of temperature history (then in turns, thedisplacement history), while the strain rate-dependence of theyield stress greatly influences the model convergence. For a simu-lation that is stable and realistic, correct definitions of the temper-ature and strain rate dependence of the properties of the materialsare of crucial importance.
3. Synchrotron X-ray diffraction measurements
3.1. Experimental setup
Strain measurements of LFW samples were carried out on beam-line I12 (JEEP) at the Diamond Light Source using multi-detectorEnergy Dispersive X-ray Diffraction (EDXRD) setup. Residual stres-ses in the LFW work pieces have been measured previously usingdiffraction methods [26,27], but not in samples with strong initialtexture such as present in the extruded aluminium alloys and com-posite used in the current study. The experimental setup was sim-ilar to that described in [28]. However, in the experiment describedin [28], monochromatic synchrotron X-ray and an area detectorwere used and the collected Debye–Sherrer patterns were ‘‘caked’’into equal azimuthal-width portions. Each portion gave the straininformation along the corresponding azimuthal direction. There-fore, all the directional normal strains within a plane perpendicularto the incident beam were evaluated. In contrast, in the setup usedin the present study diffraction patterns were collected by a ‘‘horse-shoe’’ detector which consists of 23 independent energy dispersiveX-ray elements [22,23]. This allows simultaneous collection of datafor the evaluation of linear strain in 22 different directions in theplane perpendicular to the incident beam. A collimator was usedto define the incident beam size of 0.8 mm � 0.8 mm. Another col-limator with 23 small windows was placed behind the sample tocollimate the diffracted beams size to less than 1 mm � 1 mm).The angle between each diffracted beam and incident beam wasaround 5�. An illustration of the experimental setup configurationis given in Fig. 4. This setup gives a total length of the gauge volume�24 mm along the incident beam direction, which was calculatedbased on the equations provided by Rowles [29]. Comparing withthe sample thickness of 15 mm in our case, strain can be averagedthrough the entire sample thickness. It is sufficient for 2D simula-tion validation as the model does not require depth resolution.
Fig. 4. Illustration of the experimental setup.
364 X. Song et al. / Materials and Design 50 (2013) 360–369
During the experiment, a NIST standard silicon powder samplewas firstly used to calibrate the detector and to obtain the exactdiffraction angle and channel-to-energy conversion for each ele-ment. Following this, the LFW samples were mounted on the stageand the diffraction pattern from each element was subjected to aPawley refinement [30] implemented within the General StructureAnalysis System (GSAS) software [31] to determine the latticeparameters. Maps were collected by scanning along five horizontallines (Fig. 4) in the middle of the sample. Each scanning line wasdivided into three parts: parts one and three had the step of1 mm; part two (the middle) had a finer step of 0.333 mm. Theedge of the sample was subsequently cut out and made into acomb sample (with 1 mm � 1 mm cross section sticks) for strain-free lattice parameter evaluation.
3.2. A novel data interpretation procedure for textured samples
A strongly textured sample (with preferred orientation) pro-duces varying diffraction intensities in different directions. If thisdirectional variation of scattered intensities was ignored, the accu-racy of interpretation would be compromised. The LFW samplesstudied were made from extruded bars and possessed strong tex-ture. The variation in scattered intensity with azimuthal angle isillustrated in an integrated plot for a given gauge volume withinthe plane perpendicular to the incoming beam in Fig. 5. For almostall peaks, the intensity varies with the azimuthal angle.
In order to quantify the textured nature of the sample, two dif-ferent plotting methods were employed (Fig. 6a and b). Fig. 6a
Fig. 5. Azimuthal diffraction patterns combined. Diffraction energy is s
shows the dependence of the integrated intensity of the (111)peak on the azimuthal angle. The data can also be transformed toa logarithmic scale and normalised, and then fitted by a curve.The result can be displayed on a polar plot (‘‘cake’’ in Fig. 6b).The curve used for fitting was a series of harmonics,
IðhÞ ¼ m0 þm1 cos 2hþm2 cos 4hþ � � � þmi cosð2ihÞ ð1Þ
Here mi is the coefficient to be determined by least square fitting.For the purposes of fitting, the series was truncated at the term con-taining coefficient m3. If the diffracting volume possesses no pre-ferred orientation, the polar plot appears to be close to a unitcircle, which is clearly not the case for the LFW samples.
One of the principal effects of texture is that in the cases whenpeak intensity is low, the evaluation of the lattice parameter (andhence strain) becomes less reliable. On the other hand, strongerpeaks are typically associated with lower fitting error, and providebetter data. The proposed procedure to reduce the influence of tex-ture-induced intensity variation on the lattice parameter determi-nation is as follows. Firstly, for a two-dimension strain state, therelationship between the normal strains and the principal strainscan be written as,
eðhÞ ¼ e1 cos2ðh� h0Þ þ e2 sin2ðh� h0Þ ð2Þ
Least squares was used to determine the parameters appearingin Eq. (1) (the principal strains e1 and e2, and the principal strainorientation angle h0) by fitting it to the experimental data for linear
hown in keV, and arbitrary units (counts) were used for intensity.
Fig. 7. (a) Weighted fitting and (b) un-weighted fitting.
Fig. 6. (a) The azimuthal distribution of integrated of the 111 peak intensity (without intensity correction). (b) Polar plot of the comparison value 1 + log(I/I0), and thespherical harmonics fitting.
1 For interpretation of colour in Fig. 7, the reader is referred to the web version othis article.
X. Song et al. / Materials and Design 50 (2013) 360–369 365
strains measured in 22 different directions. The contribution fromeach data point to the overall functional can be weighted in orderto optimise the fitting result.
The weighted least squares fitting approach seeks to minimisethe functional S, given by
S ¼Xn
i¼1
Wiðyi � f ðxiÞÞ2 ð3Þ
Here yi denotes the experimentally measured value, and f(xi) is thevalue calculated from the fitting function (2). Therefore, the larger
the weighting factor, the more significance a data point has in thefitting. In the present analysis, the following weights were usedfor each data point i:
Wi ¼ jlog10rij ð4Þ
Here ri is the absolute error of the lattice parameter determinationfrom GSAS fitting: it is smallest for the most accurate measure-ments, and larger for less precise ones. Since this error is less thanunity, the weight Wi in Eq. (4) becomes large for a data point corre-sponding to an accurate measurement, and small for measurementswith larger errors.
Fig. 7 shows the difference between un-weighted fitting andweighted fitting. In the weighted fitting case, the data points areplotted using three colours: red1 indicates the data associated withleast error, green points have larger errors, and blue points havethe biggest errors. The weighted fitting curve passes more closelyto the red points (most accurate data). Thus, reliable strain datacan be obtained from textured samples, and allows not only prob-ing the strain in the LFW TMAZ (Thermal–Mechanically AffectedZone) regions more accurately, but also extract additional informa-tion, such as the principal strain orientation and magnitude.
4. Results and discussions
4.1. Model simulation and validation through upset time history
The von Mises stress contours at different stages of the simula-tion are shown in Fig. 8. The sequence provides an illustration ofhow the process evolves and what significant shape changes occur.The material is ‘‘squeezed out’’ from the weld zone and into theflash. It is seen in Fig. 8a that at �0.66 s, visible flash generationand axial shortening (upset) have begun. At �0.79 s, a greateramount of flash was generated in both bars, with some noticeableasymmetry (right to left), and with similar flash produced on bothsides. This remains true at 4.0 s (the end of the process, Fig. 8c):significant amount of the flash is seen on both sides of the weld-ment, while highly curved flash can be found with similar geome-try on both sides. It is worth noting that as the welding progresses,the fine mesh regions expand accordingly to accommodate thegeneration of flash and the axial shortening. Therefore, the TMAZcorresponds to the final fine mesh region (rather than the initialone) [32]. Fig. 8c represents the distribution of von Mises stresswithin the assembly at the end of the process. At this stage theexternally applied compressive force is already removed, so theresidual stress state can be evaluated.
Model post-processing was used to extract the upset and max-
f
t = 0.66s t = 0.79s t = 4.0s
(a) (b) (c)
Fig. 8. von Mises stress contours (in units of Pa) in the weld joint at different frame times of: (a) 0.66 s, (b) 0.79 s and (c) 4.0 s.
Time history of Upset in LFW Simulation
00.0010.0020.0030.0040.0050.0060.0070.0080.0090.01
Time (s)
Axi
al S
hort
enin
g (m
)
Time history of Max Temperature in LFW Simulation
050
100150200250300350400450
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4Time (s)
Max
Tem
pera
ture
(o C)
(a) (b)
Fig. 9. (a) Time history of the axial shortening (upset) in the LFW simulation. (b) Time history of the maximum temperature in the LFW simulation.
0
0.002
0.004
0.006
0.008
0.01
0 0.5 1 1.5 2 2.5 3 3.5 4
Ups
et (m
)
Time (s)
Upset in LFW Simulation and experiment
ModelExperiment
Fig. 10. Comparison of the upset (axial shortening) between the FE model andexperiment.
366 X. Song et al. / Materials and Design 50 (2013) 360–369
imum temperature across the entire sequence of simulation steps.Fig. 9 illustrates the time history of the model upset and maximumtemperature during the LFW simulation. It is worth noting that thefinal upset value for the model (9.47 mm) lies very close to the
experimental value (9.51 mm). In order to have a closer look atthe entire cycle of the LFW process, FE model simulation resultof axial shortening (upset) is plotted alongside with the experi-mental one in Fig. 10. We can clearly see that, apart from the matchof the final total upset, the entire histories of the upset for both areremarkably close along the timeline, which has also been achievedby other researchers using A285 steel [32]. At the time of 0.65 s,the upset predicted by the model is �1.5 mm, i.e. in precise agree-ment with the burn-off value when the oscillation amplitude be-gan to decline. It is worth pointing out for clarity that theramping down of oscillation amplitude in the simulation is time-triggered at t = 0.65 s. It is exactly the case in the experiment that,the burn-off reaching u = 1.5 mm acts as a trigger for reducingoscillatory amplitude. It is also worth noting that the maximumprocess temperature within the model falls into the expectedrange, not exceeding the solidus temperature (502 �C) yet closeto that typical for warm forging (400 �C) [11].
4.2. Residual stresses in the model and experiment for LFW of AA2024/AA2024
The process model can also be validated using the residualstress data obtained from synchrotron X-ray diffraction. Usingthe new setup and interpretation procedures, the principal strains
-17 -12 -7 -2 3 8 13 18Y (mm)
15
10
5
0
-5
-10
X (m
m)
Fig. 11. Quiver plot of principal strain in x–y plane (defined in Fig. 1). The red lineindicates the boundary of the TMAZ region. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)
-400
-300
-200
-100
0
100
200
300
400
500
600
-80 -60 -40 -20 0 20 40 60 80Mic
rost
rain
Position (mm)
Residual strain profile in the direction of oscillation
X-ray experimentFE model
Fig. 13. Residual strain profiles from the FE model and X-ray diffraction experimentin x direction.
X. Song et al. / Materials and Design 50 (2013) 360–369 367
for all scanning points can be obtained. In Fig. 11, a quiver map isplotted in which an arrow represents a principal strain vector. Bluearrows indicate the maximum in-plane principal strain while thegreen ones indicate the minimum principal strain. The length scaleis given at the top of the diagram by the arrow corresponding to500 microstrain (5 � 10�6). For each line, vectors above the lineindicate tensile strain and below indicate compressive. We canclearly see that except the central welding region, strain vectorschange gradually along with position along each scanning line.The central welding region boundary is identified by the positionwhere strain vectors are smaller, but their magnitude undergoesrapid change. This observation has also been made by other inves-tigators [27].
Although the map in Fig. 11 is informative, it does not lend itselfnaturally to comparisons between the model and the experiment,as it is hard to gauge the quality of agreement. A better way tocompare the two maps is to consider the contours for certain scalarquantities, rather than vectors or tensors. The contours of vonMises equivalent stress are shown in Fig. 12, providing the compar-ison between finite element simulation result and X-ray diffractionexperimental data.
Bond
F(a)
(b)
y
x
Fig. 12. Contour plots of von Mises equivalent residual stresse
From Fig. 12, it is apparent that the model captures qualitativelycorrectly the residual stress distribution in the LFW joint in theAA2024/AA2024. The magnitude of residual stresses is only approx-imately correct, and the stress concentration points at four extremepoints of the thermo-mechanically affected zone (TMAZ) are cap-tured correctly. For the purposes of fatigue life prediction in LFW-joined pieces, attention must be focused on the stress concentrationpoints, particularly those located near the sample surface, as theyare likely to correspond to the location of crack initiation.
However, some discrepancy between the model and experi-ment also must be noted. Disagreement may be due to the tex-ture-induced material property variation and inhomogeneity[27]. Further errors may be associated with the choice of materialdeformation model and its parameters [32]. Improvement of themodel is likely to be achievable by refining the model parameters,and taking into consideration grain morphology and orientation(texture) effects on stiffness and plastic flow.
Another way to compare the simulation with the experimentaldata is by plotting line profiles of key parameters, so that the dif-ference between the two can easily be spotted. Fig. 13 gives theresidual strain in the transverse direction, plotted against positionalong the long axis of the LFW piece. The model captures the resid-ual strain distribution reasonably well, especially the tensile strainat the bond line [27]. The model also predicts the correct locationof the maximum tensile strain point, and approximately correctly
Line
inite Element simulation
X-ray diffraction data(MPa)
(MPa)
s (a) FE simulation result and (b) XRD experimental data.
2 4 21( ) (1 ( ) ( ) ) exp( ( ) )
2
x x xx A B C
w w wσ = + + −
Fig. 14. Analytical function’s fitting to the experimental data in LFW of AA2024/AA2024.
368 X. Song et al. / Materials and Design 50 (2013) 360–369
the maximum tensile strain magnitude that is crucial to fatigue lifeprediction. While the model also predicts the location of the max-imum compressive strain correctly, its magnitude is underesti-mated compared to the experiment. In the model results, thecompressive strain region appears more extended, suggesting thatthe model does not capture correctly the localised nature of thedeformation that takes place near the welding zone.
4.3. Analytical function for the residual stresses distribution in LFWpieces
In terms of post-process modelling for the purposes of fatiguelife prediction, considerable advantages are associated withobtaining an analytical function to describe the residual stress dis-tribution in LFW work pieces. Not only does this simplify model in-put, but, even with limited experimental data available, fullinformation about the residual stress state can be used, includingthe highest tensile stress value and location. This development isparticularly relevant for application in industry.
The analytical expression proposed here is derived from thatpreviously postulated for the residual stresses in a welded plate[33,34]. In these earlier approaches, the maximum residual tensilestrain is predicted always to lie at the bond line. However, as this isobviously not the case for residual stress in LFW work pieces, amodified function is proposed:
rðxÞ ¼ A 1þ Bx
W
� �2þ C
xW
� �4� �
exp �12
xW
� �2� �
ð5Þ
Here A, B, C and w are the four parameters to be found for the curve.Firstly, it requires a minimum of four experimental data points toobtain a unique solution. Since in the current dataset shown inFig. 14 has more than 40 data points, a good fit with relatively smallerrors can be expected, provided the functional form is correct. Thefitting results were A = 95.9450, w = 8.6984, B = 9.4804,C = �3.4248, with the standard deviations for each parameter givenby DA = 16.3590, Dw = 0.1483, DB = 1.9081, DC = 0.6310, and theoverall quality of fit parameter R = 0.9468. In this general equation,A is the residual stress value at the bond line, and w is the half-width of the tensile zone. The stress magnitude is damped using
the term exp � 12
xW
� �2� �
, which makes the stress value at the far
field decay to zero. This term also ensures self-equilibration (stress
balance) of the model, as the definite integral (from zero to infinity)
of the product of exponential term exp � 12
xW
� �2� �
and the polyno-
mial is zero. This works well for long work pieces, while a finitelength of the work piece can be accounted for by superimposinguniform constant stress.
By differentiating Eq. (5), the locations and values of the maxi-mum (most tensile) and minimum (most compressive) stresses canbe found. This is helpful in terms of analysing the stress concentra-tion effects. The most tensile stress location is given by:
x ¼ �w
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� B
C
� ��
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� B
C
� �2 þ 4 2 BC � 1
C
� �q2
vuutð6Þ
and the most compressive stress location is:
x ¼ �w
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� B
C
� �þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� B
C
� �2 þ 4 2 BC � 1
C
� �q2
vuutð7Þ
For convenience, Eqs. (6) and (7) can be re-formulated in terms ofcoefficients B
C and 1C. However, the form of Eq. (5) reflects better
the physical meaning of the parameters. The form of Eqs. (6) and(7) show that the maximum and minimum stress locations do notdepend on the absolute value of the stress magnitude A, but ratheron the shape of the function that is determined by both coefficientsB and C, or, alternatively, by 1
C and the ratio BC. These equations yield
maximum tensile stress location of 8.2 mm from the bond line, andthat of the most compressive stress at 21.1 mm from the bond,which match the experiment data very well (Fig. 14).
This master function is in fact applicable to the description ofgeneral welding residual stress profiles. By setting B = �1 andC = 0, the function is reduced to the original Masubuchi function[33,34], with the maximum tensile stress reached at the bond line.
5. Conclusions
In the present study, simulation was successfully carried out forLFW of similar materials AA2024/AA2024. A fully coupled implicitthermo-mechanical analysis procedure was used for the first time,with semi-automatic re-meshing to control the element distortion.From the simulation, it is shown that the temperature and strainrate dependence of the material yield stress exerts crucial influ-
X. Song et al. / Materials and Design 50 (2013) 360–369 369
ence on the LFW process in terms of temperature and displacementhistory, which two are also closely linked. A satisfactory predictionof the LFW displacement history was obtained by the simulation,with threshold values like burn-off and final total upset correctlypredicted at the right time step. This in turn provided a reasonableprediction of the temperature history as well, which indicated thatthe maximum temperature of the entire process should not exceedthe melting temperature of the material, but close to that of the hotforging. The simulation has shown satisfactory agreement with thesynchrotron X-ray diffraction measurements using the novel poly-chromatic multi-detector setup and an interpretation procedurethat involved weighted data fitting for textured samples. Basedon this experimental result, a function was proposed to describethe distribution of the residual stresses in the LFW piece, helpingto predict the location of the max tensile strain and its magnitude.A minimum of only four data points is required to obtain the fullset of parameters in the function, and the location of the max ten-sile strain and its magnitude can be easily determined using theseparameters and additional equations derived from the proposedresidual stress distribution master function.
Acknowledgements
Authors would like to acknowledge the funding support of theEPSRC under Projects EP/G035059/1 and EP/H003215/1, and Dia-mond Light Source for the provision beam time under allocationsEE6974 and EE7016.
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