Analysis of the Unstressed Lattice Spacing, d0, For the Determination of the Residual Stress in a...

8/11/2019 Analysis of the Unstressed Lattice Spacing, d0, For the Determination of the Residual Stress in a Friction Stir Welded Plate of an Age-hardenable Aluminum Al

1/11

Analysis of the unstressed lattice spacing, d0, for the determinationof the residual stress in a friction stir welded plate of an

age-hardenable aluminum alloy Use of equilibriumconditions and a genetic algorithm

F. Cioffi a, J.I. Hidalgo b, R. Fernandez a,T. Pirling c, B. Fernandez a, D. Gesto d,I. Puente Orench c,e, P. Rey d, G. Gonzalez-Doncel a,

a Dept. of Physical Metallurgy, Centro Nacional de Investigaciones Metalurgicas, CENIM, C.S.I.C., Av. de Gregorio del Amo 8, 28040 Madrid, Spainb

Dept. de Arquitectura de Computadores y Automatica, Universidad Complutense de Madrid, 28040 Madrid, Spainc Diffraction Group, Institut Laue-Langevin, ILL, BP 156, F-38042 Grenoble Cedex 9, Franced Centro Tecnologico AIMEN, C. Relva 27, E-36410 O Porrino, Spain

e Instituto de Ciencia de Materiales de Aragon, C.S.I.C.-Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009 Zaragoza, Spain

Received 25 March 2014; accepted 16 April 2014

Abstract

Procedures based on equilibrium conditions (stress and bending moment) have been used to obtain an unstressed lattice spacing,d0, asa crucial requirement for calculating the residual stress (RS) profile across a joint conducted on a 10 mm thick plate of age-hardenableAA2024 alloy by friction stir welding (FSW). Two procedures have been used that take advantage of neutron diffraction measurements.

First, equilibrium conditions were imposed on sections parallel to the weld so that a constantd0value corresponding to the base materialregion could be calculated analytically. Second, balance conditions were imposed on a section transverse to the weld. Then, using thedata and a genetic algorithm, suitable d0 values for the different regions of the weld have been found. For several reasons, the combmethod has proved to be inappropriate for RS determination in the case of age-hardenable alloys. However, the equilibrium conditions,together with the genetic algorithm, has been shown to be very suitable for determining RS profiles in FSW joints of these alloys, whereinherent microstructural variations ofd0 across the weld are expected.2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Residual stress; Genetic algorithms; Neutron diffraction; Aluminum alloys; Friction stir welding (FSW)

1. Introduction

Probably the most critical issue for the determination ofthe residual stress (RS) state of a given sample/componentby diffraction methods (usually X-rays or neutrons) is theavailability of an accurate and reliable value of theunstressed lattice spacing, d0 [14]. It is well known that

the procedure requires the determination of the spacing,dhkl, of a given crystallographic hklplane in the stressedsample using Braggs law, 2dhkl sinh= k, where h is theBraggs angle andk is the wavelength of the incident beam.Then, by using the formula

e dhkl d0

d01

the elastic strain,e, in a given gauge volume is calculated inthe same way as macroscopic strains are obtained in

http://dx.doi.org/10.1016/j.actamat.2014.04.035

1359-6454/2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +34 91 5538900x337; fax: +34 915347425.E-mail address:[email protected](G. Gonzalez-Doncel).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

ScienceDirect

Acta Materialia 74 (2014) 189199
http://dx.doi.org/10.1016/j.actamat.2014.04.035mailto:[email protected]://dx.doi.org/10.1016/j.actamat.2014.04.035http://crossmark.crossref.org/dialog/?doi=10.1016/j.actamat.2014.04.035&domain=pdfhttp://dx.doi.org/10.1016/j.actamat.2014.04.035mailto:[email protected]://dx.doi.org/10.1016/j.actamat.2014.04.035http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


2/11

conventional mechanical testing with the help of a suitableextensometer. Direct use of the generalized Hookes law ofelasticity, namely,

ri E

1 m1 2m1 mei mej ek 2

where r iis the icomponent of the stress tensor, related tothe three strain components, ei, ej and ek, E is the elasticmodulus and m is the Poissons ratio, provides the stressstate in the same way as a load cell measures applied loadsin a testing machine. The attractiveness of using neutrons(or X-rays) is that a triaxial stress state can be evaluatedin a non-destructive way. The data obtained can then becompared with the predictions derived from elaboratedfinite element codes or proposed models for subsequentassessment.

The crucial issue that makes the need for a very precisevalue ofd0 is that small fluctuations or experimental errorsin the peak position can lead to huge variations of the cal-

culated strain and resulting stress. For example, a variationof a few 1014 m (or 104 A) in the value ofd0 (or d), say2 1014 m, can lead to a variation in the stress of theorder of 10 MPa for the case of aluminum alloys. The stif-fer the material, the wider the variation (titanium, steel,etc.).

Usually, a guaranteed d0value is obtained from powderof the same material, provided that the powder does nothold any macroscopic stress. For this purpose, powderwith a particle microstructure similar to that of the sampleon which the RS is to be measured is required. In the caseof many age-hardenable aluminum alloys, it is thus impor-

tant that both the powder and the sample have undergonethe same temperature cycle to avoid the well-known stronginfluence of the precipitation state on the lattice spacing[5].

However, in many real cases, such as welds of these alu-minum alloys, the above procedure is impractical since thetemperature cycle undergone by the material is not known.Rather, it differs across the welded region, i.e. differentmicrostructures can be found across the weld. Such varia-tions make it impossible in practice to apply the powdermethod to determine d0. In cases like these, the so-calledcombsample method is used[3,610]. This method con-sists in sectioning the weld into small pieces or coupons sothat the macroscopic RS(M-RS) is relieved. The peak posi-

tions obtained in each piece before and after sectioning canthen be compared and the M-RS profile calculated, pro-vided that the microstructures in the different pieces arenot modified by the cutting operation. To ensure that theRS is relieved after sectioning, it is necessary for the dimen-sions of each piece to be significantly smaller than the ini-tial sample size or the "wavelength" of the M-RS to bedetermined. For practical purposes, the different piecesare not separated from each other; instead, they are kepttogether in the form of a comb, in which each tooth keepsthe specific orientation and d0value at that portion of theoriginal weld with respect to a given coordinate system [6].

There are, however, several limitations to the combmethod. First, one has to take into account the procedureitself. The measurements must first be conducted on thestressed sample. Then the comb must be machined out toconduct further measurements for the d0 determination.In this process, two set of measurements must be carried

out, in two separate experiments. This implies a differentconfiguration of the instrument, which can significantlymodify the positions of the diffracted peaks. To avoid theseinconveniences, it is common to extract a comb piece fromthe sample under study, so that both the sample and thecomb can be studied in the frame of the same round ofmeasurements using the same experimental set up[11]. Thisoperation nevertheless inevitably leads to an alteration ofthe initial RS state to be studied. Alternatively, one canprepare two welds under identical conditions such that asimilar RS state is assumed in both cases: one of the weldswould be used to measure the lattice spacing across theweld and the second one to extract a comb to be used as

the unstrained reference. Both samples could be measuredduring the same experiment. Even in this case, it is possiblethat the precipitation and solid solution state differ fromone weld to the other. As has been shown[5], very smallvariations in Cu content greatly modify the lattice spacingof aluminum and, as mentioned above, it is unlikely thatthe temperature cycle undergone in both welds is identical.For example, a slight difference in the temperature of thebacking plate in contact with the plates between the firstand second friction stir welding (FSW) experiments canlead to different cooling cycles. Therefore, the possibilitythat the same microstructural gradient across the weld

is obtained in both experiments is minimal. This makesthe use of a reference comb extracted from a replica weldrisky, even when the welding parameters are ostensiblythe same.

Furthermore, several other considerations should beborn in mind when using the comb method. Gangulyet al.[2]suggest that a retained macrostress along the direc-tion of the maximum dimension of the teeth is still presentin the comb sample and, to use the sample as a stress-freereference sample, the M-RS should first be determined.They propose, in agreement with Ref. [3], to measure nearthe tooths end using the sin2w method. They also foundthat inter-granular strains were retained along the trans-verse direction of the comb teeth. Hughes et al.[3] scannedalong the teeth of a comb specimen machined out from a100 mm diameter quenched nickel superalloy cylinder.Measurements were conducted on several teeth for thestrain direction parallel to the long tooth dimension andfor a perpendicular direction. They found no variationalong almost the whole length of a tooth for the perpendic-ular strain direction, but found significant disparity alongthe teeth for the parallel direction. It was also found thatthere was convergence in data for the parallel direction atall tooth ends, and that the value coincided with the valueseen for the direction perpendicular to the teeth.

190 F. Cioffi et al./ Acta Materialia 74 (2014) 189199


3/11

In summary, the comb method may be of interest from ascientific point of view as it gives further understanding ofthe RS evolution/relaxation upon sample sectioning and/ormachining. It is, however, undesirable from an engineeringor practical approach, when the RS state of realcompo-nents is sought. As a consequence of the sample sectioning

to obtain the comb, the diffraction method becomes unat-tractive as it turns into a destructive technique. Nobodyconsiders machining out a comb from a crucial componentof an expensive device to study its RS state, particularly ifit is to be used in service . Alternative procedures whichprovide reliable d0 data to determine the RS state in realcomponents such as welds of age-hardenable Al alloys,e.g. 2xxx, are therefore desirable.

In the present study, the RS state resulting from a jointconducted in an AA2024 plate by FSW has been analyzed,making use of neutron diffraction data and equilibriumconditions (stress and bending moment) to obtain d0.FSW is an excellent method for joining aluminum alloys

in many industrial uses[12]and, as in other welding tech-niques, residual stresses are developed, affecting compo-nent performance. Two methods have been used toobtain d0: first, equilibrium conditions have been imposedon sections parallel to the weld so that a d0 value corre-sponding to the microstructure of the base material regioncan be calculated. Second, the same conditions wereimposed on a cross-section transverse to the weld. An ana-lytical solution is not possible here because d0 values varyacross the weld, i.e. a large number of variables is involved.The differentd0values across the weld were obtained usinga genetic algorithm (GA), which is a simplified evolution-

ary algorithm (EA). Such a method is able to handle prob-lems in which many variables and possible solutions areinvolved. EAs operate in the field of evolutionary compu-tation, which emerged in the late 1960s, when the possibil-ity of incorporating the natural mechanisms of selectionand survival, in accordance with Darwinian evolution,was proposed to solve artificial intelligence problems [13].An important factor of natural selection is the appearanceof small variations, which are both random and direction-less in phenotypes. These variations, usually referred to asmutations, can survive the selection process if they provetheir value in the environment of the species. The capacityof these search and optimization algorithms has recentlybeen extended to solve very different problems in materialsscience [14]. For example, a GA has been used togetherwith a finite element code to simulate the FSW process ofaluminum alloys 6005A-T6 and 2024-T3[15].

2. Materials and experimental procedure

The aluminum alloy used in this investigation wasAA2024 in the form of a 10 mm thick plate, supplied byAlustock in a T351 condition. In this condition the mate-rial is solution treated, quenched, stress relieved by 1.52% stretching and naturally aged. In its naturally aged

condition, the alloy exhibits high ductility crack resistance

and high fracture toughness. For this investigation, two296 150 mm2 pieces were cut in order to perform a buttweld, of about 250 mm in length.

The weld was made with no additional mechanical/chemical preparation of the plates butt surfaces: just theusual finishing from the mill was made to eliminate defects

from the cuts. The welding experiment was conducted atAIMEN (Porrino, Spain), making use of a PDS-4 Intelli-gent Stir FSW machine from MTS. The welding conditionswere: 400 rpm rotational speed and 100 mm min1 advanc-ing speed. The tilt angle was 1.5. A two-piece fix tool witha threaded pin (7.62 mm long) which included three groundflats was used. The shoulder diameter was slightly morethan 20 mm while the pin diameter varied from 7.95 to6.35 mm. The internal face of the shoulder was not flat,but inclined by 7to create a conical surface. The shoulderand the pin were made of H13 steel (48 HRc) and MP159alloy, respectively.

To investigate the RS state across the weld, d311(where

the sub-index denotes the crystallographic plane) wasextracted from neutron diffraction experiments at three dif-ferent depths: 2.5, 5 and7.5 mm from the surface onwhich the tool pin entered the plate (top surface), denotedhereafter as the front, center and back regions. The regionmeasured covered up to 85 mm on both sides from thecenter of the weld, in the transverse direction of the plate.In this way, a complete scan across the weld, including thenugget, the thermomechanically affected zone (TMAZ), theheat-affected zone (HAZ) and the base material (BM) wasperformed. Since the plate was not sectioned, it is guaran-teed that no modification of the M-RS had occurred.

Furthermore, measurements ofd311across two differentlongitudinal sections, located 52 and 120 mm from the cen-ter of the weld, on the retreating side, at five locationsalong the plate and at six depths, namely 1.2, 2.7,4.2, 5.7, 7.2 and 8.7 mm, were conducted.Fig. 1isa scheme of the welded plate showing the location acrossthe weld where the M-RS state was studied and the twolongitudinal sections on which d311 measurements for thethree principal directions were carried out. As seen in thefigure, both sections correspond to the base material where

LN

T 52 mm

120 mm

Fig. 1. (a) Scheme of the welded plate showing the location on which theRS state was studied and the two longitudinal sections on which d311measurements were conducted. Directions T, L and N of the reference

system coincide with axes X, Yand Z, respectively.

F. Cioffi et al./ Acta Materialia 74 (2014) 189199 191
http://-/?-


4/11

microstructural variations are insignificant. Measurementswere made on two different sections to ensure the reproduc-ibility of the results. A Cartesian reference system wasadopted whereby the origin is located at the center of thewelded plate, on the top surface, and the axes coincide withthe longitudinal (rolling,L), transverse (T) and normal (N)

directions of the plate (Fig. 1). Due to the limited beamtime allocated for this kind of experiment and the typicallylong time required to obtain reasonable peak signals byneutron diffraction, only measurements for the principaldirections were possible. Finally, sin2W scans (with W theangle between the scattering vector and the N direction)were conducted at different locations of the weld to ensurethat the sample reference system is a principal directionone.

After the neutron diffraction experiment, the plate wasused to study the microstructure of the plate and the differ-ent zones of the weld. Conventional metallographic proce-dures were used to examine the microstructure by optical

microscopy. Texture measurements were conducted bythe Schulz reflection method using laboratory X-rays, asdescribed elsewhere[16].

The d311measurements were performed on the SALSAdiffractometer of the ILL (Institute Laue Langevin, Greno-ble, France). This instrument is most suitable for measur-ing RS in real components used in engineering, with ahexapod that can hold, move and rotate large and heavysamples (up to some 1000 kg) with high precision (5 lm)[17]. Fig. 2a shows a general overview of the instrumentwith the AA2024 welded plate mounted on the holder,andFig. 2b shows the details of the plate studied. Further

details of this diffractometer are provided in Refs. [17,18].The wavelength of the neutron beam was k= 1.699 A,selected by a double-focusing silicon bent crystal mono-chromator. The Al(311) was used because it representsthe bulk behavior of the material well and is little affectedby crystallographic texture[19]. The resulting Bragg anglefor the diffracted peak used was around 2h= 88. The scat-tering plane, defined by the incident and diffracted beams,lies in the horizontal plane of the instrument. The elasticmodulus used for the RS calculations was taken as theaverage between the tension and compression values(72.4 and 73.8 GPa, respectively), and the Poisson ratiowas 0.33[20].

Due to the limited rotations of the hexapod and thesample shape, the measurements were made in two steps.The plate, placed in a vertical position, was measuredunder two configurations: first, with the weld in the verticaldirection so that a 90 rotation about a vertical axis (per-pendicular to the scattering plane) allowed for the measure-ment in theNand Tdirections; and second, with the weldlocated horizontally, in which case the 90 rotation aboutthe same vertical axes makes it possible to determine thed311values for theL and Ndirections. Two different gaugevolumes were used to optimize the beam time allocated forthis experiment. For the first case (with the weld in the

vertical position), a gauge volume of 2 2 24 mm3

,

where the longest direction coincided with the verticalone (rolling direction), was used. It was possible to use sucha large gauge volume because the microstructural variationalong the weld and corresponding RS state is minimal. Forthe second case, a smaller gauge volume, of 2 2 2 mm3,was used to avoid possible microstructural gradients andRS variations inside the gauge volume. The gauge volumeswere defined by radial focusing collimators made of Gd-coated Mylar foils.

3. Experimental results

3.1. Microstructure

The resulting microstructure of the weld is shown fromthe transverse metallographic section inFig. 3. In this fig-

ure, the microstructures of the different regions developed

(a)

(b)

Fig. 2. Experimental set-up of the SALSA instrument, showing (a) theAA2024 welded plate mounted on the hexapod and (b) the welded plateused for RS determination. The welding line is parallel to the rollingdirection.



5/11

during FSW are clearly discernible: in the nugget, orrecrystallized region, an even distribution of small equiaxedgrains, of about 5lm, is seen. The boundary between thenugget and the TMAZ is very sharp on the advancing side,but becomes more diffuse in the retreating one, whichreveals the non-symmetrical nature of the FSW process.

The microstructure developed on the TMAZs results fromthe combined effect of temperature and deformation. Thisleads to elongated grains in the direction of the materialflow and an overaged, coarse, precipitation. The micro-structure of the HAZ is characterized by the same grainstructure as that of the base material: a bimodal distribu-tion, with average grain sizes of 30 and 110 lm. This micro-structure results from the cold rolling process undergoneby the original plates. Additionally, the temperaturesreached in these zones are insufficiently high to cause re-solution, but still high enough to cause substantial overag-ing of the parent-metal precipitates [21,22].

Despite attempts to determine clear pole figures of the

BM in the outer and inner regions of the plate beingunsuccessful (the grain size was too large for the gaugevolume used), differences between the resulting texture inthese two regions were evident. The development of a

through-thickness texture gradient is commonly observedin rolled aluminum alloys [23]. The presence of a texture gra-dient suggests that not only does the inter-granular RS varythrough the plate thickness, but also that the usual approx-imation of plane stress conditions for RS calculation is notrecommended.

3.2. Neutron diffraction measurements

The results of the d311values obtained across the weldedplate for the three components L,NandT, and at the threedepths front, center and back, are summarized in the plotsofFig. 4. For any component, the shape of the profile doesnot vary substantially with plate depth. Also, the profiles ofthe L component are M-shaped, in agreement with otherinvestigations [7,2426], with a total variation ofDd311 0.0025 A, which corresponds to some 2450 l-strains. Thed311 value remains nearly constant in regions away fromthe weld, but increases rapidly to a maximum as the weld

is approached, then decreases to a minimum at the center(nugget) of the weld (X= 0). The N and T components,however, are W-shaped, and the total variation in d311 isslightly smaller than for the L component: the total

Fig. 3. Transverse metallographic section of the weld showing the microstructure developed in the different regions.

http://-/?-


6/11

variation corresponds to about 1800 and 1500l-strains forthe Nand Tcomponents, respectively. These profiles are,again, commonly observed in FSW of aluminum alloys[25]. The significant d311variation across the weld for theNcomponent should be attributed not only to the micro-structural changes caused by the FSW process[5], but alsoto the presence of an RS. This confirms that a biaxial stressapproach is not valid here, as suggested above.

The results of the d311values in the longitudinal sectionsatX= 52 mm, for the three directions (L,Nand T) and at

different plate depths, are summarized in the plot ofFig. 5.

The close values obtained for the different depths suggestthat the through-thickness gradient is not relevant. It canalso be seen that the d311variation alongL is very smooth.No significant differences between the profiles obtained atX= 52 mm and X= 120 mm (not shown in this figure)were noted.

The sin2

Wscans led to reasonable linear 2hdependences,which confirm that the coordinate reference system selected that of the rolled plate is a system of principal axes. Forthe sake of simplicity, plots of these data are not presented.

4. The genetic algorithm

A GA works with a population of individuals, wherebyeach individual represents a potential solution to a prob-lem. The basic idea is to evolve the initial randomly gener-ated population (or representations of solutions), forcing itinto a repeated process of selection, crossover and muta-tion. Each cycle or process gives rise to a new generation.

The objective is to find the individual that, over a certainnumber of generations, best approaches the optimal solu-tion. As follows, a multi-objective genetic algorithm hasbeen implemented to obtain d0 profiles across the weldwhich minimizes both the sum of the longitudinal stressesand the bending moments in the transverse section, accord-ing to a protocol described in this section.

Using the smallest distance between neutron diffractionmeasurement points, the whole transverse section of thewelded plate was divided into 181 zones. The actual mea-surements made in the regions away from the weld weremore spread out than in regions close to the nugget, and

the very extreme regions were not scanned since the timeavailable for the experiment at SALSA had to be opti-mized. Since these regions were not affected by the FSWprocess, it was assumed that their d311values were similar(the average of the measurements made in the base mate-rial). In principle, the procedure was as follows: each zonewas specified by nine parameters, corresponding to thethree depths (front, center and back) and the three compo-nents of the stress (L,Nand T). Therefore, a chromosome(string of numbers representing the solution) composed of181 9 = 1629 genes (positions) was used. The first ninegenes represent the parameters for the first zone, i.e.d0LFront, d0TFront, d0NFront, d0LCenter, d0TCenter, d0NCenter,d0LBack, d0TBackand d0NBack), the next nine the parametersfor the second one and so on. For the purpose of the pres-ent research, however, it was assumed that no anisotropiceffect on the d0 value exists, i.e. the same d0 value isobtained irrespectively of the sample direction. This is inagreement with the results of other investigations on age-hardenable 2xxx (and also 7xxx) aluminum alloys[2,6,27]. Therefore, the final chromosome was composedof 543 genes. A randomly generated set of solutions, or aset of strings of numbers, was used as the starting point.Finally, a population size of 8000 individuals, 2000 gener-ations, 0.1% mutation probability and 80% crossover prob-

ability was used in the present analysis.

-100 -80 -60 -40 -20 0 20 40 60 80 1001.2205

1.2210

1.2215

1.2220

1.2225

1.2230

1.2235

1.2240

1.2245retreating side advancing side

longitudinal

d311(

x10-10

m)

distance from weld center, mm

front

center

back

-100 -80 -60 -40 -20 0 20 40 60 80 1001.2205

1.2210

1.2215

1.2220

1.2225

1.2230

1.2235

1.2240

1.2245

d311

(x10-10

m)


front

center

back

normal

-100 -80 -60 -40 -20 0 20 40 60 80 1001.2205

1.2210

1.2215

1.2220

1.2225

1.2230

1.2235

1.2240

1.2245

d311

(x10-10

m)

front

center

back


transverse

Fig. 4. Inter-planar distance data,d311, as a function of distance from thecenter of the weld, at three different depths, front, center and back, forthree directions, L, Nand T. The error is 0.0007 A.



7/11

5. Discussion

As mentioned in the introduction section, the combmethod does not provide a reliable reference for the caseof age-hardenable aluminum alloys. Besides the arguedreasons, it is important to emphasize the additional effect

of the microscopic (inter-granular) residual stress stateresulting from the cold rolling process. It has been arguedthat, despite the machining operation aimed at relieving thestresses in the comb samples, relaxation of the microscopicinter-granular stress (m-RS) is not possible[2,3,27]. This isbecause the comb teeth are typically significantly largerthan the grain size. The presence of a m-RS is revealed asdifferent lattice spacing values in theL,Nand Tdirections:a circumstance that is an obstacle for the determination ofa reliable d0value from a comb sample for the subsequentRS measurements. Procedures based on equilibrium condi-tions of macrostresses[28]were therefore used to calculateappropriate d0 values for RS determination.

On the basis that the AA2024 alloy can be considered asingle-phase material, like all 2xxx alloys, the contributionof second-phase particles and the required stress equilib-rium with them are ignored in the analysis. Therefore, itis assumed that the measurements conducted correspondto actual macroscopic stresses, as assumed in other works[46,29].

The use of equilibrium conditions is a very rigorous pro-cedure, which furthermore avoids the tedious operationsand approximations involved in the comb method. It is sur-prising, however, that it has not been exploited morewidely in the past [30,31]. The condition of equilibrium

implies that, on any section of the component, S, the totalforce perpendicular to it and the bending moment of the in-section forces with respect to the neutral plane, N, shouldbe zero. For the present case, a plate in which one of thedimensions, Z, is significantly smaller than the other twodimensions, X and Y (Fig. 1), bending is possible onlyabout an in-plane axis. In these circumstances, the equa-tions describing this equilibrium in a general form are inte-grals [28], which can be substituted by summations sinceonly discrete measurements were taken in the experiment.Therefore, the equilibrium is given by

X r? DSjS 0 3and for the bending moment, M,X

MjSX

rTDSZnjS 0 4

where Zn refers to the distance from the neutral plane ofthe plate.

Local equilibrium within the plate should also be satis-fied. To apply this condition, however, a description ofthe stress variation along the three spatial directions isneeded. Since measurements on specific sections of theplate were conducted, stress variations in only two direc-tions are possible. Therefore the above condition cannot

be applied.

As mentioned above, two different approaches are pre-sented here. First, taking advantage of the measurementsconducted on the longitudinal sections (retreating side),at X= 52 mm and X= 120 mm (Fig. 5), balance condi-tions are imposed to seek a constant d0 corresponding tothe BM. Secondly, equilibrium is applied to the transverse

section, across the weld, where a variation in the unstressedlattice parameter is expected due to the aging nature of theAA2024 alloy.

5.1. Equilibrium on longitudinal sections of the plate

In this case, equilibrium, in accordance with Eq. (3),should be found for rT. Using Eqs.(1) and (2), this must be

X E1 m1 2m

1 mdT d0

d0

m dN d0

d0

dL d0d0

DS

X52;120

0 5

Since a unique value ofd0is sought, an analytical solutionof Eq.(5) is possible. Solving for the two sections, this isfound to be d0= 1.22228 A at X= 52 mm andd0= 1.22232 A at X= 120 mm. The small differencebetween these values accounts for their validity. Theirsoundness is further supported by imposing the equilibriumof the bending moment, as dictated by Eq. (4). For thiscase, Eq.(4), as applied to the longitudinal sections, turnsinto

X E1 m1 2m

1 mdL d0

d0

m dN d0d0

dT d0d0

DSZn

X52;120

0 6

Using the averaged0value resulting from those obtained atX= 52 mm and X= 120 mm of 1.22230 A, it is found thatthe total bending moment, M, for these two sections is110.8 Nm at X= 52 mm and 22.3 Nm atX= 120 mm. The values are not null, but are reasonablylow considering that a discrete summation of data,obtained from the experimental results, has been made,rather than the integral of a continuous description ofthe stress. It should be borne in mind that it is not analyt-ically possible to find a unique solution which satisfies both

Eqs.(5) and (6)simultaneously. It is also worth mentioningthat this unstressed lattice parameter is not influenced bythe presence of a microscopic inter-granular stress, usuallyrevealed in comb samples[2,3,27]. This microstress, with amuch smaller wavelength than that of the macroscopicstress, overlaps the latter, but does not modify its profilethrough the sample and must be self-equilibrating.

The averaged0, together with the d311data ofFig. 4andEqs. (1) and (2), were used as a first approximation toobtain the RS profiles across the weld. The results obtainedare shown in the plots ofFig. 6, where the RS profiles forthe L, N and T components at the three depths are

presented. As can be seen, profiles similar to those obtained

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


8/11

in other research works are obtained[6,18,29]: the stress inthe L direction shows the largest absolute value (positive)and is similar for the three depths.

With this value, it is possible to determine if theL com-ponent of the stress for the section perpendicular to theweld also satisfies the balance condition. The total stresscalculated is above 10,300 MPa and the bending momentis 104 Nm, the former value clearly away from equilibrium.

This result is not surprising, since the expected variation in

d0across the weld, nugget, TMAZ and HAZ regions is nottaken into account in this calculation.

5.2. Equilibrium on a transverse section of the plate using a

genetic algorithm

Despite the above method producing a reasonable RSprofile (Fig. 6), equilibrium across the weld is not reached.For a rigorous description of the RS state, it is thereforenecessary to take into account that d0 is not constantthrough the weld. The BM has a uniform microstructure,

and therefore thed0calculated in Section5.1for this region

-150 -100 -50 0 50 100 1501,2205

1,2210

1,2215

1,2220

1,2225

1,2230

1,2235

1,2240

1,2245

X=52 mm

d311

(x

10-10m)

distance from plate center (mm)

-1.2 mm

-2.7 mm

-4.2 mm

-5.7 mm

-7.2 mm

-8.7 mm

Longitudinal

-150 -100 -50 0 50 100 1501,2205

1,2210

1,2215

1,2220

1,2225

1,2230

1,2235

1,2240

1,2245

Normal

d311

(x10-10m

)

-1.2 mm

-2.7 mm

-4.2 mm

-5.7 mm

-7.2 mm

-8.7 mm


X=52 mm

-150 -100 -50 0 50 100 1501.2205

1.2210

1.2215

1.2220

1.2225

1.2230

1.2235

1.2240

1.2245

X=52 mm

Transverse

d311

(x10-10m)

-1.2 mm

-2.7 mm-4.2 mm

-5.7 mm

-7.2 mm

-8.7 mm


Fig. 5. Inter-planar distance data, d311, for the L, Nand T directions inthe section atX= 52 mm, parallel to the welding line and at different platedepths (error is of about 0.0007 A).

-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

150

retreating side advancing side


Longitudinalstress,

MPa

front

center

back

-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

150


Normalstress,

MPa

front

center

back

-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

150


Transversalstres

s,

MPa

front

center

back

Fig. 6. Residual stress profiles of the L, Nand Tcomponents across theweld calculated with a constant d0 value obtained from the equilibriumconditions in the longitudinal sections of the plate at X= 52 mm and

X= 120 mm. The maximum error is 10 MPa).



9/11

can be considered valid. In contrast, the nugget, the TMAZand the HAZ undergo microstructural changes producedby the welding process which affect the unstrained latticespacing considerably. The use of the equilibrium conditionon a section with a variable d0 implies a great number ofunknown quantities, which cannot be calculated by an

analytical method such as the one used in the previoussection.For this case, the implementation of EAs can be of great

help due to their capacity to handle problems in which alarge number of unknown quantities are to be determined.In the present study, we pursue a string ofd0values whichvary in the welded region as a consequence of the severethermomechanical process introduced by the FSW. Here,it is important to note the restrictions imposed on the pos-sible d0values in the different weld regions: they should liebetween the maximum and minimumdvalues given by theplots in Fig. 4. This restriction is correct under twoassumptions: (i) the possible hydrostatic stress component

is negligible; and (ii) the inter-granular stress is also insig-nificant. The first assumption is supported in principle bythe fact that welds do not lead to significant hydrostaticstress, as revealed by the RS profiles obtained in previousinvestigations [11,29,32]; the second is supported by theresults obtained from comb measurements in publishedinvestigations[2,3,27].

To obtain the best set ofd0 values for each one of the181 zones after the iterative procedure of the GA, Eqs.(3) and (4)should be employed using the different chromo-somes and the experimental data ofFig. 4 (the searchingprocess was simplified by using only eight intervals in the

welded region and the constant d0value calculated in Sec-tion5.1 for the BM). The iterative procedure was repeated30 times to ensure the reproducibility of the final result.The GA was implemented and run in Java, using an Intelcore i-5, 4 GB RAM memory running under Windows 7.

The average total execution time for a population of1000 individuals and 500 generations over 30 runs wasabout 100 s and the average evaluation time of one individ-ual was nearly 0.2 ms. A progressive decay of the totalstress as a function of the number of iterations wasobtained, which supports the validity and uniqueness of

the result. It is worth mentioning that once full conver-gence was reached no further change in the final result(string ofd0values) was required by the GA. The averageof the 30 solutions resulting from the GA is summarized inFig. 7, where the d0 profile across the weld is shown. Thehighest values are obtained in the BM, whereas in theHAZ, TMAZ and nugget a clear drop is evident. In sum-mary, a W-type profile in the welded region is obtained,where the minimum is reached near the TMAZ zonesand the maximum occurs in the nugget region.

Thed0values provided by the GA (Fig. 7) are consistentwith the precipitation state in the BM, HAZ, TMAZ andnugget, in agreement with other works [33]. Considering

that the plates were in a T351 state before welding, the lat-tice spacing must be increased by the coherent precipitates(S0). The temperature rise in the HAZ and TMAZ zonesmodifies the initial precipitation state to an overaged one(mainly incoherent coarse S precipitates). In these condi-tions, the lattice spacing is relaxed to its lowest value.Finally, the FSW-induced recrystallization process modi-fies the microstructure, leading to grain boundary S precip-itates and a fine dispersion of coherent precipitates insidegrains. This microstructure results in intermediate latticespacing. Therefore thed0profile across the weld is expectedto have a W shape, which somewhat agrees with the result

shown inFig. 7. It is also seen that thed0profiles across theplate (front, center and back) differ from each other, whichis an indication that the friction occurring mainly on thetop surface induces a through-thickness gradient of thethermomechanical process.

-150 -100 -50 0 50 100 150

1,2208x10-10

1,2212x10-10

1,2216x10-10

1,2220x10-10

1,2224x10-10

1,2228x10-10

1,2232x10-10

1,2236x10-10

1,2240x10-10

d0of base material

Lower limitUnstressedlatticeparameter,m


d0, front

d0, center

d0, back

Upper limit

Fig. 7. Profiles of the unstressedd0values (front, center, and back regions) calculated from the equilibrium conditions imposed to the transverse section of

the weld using the GA. The upper and lower d311values used by the GA are included.

http://-/?-


10/11

The final RS profiles calculated using the d0values giveninFig. 7are shown in Fig. 8. The profiles resemble an Mshape for the longitudinal component and a W shape forthe transverse and normal components, as reported else-where [4,25,3436]. The highest tensional RS was foundin the longitudinal component, in agreement with previousstudies [10,26,35], and reached about 74% of the yieldstrength of the BM, which exceeds the 2070% range

reported in Ref. [25]. The asymmetry of the FSW process

is also reflected in the RS profile, the maximum beinglocated in the TMAZ of the advancing side. Dependingon the material and the welding parameters, the locationof the maximum RS has been reported as being in theHAZ[4,25,26]or the TMAZ[10,34,35]. The profiles corre-sponding to the back region (Fig. 8) showed the highest RS

levels, which decay progressively, first in the center andthen in the front region. The latter region shows thesmoothest RS profile. This trend is more evident in the Tand N components, and is consistent with the idea thatthe macroscopic RS generated by FSW is greatest in theback region, where the temperature reached is not as highas in the front region, where partial RS relaxation mayhave occurred.

When comparing the RS profiles ofFigs. 6 and 8, it canbe seen that the maximum stress is higher in the secondcase. In other words, the RS obtained using a constant d0value underestimates the actual RS state in the plate, asobtained from the GA. Not taking this difference into

account in engineering design could lead to dramatic con-sequences during the welds service.

6. Summary

In this research, an analysis of the d0value needed to cal-culate the M-RS across a weld conducted by FSW onAA2024 plates has been made. The comb method usuallyemployed to obtain an unstressed reference is criticizedfor several reasons: the first disadvantage of this method liesin the fact that the sample/component on which the RS is ofinterest must be sectioned. On the other hand, if the comb is

machined from a twin

weld, it is likely that minimal anduncontrolled variations in the precipitation state of age-hardenable alloys (in particular, alloys of the 2xxx series,where small variations of Cu in solid solution lead to largechanges in the lattice spacing) will distort the real unstressedreference. Finally, the possible presence of inter-granularstresses resulting from the manufacture process (e.g. rolling)not relieved in the comb can further complicate the determi-nation of a reliable d0value. As an alternative to the combmethod, two procedures based on equilibrium principleshave been used to calculate a suitable unstressed reference.The first one makes use of a constant d0 value calculatedfrom the equilibrium conditions applied to longitudinal sec-tions, parallel to the weld. Here, an analytical solution canbe derived, given that a singled0value that correspondingto the base material is the unknown variable. Theunstressed lattice parameter obtained is not affected bythe inter-granular stress since it must self-equilibrate: itoverlaps the profile of the M-RS. Since a constant d0is used,the RS obtained across the weld (the Lcomponent) and thebending moment are not equilibrated, despite the profilebeing reasonably similar to those obtained in previous stud-ies. The second procedure uses a genetic algorithm to findthe equilibrium across the weld (transverse section). In thiscase, application of the analytical method used in the longi-

tudinal section is not possible as d0varies through the weld

-150 -100 -50 0 50 100 150-100

-50

0

50

100

150

200

250

retreating side advancing side


Longitudina

lstress,

MPa

front

center

back

-150 -100 -50 0 50 100 150-100

-50

0

50

100

150

200

250


Normalstress,M

Pa

frontcenter

back

-150 -100 -50 0 50 100 150-100

-50

0

50

100

150

200

250


Transversestress,

MPa

front

center

back

Fig. 8. Residual stress profiles across the weld calculated using theunstressed d0 values resulting from the GA (data ofFig. 7).



11/11

due to the presence of different microstructural regionswhich have undergone different thermomechanical cycles.The only requirement imposed on the GA is to minimizethe summation of stresses perpendicular to the sectionunder study and the bending moment about an axis parallelto the weld. With this method, d0 values for the different

regions, namely, the nugget, the TMAZ and the HAZ, areselected from values which range within the interval givenby the maximum and minimum values measured in eachregion. With the resulting d0profile, a final RS state acrossthe weld is obtained, where the d0 oscillations due to themicrostructural variations in the weld are taken intoaccount. The RS obtained (the Lcomponent) with the con-stant d0 value underestimates the calculated RS resultingfrom the GA.

Acknowledgments

Support from Projects MAT-05-0527 and MAT-09-

09545 from MICINN, and PIE 200960I076 fromC.S.I.C., Spain, is gratefully acknowledged. Financial helpfrom the ILL to conduct the neutron diffraction measure-ments on SALSA, experiments 1-02-6 and 1-02-10, is rec-ognized. Support from Spanish Government Grants TIN2008-00508 and MEC CONSOLIDER CSD00C-07-20811and from Spanish MINECO (projects MAT2011-23455and MAT2011-25991) is also acknowledged by J.I.H. andI.P.O., respectively.

References

[1] Altenkirch J, Peel MJ, Steuwer A, Withers PJ. J Strain Anal Eng Des2011;46(46):65162.[2] Ganguly S, Edwards L, Fitzpatrick ME. Mater Sci Eng A

2011;528:122632.[3] Hughes DJ, James MN, Hattingh DG, Webster PJ. J Neutron Res

2003;11(4):28993.[4] Prime MB, Gnaupel-Herold T, Baumann JA, Lederich RJ, Bowden

DM, Sebring RJ. Acta Mater 2006;54(15):401321.[5] Steuwer A, Dumont M, Peel M, Preuss M, Withers JP. Acta Mater

2007;55:411120.[6] Altenkirch J, Steuwer A, Peel M, Richards DG, Withers PJ. Mater

Sci Eng A 2008;488(162):4.[7] Lombard H, Hattingh DG, Steuwer A, James MN. Mater Sci Eng A

2009;501:11924.[8] Florea RS, Hubbard CR, Solanki KN, Bammann DJ, Whittington

WR, Marin EB. J Mater Proc Technol 2012;212:235870.

[9] Pratihar S, Stelmukh V, Hutchings MT, Fitzpatrick ME, Stuhr U,Edwards L. Mater Sci Eng A 2006;437:4653.

[10] Woo W, Em V, Hubbard CR, Lee H, Park KS. Mater Sci Eng A2011;528:80217.

[11] Sutton MA, Reynolds AP, Wang D-Q, Hubbard CR. J Eng MaterTechnol 2002;124:21521.

[12] Mishra RS, Ma ZY. Mater Sci Eng R 2005;50:178 .[13] Holland JH. JACM 1962;9:279314.[14] Li B, Lin J, Yao X. Int J Mech Sci 2002;44:9871002.[15] De Vuyst T, D Alvise L, Simar A, de Meester B, Pierret S. Weld

World 2005;49:4755.[16] Perez-Prado MT, Cristina MC, Ruano OA, Gonzalez-Doncel G.

Mater Sci Eng A 1998;244:21623.[17] Bruno G, Pirling T, Withers PJ, Hutta W, Rowe S. J Neutron Res

2003;11:2359.[18] James MN, Webster PJ, Hughes DJ, Chen Z, Ratel N, Ting SP, et al.

Mater Sci Eng A 2006;427:1626.[19] Bruno G, Fernandez R, Gonzalez-Doncel G. Mater Sci Eng A

2004;382:18897.[20] ASMHandbook, vol. 2. Properties and Selection; Nonferrous Alloys

and Special-Purpose Materials, 10th ed. Materials Park, OH: ASMInternational; 1990.

[21] Bussu G, Irving PE. Int J Fatigue 2003;25:7788.[22] Fratini L, Pasta S, Reynolds AP. Int J Fatigue 2009;31:495500.[23] Perez-Prado MT, Cristina MC, Torralba M, Ruano OA, Gonzalez-

Doncel G. Scripta Mater 1996;36:145560.[24] Peel M, Steuwer A, Preuss M, Withers PJ. Acta Mater

2003;51:4791801.[25] Woo W, Feng Z, Wang XL, David SA. Sci Technol Weld Join

2011;16:2332.[26] Dalle Donne C, Lima E, Wegener J, Pyzalla A, Buslaps T. In:

Proceedings of the 3rd international symposium on friction stirwelding. Kobe; September 2001.

[27] Ganguly S, Fitzpatrick ME, Edwards L. Metall Mater Trans A2006;37:41120.

[28] Noyan IC, Cohen JB. Residual stress; measurements by diffractionand interpretation. New York: Springer Verlag; 1987.

[29] Linton VM, Ripley MI. Acta Mater 2008;56:431927.[30] Withers PJ, Preuss M, Steuwer A, Pang WL. J Appl Crystallogr

2007;40:891904.[31] Ferreira-Barragans S, Fernandez R, Fernandez-Castrillo P, Gon-

zalez-Doncel G. J Alloy Compd 2012;523:94101.[32] Reynolds AP, Wei Tang, Gnaupel-Herold T, Prask H. Scripta Mater

2003;48:128994.[33] Genevois C, Deschamps A, Denquin A, Doisneau-cottignies B. Acta

Mater 2005;53:244758.[34] Masubuchi K. Analysis of welded structures: residual stresses,

distortion, and their consequences. New York: Pergamon Press; 1980.[35] Hatamleh O, Rivero IV, Maredia A. Metall Mater Trans A

2008;39:286774.[36] Buffa G, Fratini L, Pasta S. Invited paper at ICRS-8 8th international

conference on residual stresses. Denver Marriott Tech Center Hotel,

Denver, CO; 2008.

http://refhub.elsevier.com/S1359-6454(14)00286-9/h0005http://refhub.elsevier.com/S1359-6454(14)00286-9/h0005http://refhub.elsevier.com/S1359-6454(14)00286-9/h0010http://refhub.elsevier.com/S1359-6454(14)00286-9/h0010http://refhub.elsevier.com/S1359-6454(14)00286-9/h0015http://refhub.elsevier.com/S1359-6454(14)00286-9/h0015http://refhub.elsevier.com/S1359-6454(14)00286-9/h0020http://refhub.elsevier.com/S1359-6454(14)00286-9/h0020http://refhub.elsevier.com/S1359-6454(14)00286-9/h0025http://refhub.elsevier.com/S1359-6454(14)00286-9/h0025http://refhub.elsevier.com/S1359-6454(14)00286-9/h0030http://refhub.elsevier.com/S1359-6454(14)00286-9/h0030http://refhub.elsevier.com/S1359-6454(14)00286-9/h0035http://refhub.elsevier.com/S1359-6454(14)00286-9/h0035http://refhub.elsevier.com/S1359-6454(14)00286-9/h0040http://refhub.elsevier.com/S1359-6454(14)00286-9/h0040http://refhub.elsevier.com/S1359-6454(14)00286-9/h0045http://refhub.elsevier.com/S1359-6454(14)00286-9/h0045http://refhub.elsevier.com/S1359-6454(14)00286-9/h0050http://refhub.elsevier.com/S1359-6454(14)00286-9/h0050http://refhub.elsevier.com/S1359-6454(14)00286-9/h0055http://refhub.elsevier.com/S1359-6454(14)00286-9/h0055http://refhub.elsevier.com/S1359-6454(14)00286-9/h0060http://refhub.elsevier.com/S1359-6454(14)00286-9/h0065http://refhub.elsevier.com/S1359-6454(14)00286-9/h0070http://refhub.elsevier.com/S1359-6454(14)00286-9/h0075http://refhub.elsevier.com/S1359-6454(14)00286-9/h0075http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0085http://refhub.elsevier.com/S1359-6454(14)00286-9/h0085http://refhub.elsevier.com/S1359-6454(14)00286-9/h0090http://refhub.elsevier.com/S1359-6454(14)00286-9/h0090http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0105http://refhub.elsevier.com/S1359-6454(14)00286-9/h0110http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0120http://refhub.elsevier.com/S1359-6454(14)00286-9/h0120http://refhub.elsevier.com/S1359-6454(14)00286-9/h0125http://refhub.elsevier.com/S1359-6454(14)00286-9/h0125http://refhub.elsevier.com/S1359-6454(14)00286-9/h0135http://refhub.elsevier.com/S1359-6454(14)00286-9/h0135http://refhub.elsevier.com/S1359-6454(14)00286-9/h0140http://refhub.elsevier.com/S1359-6454(14)00286-9/h0140http://refhub.elsevier.com/S1359-6454(14)00286-9/h0145http://refhub.elsevier.com/S1359-6454(14)00286-9/h0150http://refhub.elsevier.com/S1359-6454(14)00286-9/h0150http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0160http://refhub.elsevier.com/S1359-6454(14)00286-9/h0160http://refhub.elsevier.com/S1359-6454(14)00286-9/h0165http://refhub.elsevier.com/S1359-6454(14)00286-9/h0165http://refhub.elsevier.com/S1359-6454(14)00286-9/h0170http://refhub.elsevier.com/S1359-6454(14)00286-9/h0170http://refhub.elsevier.com/S1359-6454(14)00286-9/h0175http://refhub.elsevier.com/S1359-6454(14)00286-9/h0175http://refhub.elsevier.com/S1359-6454(14)00286-9/h0175http://refhub.elsevier.com/S1359-6454(14)00286-9/h0175http://refhub.elsevier.com/S1359-6454(14)00286-9/h0170http://refhub.elsevier.com/S1359-6454(14)00286-9/h0170http://refhub.elsevier.com/S1359-6454(14)00286-9/h0165http://refhub.elsevier.com/S1359-6454(14)00286-9/h0165http://refhub.elsevier.com/S1359-6454(14)00286-9/h0160http://refhub.elsevier.com/S1359-6454(14)00286-9/h0160http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0155http://refhub.elsevier.com/S1359-6454(14)00286-9/h0150http://refhub.elsevier.com/S1359-6454(14)00286-9/h0150http://refhub.elsevier.com/S1359-6454(14)00286-9/h0145http://refhub.elsevier.com/S1359-6454(14)00286-9/h0140http://refhub.elsevier.com/S1359-6454(14)00286-9/h0140http://refhub.elsevier.com/S1359-6454(14)00286-9/h0135http://refhub.elsevier.com/S1359-6454(14)00286-9/h0135http://refhub.elsevier.com/S1359-6454(14)00286-9/h0125http://refhub.elsevier.com/S1359-6454(14)00286-9/h0125http://refhub.elsevier.com/S1359-6454(14)00286-9/h0120http://refhub.elsevier.com/S1359-6454(14)00286-9/h0120http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0115http://refhub.elsevier.com/S1359-6454(14)00286-9/h0110http://refhub.elsevier.com/S1359-6454(14)00286-9/h0105http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0095http://refhub.elsevier.com/S1359-6454(14)00286-9/h0090http://refhub.elsevier.com/S1359-6454(14)00286-9/h0090http://refhub.elsevier.com/S1359-6454(14)00286-9/h0085http://refhub.elsevier.com/S1359-6454(14)00286-9/h0085http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0080http://refhub.elsevier.com/S1359-6454(14)00286-9/h0075http://refhub.elsevier.com/S1359-6454(14)00286-9/h0075http://refhub.elsevier.com/S1359-6454(14)00286-9/h0070http://refhub.elsevier.com/S1359-6454(14)00286-9/h0065http://refhub.elsevier.com/S1359-6454(14)00286-9/h0060http://refhub.elsevier.com/S1359-6454(14)00286-9/h0055http://refhub.elsevier.com/S1359-6454(14)00286-9/h0055http://refhub.elsevier.com/S1359-6454(14)00286-9/h0050http://refhub.elsevier.com/S1359-6454(14)00286-9/h0050http://refhub.elsevier.com/S1359-6454(14)00286-9/h0045http://refhub.elsevier.com/S1359-6454(14)00286-9/h0045http://refhub.elsevier.com/S1359-6454(14)00286-9/h0040http://refhub.elsevier.com/S1359-6454(14)00286-9/h0040http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0035http://refhub.elsevier.com/S1359-6454(14)00286-9/h0035http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0030http://refhub.elsevier.com/S1359-6454(14)00286-9/h0030http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0025http://refhub.elsevier.com/S1359-6454(14)00286-9/h0025http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0020http://refhub.elsevier.com/S1359-6454(14)00286-9/h0020http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0015http://refhub.elsevier.com/S1359-6454(14)00286-9/h0015http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0010http://refhub.elsevier.com/S1359-6454(14)00286-9/h0010http://-/?-http://refhub.elsevier.com/S1359-6454(14)00286-9/h0005http://refhub.elsevier.com/S1359-6454(14)00286-9/h0005http://-/?-http://-/?-http://-/?-

Analysis of the Unstressed Lattice Spacing, d0, For the Determination of the Residual Stress in a...

Documents

Transcript of Analysis of the Unstressed Lattice Spacing, d0, For the Determination of the Residual Stress in a...