Marin ActaMat 2008

8/13/2019 Marin ActaMat 2008

1/17

Diffraction measurements of elastic strains in stainless steel subjectedto in situ biaxial loading

T. Marin a, P.R. Dawson a,*, M.A. Gharghouri b, R.B. Rogge b

a Sibley School of Mechanical and Aerospace Engineering, 196 Rhodes Hall, Cornell University, Ithaca, NY 14853, United Statesb Canadian Neutron Beam Centre, National Research Council, Chalk River, ON, Canada K0J 1J0

Received 29 February 2008; received in revised form 18 April 2008; accepted 19 April 2008Available online 27 June 2008

Abstract

The lattice strains in AL6XN stainless steel specimens subjected to in situbiaxial stress states have been measured by neutron diffrac-tion. The biaxial stress states were generated in tubular specimens using a specially designed loading apparatus that is capable of applyingaxial loads to specimens under internal pressure. Lattice strains in the axial and hoop directions were measured for different levels ofstress biaxiality in numerous loaded and partially unloaded states. The results revealed the role of the biaxial stress state in the differencesin average lattice strains between various crystallographic fibers. These trends are examined in light of the orientational dependencies ofthe lattice strains on the stress biaxiality under an assumption of uniform stress. Possible additional factors contributing to the observedtrends are discussed.2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Austenitic steel; Neutron diffraction; Elastic behavior; Plastic deformation

1. Introduction

The deformation of metallic materials under the actionof applied loads typically is heterogeneous at the crystalscale. There are many reasons for this. If the material ispoly-phase, the topology of the phases and the contrastin their properties are contributing factors. In single-phasesystems, the degree of heterogeneity depends directly onthe strength of the anisotropy in the mechanical propertiesof the constituent crystals. Whether the system is single-

phase or poly-phase, non-equiaxed grain morphology addsto the heterogeneity of deformation. Conjugate to thedeformation is the stress, which is similarly heterogeneousover an aggregate of grains. To facilitate better utilizationof load-bearing materials, a deeper understanding of thedistribution of stress is essential, as behaviors like yieldingand failure are closely associated with stress levels experi-enced at the crystal level. Measuring the mechanical behav-

ior at the crystal scale is critical to developing aquantitative understanding of it.

Only a few experimental methods offer the capability tomeasure crystal level mechanical behaviors directly. Dif-fraction-based methods have proven effective for measur-ing lattice distortions [13], from which lattice, or elastic,strains may be deduced. Both high-energy synchrotronX-rays and neutron beams have been used to determinethe pole distributions of lattice strains in thin specimens[4,5]. Recently, X-ray methods also have been developed

to probe single crystals inside a loaded, bulk material[6,7]. Neutron diffraction offers an attractive choice whenmeasuring lattice strains in comparatively thick samples[8]. The successful application of neutron diffraction forthe measurement of lattice strain in metallic materials sub-jected to uniaxial tests is documented well in the literature.Measurements have been made on face-centered cubic (fcc)metals, including inconel [9], aluminum [5], copper [10],and stainless steels [1012]. Of the body-centered cubicmetals, steels have received a great deal of attention[13,14]. Hexagonal close-packed (hcp) zirconium alloys

1359-6454/$34.00 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2008.04.048

* Corresponding author. Tel.: +1 607 255 1222; fax: +1 607 255 3466.E-mail address:[email protected](P.R. Dawson).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 56 (2008) 41834199
mailto:[email protected]:[email protected]


2/17

have been studied extensively because of their importanceto the nuclear industry[15]. Measurements have also beenmade on hcp a-titanium[16]and magnesium alloys[17]. Inthe case of hcp metals, changes in the diffraction peakintensity have provided valuable insights into the contribu-tion of twinning to plastic deformation[17,18]. The ability

of diffraction methods to phase discriminate has been usedto study the stress distribution in metal-matrix composites[19,20], to study stress-induced austenite to martensitetransformation under load in TRIP steels[21,22], and alsoto study the formation of martensite and the variant selec-tion in shape memory alloys [23]. Recently, results havebeen reported on the influence of microstructural featureson material response: Hutanu et al. explored the Ludersbands phenomenon in mild steel [24]; Pang et al. studiedthe grain size effect on intergranular strains in nickel [25];Tomota et al. investigated microstructurally related phe-nomena in heat-treated ferritecementite steel [26]; andDawson et al. correlated peak broadening with strain hard-

ening[27].Samples subjected to uniaxial loading provide informa-

tion related only to one simple, macroscopic, deformationmode. Obviously, other simple modes are important aswell, not to mention more complex situations with variableloading paths. Data for these cases are crucial for develop-ing a comprehensive understanding of how stress anddeformation are distributed over loaded polycrystallinematerials. Performing multiaxial loading experiments whileconducting diffraction measurements is far from trivial,however, and examples of such are few. Imbeni et al. [28]performed in situ tension and torsion experiments (not

combined) on thin tubular specimens of Nitinol using syn-chrotron X-rays, measuring only the lattice strains normalto the tube axis. The torsional strain that could be appliedwas limited to 2.5% strain due to buckling of the specimen.Other work on Nitinol includes that of Mehta et al. [29],who used X-ray microdiffraction to map the strain fieldin specimens designed to develop a multiaxial stress statewhen tested in uniaxial tension. Gnaupel-Herold et al.[30] performed in situ measurements on cruciform speci-mens of pipeline steels to investigate the effects of biaxialloading on fatigue failure.

Here we document the design of an experimental set-upthat enables the measurement of lattice strains by in situneutron diffraction in samples subjected to specified levelsof biaxial loading. The set-up achieves a biaxial stress stateusing tubular specimens that are simultaneously pulled intension and pressurized internally. Further, it facilitatescontrolling the level of biaxiality from uniaxial tension tobalanced biaxial tension. An advantage of the design is thatthe level of biaxiality can be altered without changing theorientation of the principal directions of the stress withrespect to the materials microstructure. A disadvantageof the design is that the stress is not uniform over the gagevolume: the circumferential component of the stress variesthrough the tube wall and the radial component of the

stress, although generally small in comparison to the other

stress components, is not identically zero anywhere exceptat the outer diameter of the tube. This concern is addressedthrough examination of the variation of the stress withinthe diffraction volume.

We illustrate the influence of the loading mode throughthe measured responses of a stainless alloy under varying

levels of stress biaxiality. The levels of single crystal elasticand plastic anisotropies have a direct bearing on the differ-ences between the average lattice strains associated withobserved crystallographic fibers (namely, those corre-sponding to the combinations of crystallographic planesand scattering vectors measured in the experiments). Inter-esting trends are apparent in these differences as a functionof the stress biaxiality. By considering the orientationaldependence of the lattice strain under prescribed stress,assuming as a first approximation that the crystal stressesequal the nominal stress, the trends can be explained inpart in terms of the relative contributions of the deviatoricand mean portions of the stress to the lattice strain. How-

ever, some contradictions arise, which are discussed interms of roles of plasticity and spatial heterogeneity ofthe deformation.

The paper is organized as follows. Section2 describes theexperimental apparatus, which is a combination of the spec-imen and associated fixtures. This section also presentsdetails of the diffraction measurements. Section3 providesdetails of the study material and the loading history used inthe experiments. Section4 presents the data collected forloading under several levels of stress biaxiality. In Section5, we examine the individual rolesof the mean and deviatoricportions of the stress together with the elastic anisotropy to

partially explain the observed trends. Finally, Section 6 sum-marizes the important findings from the experiments.

2. Experimental methodology

In the past, several approaches have been developed forgenerating biaxial stress states under laboratory conditionsthat provide independent control of the active stress com-ponents. Popular choices include tensiontorsion loadingof rods or cylinders, axial loading of pressurized tubes,and dual axis tensile loading of samples with cruciformgeometries. Each approach has advantages and limitations.

The tensiontorsion test guarantees a fairly uniformbiaxial stress state in a thin-walled specimen, but may sufferlocalized buckling under torsion[31]. Moreover, the direc-tions of the principal axes of stress depend on the level ofbiaxiality. In tensiontorsion testing, the radial directionis a principal direction and its associated stress is zero.The directions of the two non-zero principal stresses rotatein the plane normal to the radial direction as a function ofthe level of stress biaxiality. As a result, principal directionschange with respect to the material properties and micro-structure during a test [32]. This introduces anotherunknown in the experiment, which is especially unwantedin the testing of anisotropic materials such as composites

or textured polycrystals.

4184 T. Marin et al. / Acta Materialia 56 (2008) 41834199


3/17

Two main drawbacks characterize the cruciform test:the difficulty in obtaining a homogeneous macroscopicstress state in a reasonably wide area in the center of thespecimen and the high forces required in both directions.The former requires the careful design of a suitable speci-men geometry [33,34], while the latter can necessitate the

use of cumbersome loading arrangements that are not eas-ily adapted to diffraction experiments. An excellent descrip-tion of the issues involved in biaxial testing of cruciformsheet specimens is provided by Makinde et al. [35].

Axial loading of pressurized tubes avoids two limita-tions of the other methods: constant principal stress direc-tions can be obtained for different biaxial ratios and stressstates into the plastic regime can be achieved with straight-forward modifications of a single-axis load frame. Theseare gained at the expense of introducing spatial gradientsin some stress components. Nevertheless, the advantagesmotivated its choice for these experiments. Consequently,a custom loading apparatus and associated specimen con-

figuration were developed to allow diffraction measure-ments of lattice strains while loading to various biaxialstress ratios. The criteria used in the design and the mainfeatures of the loading apparatus are described in thissection.

2.1. Loading apparatus

To proceed with this approach using an existing single-axis load frame, a set of fixtures and a specimen geometrywere designed to transmit the axial loading while applyingan internal pressure, as shown in Fig. 1. From a loading

perspective, a smaller cross-sectional area will lessen thedemand on the capacity of the load frame. At the sametime, in a diffraction experiment the size of the samplingvolume is crucial because it must encompass a sufficientnumber of grains to ensure that the intensity of the dif-fracted peaks is satisfactory and that the center of the scat-tering volume coincides with the center of the sampling

volume. To balance these considerations, the wall thick-ness, t, of the specimen was specified to be 1 mm.

The biaxial stress state in the thin-walled region of thespecimen was achieved by applying an axial load with anelectromechanical load frame and an internal pressure viaa hydraulic system. The pressure was monitored using a

high precision transducer (OMEGA Model PX01C0-7.5KG5T); the axial load was measured by a standard loadcell. The hydraulic system consisted of a pneumatic pres-sure intensifier (not shown in Fig. 1). Before each test,the circuit was filled with hydraulic fluid. A valve, posi-tioned at the end of the circuit, was left open to allow thefluid to expel any air in the system, then it was firmlyclosed. When the pump was activated, the fluid inside thespecimen did not flow because of the closed valve, butinstead underwent a static pressurization. This avoidedflow-induced vibrations that might affect the diffractionmeasurements. The system was sufficiently effective atautomatically compensating for fluctuations in the pressure

caused by the presence of some residual air, by minor leak-ing at the seals and by small changes in the specimen geom-etry under load.

Any bending of the specimen is deleterious to the unifor-mity of the axial stress. Therefore, all the components inthe loading apparatus were machined and assembled withhigh precision to minimize the misalignment and the eccen-tricity of the specimen axis with respect to the loadingdirection. Four strain gages, equally spaced around the cir-cumference, were mounted on the central part of the spec-imen to monitor the uniformity of axial strain. Themacroscopic axial strain was also measured over the entire

test using an extensometer with a gage length of 8 mm.

2.2. Specimen configuration

Fig. 1depicts a differential element of material extractedfrom the central part of the specimen and showing the idealbiaxial stress state. In a closed-end cylinder, when the

Fig. 1. The main components of the loading fixture designed to create a biaxial stress state in the tubular specimen.

T. Marin et al. / Acta Materialia 56 (2008) 41834199 4185


4/17

radius to thickness ratio is sufficiently high, the wallbehaves like a membrane so that the axial rzz and hooprhhstress components have nearly constant values throughthe thickness [36]. Achieving conditions close to ideal isdesirable because the diffraction volume spans all of thewall thickness for the axial strain measurements and a large

fraction of it for the hoop strain measurements. However,applying a hoop stress by pressurizing a tube inevitablyresults in a variation in stress through the tube wall.Assuming that the axial load superimposes an axial compo-nent of stress, given by F/A, on the stress due to the pres-sure, p, the expressions for the stress components far fromthe tube ends are[36]:

rzz r2ip

r2o r2i

F

A 1

rhh r2ip

r2or2i

1 ro

r

2 2

rrr r2

ip

r2o r2i

1 r

or

2 3

whereriis the inside radius,rois the outside radius,Ais thecross-sectional area and cylindrical coordinates, (r,h, z) areused. Note that rzzdoes not depend on the radius. Neitherdoes the sum ofrrrand rhh, which implies that the axial lat-tice strain will be independent of radius for any materialwith properties exhibiting at least axisymmetry. The varia-tions in rhhcan be minimized by specifying that the wall issmall relative to the diameter, thereby approaching theideal condition. The choice of outer radius was dictatedby a compromise between maximizing the ratio, ro

t, where

t= rori, and limiting the cross-sectional area, A, basedon the load frame capacity as discussed in Section 2.1.The dimensions chosen were:ri= 7.5 mm andro= 8.5 mm.These give a value of ro

tof 8.5. From Eq.(2), this ratio im-

plies a difference of about 12% between the hoop stress atthe inner and outer radii. Finite element analyses assumingelastic isotropy and a von Mises yield condition were per-formed to aid in the specimen design. Stress distributionswere computed for the expected combinations of axial loadand internal pressure. The analyses showed that the stressdistribution becomes more uniform once the axial load issufficient to cause plastic flow. In particular, the maximumdifference for any of the stress components within the dif-

fraction volume (discussed in greater detail in Section2.3) occurs during the initial pressurization and is less than6% of the average effective stress for all levels of stress biax-iality. This percentage drops as the axial load is applied,reaching approximately 2% when the target stress biaxialityis achieved. It continues to drop for the duration of theloading. Fairly strict tolerances were imposed on the spec-imen diameters (0.05 mm) to minimize the scatter fromthe nominal values. By adopting a generous gage length(32 mm) and a proper radius at the fillets (4 mm), the spec-imen was designed to provide, at least in the gage section,reasonably uniform values for the hoop and axial compo-

nents of the stress.

2.3. Neutron diffraction

In measuring lattice strains, neutron diffraction is thepreferred experimental technique when large penetrationdepth is desired. Only nuclear reactor and spallationsources provide sufficient flux for this purpose, however.

All the experiments described in this paper were carriedout on the L3 diffractometer of the Canadian NeutronBeam Centre, Chalk River Laboratories, Ontario, Canada.For these experiments, the {115} lattice planes of a germa-nium monochromating crystal were used to select neutronswith a nominal wavelength of 0.168 nm. Braggs law givesthe condition for diffraction when the highly collimatedbeam impinges on the specimen:

k 2dhkl sin hhkl 4

wheredhklis the interplanar spacing and 2hhklis the scatter-ing angle corresponding to a specific family of planes, {h k

l}.The elastic strain component, hkl, is a result of stretch-ing or compressing the lattice in the direction normal to theplane {h k l} and can be computed as

hkldhkld

0hkl

d0hkl

sin h0hklsin hhkl

1 5

whered0hklis the reference plane spacing and h0hklis the asso-

ciated reference scattering angle. The component given byhklis a normal component directed parallel to the scatter-ing vector and represents an average value over the sam-pling volume.

Incident and scattering beam slits were used to define

the geometry of the sampling volume inside the specimenwall. The size and shape of this volume were different foraxial and hoop configurations because of the tubular nat-ure of the specimen. In Fig. 2, the gage volumes aresketched together with the incident beam, I, the diffractedbeam,S, and the scattering vector, Q. As shown in the fig-ure, the entire specimen wall is sampled during the axialmeasurements. However, material close to the inner radiuscontributes less to the diffraction signal than material closeto the outer radius. This is due to a simple path lengtheffect: the neutrons must pass through more material toaccess the highlighted sampling volume for points closeto the inner radius compared with points close to the outerradius. As noted previously, the axial lattice strain will beindependent of radius for any material with propertiesexhibiting at least axisymmetry. In addition, the effects ofthe hoop and radial stress components on the axial latticestrain are secondary, being mitigated by Poissons ratio.Based on these considerations, the uneven sampling ofthe axial lattice strains is not expected to lead to a signifi-cant bias in the measured axial lattice strains.

For the hoop lattice strain measurements, Fig. 2showsthat the cross-section of the instrumental gage volume isdiamond shaped. The exact shape and dimensions of thediamond depend on the width of the incident and scatter-

ing slits (which was held constant for all of the measure-

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


5/17

ments), and on the scattering angle. As noted previously,there is a maximum 12% variation in the hoop stressthrough the wall when the material is in the elastic regime.However, for the range of scattering angle used, 75% of thediffraction signal comes from a band of material about0.70 mm wide, centered in the wall. This corresponds to amaximum variation in the hoop stress in the elastic regimeof about 9%. For the hoop measurements, there are no sig-nificant path length effects, and the stress gradient is sam-

pled symmetrically about the mid thickness.Furthermore, the variation in the hoop stress through thethickness (Eq.(2)) is almost linear, such that the value atmid-thickness is very close to the average taken over theentire thickness (a 0.2% difference). These considerationssuggest that the measured hoop lattice strains represent avery good average over the entire wall, and that the maxi-mum variation is relatively small. Furthermore, as thematerial deforms plastically, this variation will decrease.

A multiwire detector was used to acquire the count dis-tribution and a Gaussian function superimposed on a con-stant background was fitted to the measured distribution.Neutron counting intervals were sufficiently long to pro-vide the desired level of uncertainty in the Gaussian param-eters. The uncertainty in the calculated lattice strains wastypically on the order of 12104 [37,9].

A concentric cylindrical insert was devised to reduce thequantity of fluid in the vicinity of the diffraction volumeand through which the incident neutron beam might pass(Fig. 1). The insert was made of aluminum and had a diam-eter of 13 mm, which left a 1 mm layer of pressuring fluid inthe annulus between the insert and the specimen wall. Theattenuation of the beam is affected by the choice of pressur-izing fluid. Typical hydraulic fluids, including water, con-tain hydrogen, which is a strong incoherent scatterer of

neutrons. Tests with tap water and common hydraulic

oil, combined with the insert, resulted in weak diffractionpeaks with a low peak-to-background ratio. Deuterium isa much weaker incoherent scatterer of neutrons thanhydrogen. Heavy water (D2O), in which the hydrogen isreplaced with deuterium, was used for these experiments,and together with the insert yielded diffraction peaks ofhigh quality.

3. Experimental program

3.1. Material

The specimens were machined from a plate of AL6XN.This is an iron-based, super-austenitic stainless steel char-acterized by its high strength and excellent corrosion resis-tance. The material is nominally single-phase with an fcccrystal structure, although often some particle inclusionsare present. The grains are fairly equiaxed with an averagesize of approximately 50lm. Neutron diffraction was usedto characterize the initial crystallographic texture. The{111}, {200} and {220} pole figures, shown in Fig. 3,exhibit a texture that is typical of rolled stock and that isrelatively weak in strength. From tensile tests on specimenstaken from the plate, the initial 0.2% offset strain yieldstrength of the material is 370 MPa, the ultimate strengthis about 750 MPa and the corresponding engineering strainat the onset of necking is approximately 30%. These arecomparable to the corresponding values of 345 MPa,760 MPa and 45% given in a data sheet for this alloy[38].

All of the specimens had the same orientation withrespect to the plate: the longitudinal axis z was parallel tothe rolling direction (RD). The portion of the thin wall inthe transverse direction was selected as the gage volumefor all of the tests (Fig. 2). This ensured that the material

interrogated always had the same crystallographic charac-

Fig. 2. Schematic of the diffraction volumes inside the wall thickness drawn in the gage part of the specimen. (left) Axial strain diffraction volume. (right)

Hoop strain diffraction volume.

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


6/17

teristics and was well away from the plate surfaces. Given

the chosen orientation of the specimens with respect tothe scattering vectors, the measured axial strains are alongthe RD while the hoop strains are along the normaldirection.

3.2. Loading sequences

The stress state in the thin wall of a specimen was dic-tated by the relative magnitudes of the axial load and theinternal pressure.Fig. 4a shows the four loading paths inbiaxial stress space that were examined. The axes corre-spond to the two active principal components of the stress,

rzzand rhh(the coordinate system is shown inFig. 2). Theloading procedure was executed in two steps. First, thesample was pressurized, without exceeding the yield stress,inducing a biaxial stress state with a ratio rhh/rzzof about2.13. Note from Eqs.(1) and (2)that this fraction tends to2 as ro/tincreases towards infinity. InFig. 4a, this step cor-responds to the initial ramp from the origin. The desiredpressure was set manually by adjusting the regulator ofthe pneumatic pressure intensifier.

In the second step, the axial load was applied contribut-ing to an additional axial stress rzz but not subsequentlyaffecting the hoop stress component, rhh. Consequently,

the stress state proceeded along the horizontal lines in

Fig. 4a. For reference, the Von Mises and Tresca yield loci,assuming a yield stress in tension of 370 MPa, are plottedas well to show approximately the onset of plastic flowfor the four paths. For the Tresca criterion, the paths inter-sect the yield surface at

rhh

rzzratios equal to 0, 0.4, 0.7 and 1.

In the following text, the superscript* refers to the biaxialstress ratio at yield, according to the Tresca yield condition,and therefore serves to unambiguously specify the loadingpath. The case

rhh

rzz 0 clearly represents a pure uniaxial

stress state, while the case r

hh

rzz1 represents balanced biax-

iality at yielding. The loading is not radial, meaning that rzzand rhh are not always in the same proportion. There is only

one point in any loading sequence where the stress compo-nents correspond to the nominal value for the test: that cor-responding to yielding as given by the Tresca criterion. Forthe purpose of displaying the results inFig. 4a and in thefollowing sections, the hoop stress was estimated from Eq.(2)and the mean radius, rm, of the tube wall, i.e. at the cen-ter of the diffraction volume. Estimates of the variation inthis stress component over the wall thickness were discussedpreviously in Section2.2.

The loading proceeded in steps of increasing axial load,ultimately to well into the elastoplastic regime. Diffractiondata were acquired at each loading step. The total macro-

Fig. 3. Smoothed pole figures obtained through neutron diffraction for the tested AL6XN. Contours drawn in solid bold lines correspond to the intensityexpected from a uniform texture (i.e. 1uniform). Contours corresponding to intensities greater/lower than expected in a uniform texture are indicated bysolid/dashed lines. The maximum intensity contours are labeled with crosses, while the minimum intensity contours are labeled with closed circles.Contour intervals are 0.1.

Fig. 4. (a) Loading paths in biaxial stress space for the different biaxial stress ratios r

hh

rzz. Tresca and Von Mises yield loci are plotted in dashed lines. (b)

Measured axial stress histories plotted as functions of the axial strain over the specimen gage length.

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


7/17

scopic axial strain was approximately 34% at the end of atest. The axial stress, without the unloading episodes dis-cussed in the next paragraph, is shown in Fig. 4b for eachbiaxial ratio. At larger strains, the axial stress correspond-ing to a given macroscopic axial strain is higher for higherlevels of stress biaxiality. A first-order explanation of this

lies in the dependence of the effective stress (taken here asthe second invariant of the stress deviator) on the biaxialratio. To produce equal values of the effective stress, theaxial stress must be larger for a biaxial ratio of 1

2than for

either 0 or 1. This is evident from the von Mises surfaceinFig. 4a. Because the loading is not proportional in theseexperiments, the actual biaxial ratio by the end of the load-ing for the case of

rhh

rzz 1 is approaching 1

2 while all the

other cases have lower ratios. Thus, the axial stress wouldbe expected to demonstrate the order observed.

The loading steps were interrupted at prescribed targetloads to conduct unloading episodes, during which diffrac-tion measurements also were made. Once into the plastic

regime, diffraction peaks were recorded only after loweringthe load to 90% of the target load. This was done to avoidstress relaxation from inelastic deformation during the dif-fraction measurements. This stress level (90% of targetstress) is denoted as rHzz. Lattice strains also were measuredafter further reducing the axial force to a level whererzz= 200 MPa (denoted as r

Lzz). This stress level was cho-

sen because it is slightly higher than the rzzdue to the inter-

nal pressure alone (about 175 MPa) for the case r

hh

rzz1.

Unloading rzz to a value lower than this, for the caserhh

rzz1, would have required the application of a compres-

sive axial load to overcome the axial tensile stress from theinternal pressure. Lattice strain data were collected at athird stress level, rMzz, midway between r

Hzz and r

Lzz. Fig. 5

presents the macroscopic stressstrain curve for the sampletested at r

hh

rzz0:7 for measuring the axial lattice strains.

This plot shows several unloading and reloading episodes.The circles correspond to points in the loading cycles atwhich diffraction measurements were made.

3.3. Data acquisition

Diffraction data were acquired for several crystallo-graphic planes ({20 0}, {31 1}, {22 0} and {22 2}) withthe scattering vector aligned parallel to the axial ([001])and hoop ([01 0]) directions. However, since axial and hoopstrain components could not be measured simultaneously,

eight samples were prepared, four for each strain compo-nent. For a given stress ratio

rhh

rzz, axial and hoop specimens

were subjected to similar loading histories. Peaks for allfour crystallographic planes were also acquired in the ini-tial unloaded condition (no pressure, no axial load) toestablish reference values of the scattering angle, h0hkl. Thiswas done for each specimen at the very beginning of theexperiment and ensured that any effect of the thermome-chanical processing was excluded from the straincalculation.

An example of the loading sequence and of the evolu-tion of the lattice strains in the region close to the elas-

ticplastic transition is depicted in Fig. 6 for r

hh

rzz 0:7.Fig. 6a shows the macroscopic rzzzzresponse. Points cor-responding to diffraction measurements are labeled. Forthe sake of clarity, only the rHzz and r

Lzzpoints are shown.

The axial stress rzz is computed using Eqs. (1)(3) andthe axial strain zz is given by the extensometer. Fig. 6bshows the {200} axial and hoop lattice strains and the esti-mated errors corresponding to each of the labeled steps inthe rzzzzcurve. Data for only one of the four grain fam-ilies are presented to better explain the features of theexperimental procedure. In the unloaded state, the straincomponents are zero. At load step 1, the pressure is applied

Fig. 5. Loading and unloading episodes for the experiment with a biaxialratio

rhh

rzz0:7. The diffraction measurements were taken at loads indicated

by the circles.

Fig. 6. (a) Expanded view of the rzzzz curve showing the loading sequence in the elasticplastic transition (case r

hh

rzz0:7). (b) Axial and hoop lattice

strains for the {200} reflection during the loading step indicated in (a).



8/17

as described in Section 2 and both the axial and hoop

strains reach a positive value, i.e. they are in tension. Theaxial load is incremented in step 2, thus the axial straincomponent increases in accordance with the increase inrzz. The hoop strain component is essentially unchanged,even though a small decrement might have been expectedfrom Poisson contraction. In the steps that follow (steps3 and 4), the hoop strain does decrease while the axialstrain continues to grow linearly. Load step 5 is the firstmeasurement performed at 90% of the target load rHzz,and step 6 corresponds to the unloaded state rLzz. Uponreloading (step 7) and then unloading again (step 8), thetwo strain components continue this alternating pattern,which is maintained until the end of the loading sequence.The lattice strains for the other diffraction planes, {311},{220} and {222}, exhibit similar trends. By taking dataat several points during an unloading episode followingplastic straining we obtain information about the influ-ences of the plastic and the elastic anisotropies on the stressstate in the grains of the diffraction volume.

4. Experimental results

In this section, the lattice strain measurements are pre-sented with the aim of identifying several distinctive trendswith respect to the biaxial ratio,

rhh

rzz, and crystallographic

plane, {h k l}. First, we present the total lattice strains for

both the axial and hoop components as functions of the

axial stress for the various levels of the stress biaxiality.Second, we examine the strain changes on partial unload-ing, achieved by reducing the axial load while holding theinternal pressure constant. Finally, we show the diffractionelastic moduli deduced from the changes in stress and lat-tice strain on unloading. In most of the graphs, the axeshave limits arranged to facilitate comparison between asso-ciated sets of data.

4.1. Total lattice strains under load

Fig. 7presents the axial lattice strains hklmeasured atrH

zz

for the four biaxial ratios analyzed.1 The case r

hh

rzz

0shows that the relative magnitudes of the lattice strains cor-responding to the different crystallographic planes ({h k l}s)follow the relative magnitudes of the directional modulusEhkl as expected in a purely uniaxial stress state (e.g.E111>E110> E311>E100). These trends are in good agree-ment, both in values and in the uncertainties, with thosefound in[27]where the same material was subjected to uni-

350 400 450 500 550 600

1

2

3

4

5

x 103

*/

zz

*=0

zz

H(MPa)

Axialhkl

350 400 450 500 550 600

1

2

3

4

5

x 103

*/

zz

*=0.4

zz

H(MPa)

Axialhkl

350 400 450 500 550 600

1

2

3

4

5

x 103

*/

zz

*=0.7

zz

H(MPa)

Axialh

kl

350 400 450 500 550 600

1

2

3

4

5

x 103

*/

zz

*=1

zz

H(MPa)

Axialh

kl

{200}

{311}

{220}

{222}

typical uncertainty

Fig. 7. Axial lattice strains measured at rHzz. Each graph corresponds to a different biaxial ratio r

hh

rzz.

1 Note that in the data for a biaxial ratio ofr

hh

rzz0:4 the relative order of

the {3 1 1} and {22 0} strains is inverted with respect to the cases r

hh

rzz0

and r

hh

rzz0:7. This mismatch can be ascribed to the way the reference

stress free state was defined at the beginning of the experiment and is likely

an experimental artifact.



9/17

axial tests. At higher r

hh

rzzratios, it is observed that the spread

in the strains diminishes to the point that, for a biaxialratio of

rhh

rzz 1, the lattice strains are virtually the same

for all {h k l}s. The axial strains associated with the{222}[001] fiber, the stiffest one, are only weakly depen-dent on the level of biaxiality; no significant differences

are detected across the four graphs inFig. 7. This suggeststhat the more compliant the {h k l}, the more sensitive is itsdependence on

rhh

rzz.

The axial lattice strains obtained in the partially

unloaded state, when the axial stress was rLzz 200 MPa,are reported in Fig. 8. The abscissa of the graphs is the

stress rHzz reached before starting the unloading episode.

For r

hh

rzz 0, the partially loaded state is along the same

macroscopic path as the loaded condition (both are puretension). For all the other cases, the axial and hoop stresscomponents are in a different proportion in the partiallyloaded state than in the fully loaded state. With increasing

biaxiality, we see the increased importance of the Poissoneffect from the hoop stress, which is higher for greater biax-iality and constant during unloading, on the lattice strainsin the partially unloaded state. In fact, at a biaxiality ofr

hh

rzz 1, the relative order of the strains is reversed from

the case of r

hh

rzz0, with the {200}[001] and {311}[001]

fibers exhibiting small compressive strains. We now turn

our attention to the lattice strains in the hoop ([0 1 0]) direc-tion. In the graphs ofFig. 9, the hoop lattice strains for aparticular fiber, {h k l}, are plotted as a function ofrHzz forthe various levels of biaxiality. In general, the hoop latticestrains decrease with higher axial stress as a consequence of

the Poisson contraction. Under tensile loading r

hh

rzz0 ,

the strains are negative (compressive); under balanced

biaxial loading r

hh

rzz1

they are positive. In contrast to

the axial lattice strains, the hoop strains for all of the fibersare sensitive to the level of biaxiality.

Ideally, under balanced biaxial tension r

hh

rzz1

, the

hoop strains and the axial strains are equal from symmetry.The loading imposed in these experiments is not ideal intwo main respects: the hoop stress is not completely uni-form and the loading is not radial. Nevertheless, it is stillpossible to compare the axial and hoop strains for a biaxialratio of

rhh

rzz1 at an axial stress level of 370 MPa, where we

expect the condition of balanced biaxiality to be most clo-sely achieved. At this stress state the lattice strains in theaxial and hoop directions should be the same for each crys-tallographic plane {h k l}. From Figs. 7 and 9 atrzz= 370 MPa, the lattice strains are between 1 10

3

and 1.3103 for the axial measurements and between1103 and 1.7103 for the hoop measurements, ascan be seen more easily in a later figure (Fig. 17).

350 400 450 500 550 6001

0

1

2

3x 10

3

*/

zz

*=0

zz

H(MPa)

Axialhkl

350 400 450 500 550 6001

0

1

2

3x 10

3

*/

zz

*=0.4

zz

H(MPa)

Axialhkl

350 400 450 500 550 6001

0

1

2

3

x 103

*/

zz

*=0.7

zz

H(MPa)

Axialhkl

350 400 450 500 550 6001

0

1

2

3

x 103

*/

zz

*=1

zz

H(MPa)

Axialhkl

{200}

{311}

{220}

{222}

typical uncertainty

Fig. 8. Axial strains at the lowest stress level reached during the unloading episodesrLzz 200 MPa. Each graph corresponds to a different biaxial ratior

hhrzz.



10/17


11/17

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5x 10

3

*/

zz

*=0

zz

H(MPa)

Axia

lhkl

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5x 10

3

*/

zz

*=0.4

zz

H(MPa)

Axia

lhkl

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5x 10

3

*/

zz

*=0.7

zz

H(MPa)

Axialhkl

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5x 10

3

*/

zz

*=1

zz

H(MPa)

Axialhkl

{200}

{311}

{220}

{222}

typical uncertainty

Fig. 10. Axial lattice strain changes,Dhkl, as a function ofrHzzand linear fits. Each graph corresponds to a different biaxial ratio

rhh

rzz.

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5

x 103 {200}

zz

H(MPa)

Axial200

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5

x 103 {311}

zz

H(MPa)

Axial311

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5 x 103

{220}

zz

H(MPa)

Axial220

325 375 425 475 525 575 6250

0.5

1

1.5

2

2.5

3

3.5 x 103

{222}

zz

H(MPa)

Axial222

0

0.4

0.7

1

typical uncertainty

Fig. 11. Axial lattice strain changes, Dhkl, as a function ofrHzzand linear fits. Each graph corresponds to a different crystallographic plane {h k l}.



12/17

325 375 425 475 525

10

8

6

4

2

0

2x 10

4 {200}

zz

H(MPa)

Ho

op

200

325 375 425 475 525

10

8

6

4

2

0

2x 10

4 {311}

zz

H(MPa)

Ho

op

311

325 375 425 475 525

10

8

6

4

2

0

2x 10

4 {220}

zz

H(MPa)

Hoop

220

325 375 425 475 525

10

8

6

4

2

0

2x 10

4 {222}

zz

H(MPa)

Hoop

222

0

0.4

0.7

1typical uncertainty

Fig. 12. Hoop lattice strain changes,Dhkl, as a function ofrHzzand linear fits. Each graph corresponds to a different crystallographic plane {h k l}.

350 400 450 500 550 600100

150

200

250

300

350

400

{200}

zz

H(MPa)

E200

D

(GPa)

0

0.4

0.7

1

350 400 450 500 550 600100

150

200

250

300

350

400

{311}

zz

H(MPa)

E311

D

(GPa)

350 400 450 500 550 600100

150

200

250

300

350

400{220}

zz

H(MPa)

E220

D

(GPa)

350 400 450 500 550 600100

150

200

250

300

350

400{222}

zz

H(MPa)

E222

D

(GPa)

Fig. 13. Diffraction modulusEDhkl computed for each unloading episode. Each graph corresponds to a different crystallographic plane {h k l}.



13/17

can be noted that the diffraction moduli are fairly constantfor the various {h k l}s, regardless of the biaxial ratio.Again, this is expected because the load change is the samein all tests and the behavior is in a linear regime.

As mentioned in Section2, measuring the lattice strainsover unloading episodes subsequent to plastic strainingprovides insight into both the elastic and plastic anisot-

ropy. This is evident in the comparison of the lattice strainsfor the biaxial ratio of r

hh

rzz0 inFigs. 7 and 10. In partic-

ular, we point to the relative magnitudes of the strains. Thestrain differences inFig. 10show a clear difference betweenthe behavior of the {22 0}[00 1] and {22 2}[00 1] fibers,which is reflected as well in the diffraction moduli. Thisbehavior is dominated by the elastic response, as supportedby the linearity indicated by comparison of the variouscases. In contrast, the strains for the {22 0}[00 1] and{22 2}[0 0 1] fibers are quite similar inFig. 7. During plasticflow the crystal stress is constrained to lie on the aniso-tropic, single crystal, yield surface. Thus, the differencesbetween these strains for various {h k l}s reflects both the

plastic and elastic anisotropies.

5. Elastic behavior of polycrystals under uniform stress

A thorough simulation of these experiments requires amicromechanical model for the material behavior and theprescription of the complex loading history. Polycrystalplasticity theory implemented in self-consistent and finiteelement frameworks[40] have proven to be effective toolsfor modeling the deformation response of crystalline mate-

rials. Prior to exercising such an approach, however, con-siderable insight may be obtained using a simple modelbased on an assumption of uniform stress or uniform strainthroughout a polycrystalline aggregate. Such models arewidely used in the analysis of experimental data[41]. Herewe discuss the elastic behavior predicted under a uniform

stress assumption, known as a Voigt model [42]. We alsoexamined the behavior under an assumption of uniformstrain, or Reuss model [43], but this gives similar resultsto the Voigt model with isotropic moduli, so it will notbe discussed further.

Following this modeling approach, the material is apolycrystal that is conceptually represented by a popula-tion of crystals each being characterized by its lattice orien-tation. The population is defined by an orientationdistribution function over the fundamental domain of lat-tice orientations appropriate for the crystal symmetry, inthis case cubic. An aggregate property is determined byaveraging the single crystal responses over this space of ori-

entations. A directional property is determined by averag-ing the single crystal responses along a loci of points (fiber)passing through the space. Here we adopt a Rodriguesparametrization of orientations [44,45] and represent theorientation distribution over the fundamental domain withcontinuous piecewise polynomials (finite elements) [46,47].The mesh used for the computations is composed of 19,208tetrahedral elements and is shown in Fig. 14a.

We assume that all the crystals in the microstructure aresubjected to the same macroscopic stress state and computethe average lattice strains associated with the fibers exam-ined experimentally. In a cylindrical coordinate system

rhz, the stress tensor experienced by the gage volumeof a sample is

rmacro

rrr 0 0

0 rhh 0

0 0 rzz

264

375 8

In the ideal case, rrr= 0. We also examined the case de-fined by having rrr and rhh take on values from Eqs.(1)(3)withr = rm, which gives the values of the stress compo-nents at the midplane of the tube wall. The strain at each

Table 1Average diffraction modulus EDhkl computed in the unloading episodes ofthe axial tests

Biaxial ratio r

hh

rzzFamily of planes

{2 0 0} {3 1 1} {2 2 0} {2 2 2}

0 144 (14) 180 (14) 197 (19) 231 (24)

0.4 144 (12) 190 (17) 195 (39) 263 (32)0.7 150 (12) 181 (15) 209 (31) 267 (48)1 143 (15) 175 (9) 191 (27) 231 (34)

Standard deviation of each value given in parentheses. Values in GPa.

Fig. 14. (a) The axial lattice strain for rhh/rzz= 0.7, rzz= 370 MPa, and rrr = 0 plotted over the cubic fundamental region of Rodrigues space, as

predicted by the uniform stress model. (b) The orientations contributing to diffraction for the fiber {220}[001].



14/17

orientation is computed assuming anisotropic linear elasticbehavior at the crystal level according to the followingequation:

r C 9

where C is the crystal elastic stiffness tensor. In a framealigned with the crystallographic axes, the non-zero elasticmoduli are C11, C12 and C44. The anisotropy ofC impliesthat the strains computed from Eq.(9) depend on the lat-tice orientation and thus on the position within the funda-mental domain. There are three sets of elastic moduli listed

inTable 2. The first set are representative values used pre-viously in the modeling of neutron diffraction experimentson AL6XN under in situ uniaxial loading [27]. The othertwo sets were obtained by modifying the first set to changethe level of anisotropy. The changes were made in such away as to hold the bulk modulus and average shear modu-lus fixed while altering the anisotropic ratio[48]:

Table 2Elastic moduli for AL6XN, values in GPa

Set C11 C12 C44 rE

1 190.9 125.5 230.4 3.02 208.9 116.8 204.3 2.03 245.9 98.0 147.9 1.0

Elasticity convention used forC44.

0 0.25 0.5 0.75 10.5

1.5

2.5

3.5

4.5x 10

3

/

zz

Axialhkl

rE= 3

{100}

{311}

{110}{111}

0 0.25 0.5 0.75 12

1

0

1

2

3x 10

3

/

zz

Hoop

hkl

rE= 3

{100}

{311}

{110}{111}

0 0.25 0.5 0.75 10.5

1.5

2.5

3.5

4.5x 10

3

/

zz

Axialhkl

rE= 2

{100}

{311}

{110}{111}

0 0.25 0.5 0.75 12

1

0

1

2

3x 10

3

/

zz

Hoop

hkl

rE= 2

{100}

{311}

{110}{111}

0 0.25 0.5 0.75 10.5

1.5

2.5

3.5

4.5x 10

3

/

zz

Axialhkl

rE= 1

{100}

{311}{110}

{111}

0 0.25 0.5 0.75 12

1

0

1

2

3x 10

3

/

zz

Hoop

hkl

rE= 1

{100}

{311}{110}

{111}

Fig. 15. Lattice strains calculated using a uniform stress assumption for three levels of single crystal elastic anisotropy:rE= 3, 2 and 1. The axial stressrzzis fixed at a value equal to the yield stress (370 MPa). Two values of the radial stress rrrare considered: the value at the midplane according to Eqs.(1)(3)

(indicated by the dashed lines) and zero, which corresponds to the value at the outer wall (indicated by the solid lines).



15/17

K1

3C112C12 10

Gavg1

4C11C12C44 11

rE 12 C112C12

C11C12

12 C112C12C44

12

The parameter set with rE= 1 provides isotropic behavior,while the set having rE= 2 provides behavior between thereference set havingrE= 3 and the isotropic set. A uniformorientation distribution function was used rather than themeasured one to facilitate comparison with values reportedin the literature.

Calculations were performed for macroscopic stressstates rmacro with biaxial ratios ranging from 0 to 1. Forthis purpose, we computed the lattice strains at a singlevalue of the axial stress for each biaxial ratio. Specifically,rzz was set to a value equal to the observed macroscopic

yield stress of the material (ryield= 370 MPa). rhhwas var-ied between 0 and 370 MPa to obtain biaxial ratios atpoints within the desired range. Two cases were consideredat each biaxial ratio, one with rrr equal to zero and theother with rrr defined by Eqs. (1)(3) using r= rm. Fig.14a shows the axial lattice strain computed using the firstset of moduli as a function of orientation for the ratiorhh/rzz= 0.7, plotted on the surface of the cubic fundamen-tal region. This distribution is influenced by the type ofloading and its orientation with respect to the crystal sym-metry axes, and thus changes with the level of biaxiality.

Following[48], the lattice strains of crystals contributing

to the virtual diffraction are determined as the mean valuesof specified strain components along the appropriate crys-tallographic fibers. Specifically, the axial and hoop strainsassociated with the {h k l}[001] and {h k l}[010] fibers werecomputed[47]. A resolution angle X must be specified toindividuate the subset of crystals contributing to the dif-fraction peaks. Given a particular crystallographic plane{h k l}, only those orientations whose crystal plane normallies within the solid angle X is considered to be active. Avalue of 5 was chosen. An example for one diffractionpeak is presented inFig. 14b, where the fiber {220}[001]is mapped through the fundamental region and illustratedwith representative points that are colored according to the

magnitude of the axial strain.The average axial and hoop lattice strains for several

fibers are shown inFig. 15for the three sets of elastic mod-uli. A number of trends are apparent. Both the hoop andthe axial strains vary linearly with the biaxial ratio. At afixed biaxial ratio, the spread between the lattice strainsfor different fibers increases with the anisotropy ratio; inthe limiting case of isotropic moduli, the lattice strainsfrom all fibers are the same at fixed biaxial stress ratio.With increased biaxial stress ratio, the axial strainsdecrease in magnitude and the relative spread across thedifferent fibers diminishes. For the hoop strains, the strains

predicted for each set of parameters are equal for all fibers

when the biaxial stress ratio is 0.5 and rrr is zero. Thespread in the hoop strains across the different fibersincreases with biaxiality in either direction from a biaxialstress ratio of 0.5 if the elastic behavior is anisotropic.The effect of the radial component of stress is slight overthe full range of the biaxial ratio.

We can understand these trends, in part, by examiningthe deviatoric and mean parts of the stress as a functionof the biaxial ratio. InFig. 16observe that the mean stressincreases with increasing biaxiality, which is expected fromthis analysis because the hoop stress is increased whileholding the axial stress constant to achieve higher levels

of biaxiality. The axial component of the deviatordecreases as the biaxial ratio increases because larger val-ues of the mean stress are subtracted from the fixed axialstress component. The hoop component of the deviatorincreases with higher biaxiality because the method ofincreasing the biaxiality in the analysis consisted of increas-ing the hoop stress, which increases the deviatoric compo-nent at a higher rate than the mean component. Note thatthe deviatoric component of the hoop stress is zero at abiaxial stress ratio of 0.5 with rrr= 0 (it is zero at a slightlyhigher level of biaxiality ifrrr is not taken as zero).

For cubic crystals, the deviatoric behavior is anisotropicwhile the volumetric response is isotropic. Thus, the hydro-static stress generates the same strain for all crystallo-graphic directions. The greater its relative contribution tothe total stress, the less the influence of the deviatoric partof the stress on the net behavior and the less spread is antic-ipated across the different fibers. Because the deviatoriccomponent of the hoop stress is zero for a biaxial stressratio of 0.5 (under the rrr= 0 assumption), the latticestrain is due solely to the hydrostatic stress and the valuesfor all crystallographic directions are equal.

The simple Voigt model does not take into account theinfluence of the spatial distribution of the crystals or theirinteractions. Further, only the elastic response is explicitly

represented. Yet, it does capture one of the main trends of

0 0.25 0.5 0.75 10.5

0.25

0

0.25

0.5

0.75

/

zz

Stressc

omponents

(mean)

zz

(dev)

(dev)

Fig. 16. Values of the deviatoric and mean components for biaxial stressstates ranging from uniaxial to balanced biaxial. Axial stress componentrzz= 1 is assumed.



16/17

the experimental results, presented in Fig. 17 for strainsmeasured at rzz= ryield.

2 Namely, as the biaxialityincreases, the differences in axial strains across the various{h k l}s decrease. We observe that for uniaxial tensionrhh

rzz 0 the differences between the axial strains across

the different {h k l}s are strong, but these differences lessenas

rhh

rzz!1. The relative weighting of the hydrostatic and

deviatoric stress components as a function of stress biaxial-ity does not fully explain the trends, however. The hoopcomponents measured experimentally do not collapse toa single value at a biaxial stress ratio of 0.5, as predictedby this simple model. Further, there is much less variationobserved in either the measured axial or hoop strainsacross the {h k l}s than is computed. All of the differencescannot be rectified simply by altering the anisotropy ratio.Rather, as stated earlier, the differences observed in theseparations of the {h k l} curves between the fully loadedcondition (Fig. 7) and the strain changes on unloading(Fig. 10) suggest that transition to plastic flow alters thestress state from that expected from an elastic analysis.The data provide an excellent resource for evaluating mod-els that include both a more rigorous basis for grain inter-actions and a more complete description of the elastic andplastic behaviors.

6. Conclusions

An innovative set of in situ neutron diffraction experi-ments were performed on tubular, stainless steel specimens

to measure lattice strains under biaxial stress states. Differ-ent levels of stress biaxiality were achieved, ranging fromnearly uniaxial to nearly balanced biaxial states at yield,by varying the relative magnitudes of axial load and inter-nal pressure. An important feature of the method describedis that the principal stress directions are fixed with respectto the material microstructure. This eliminates difficultiesassociated with orientation effects when varying the levelof biaxiality.

Using this capability, lattice strains were determined inthe axial and hoop directions for several crystallographicreflections to load levels sufficient to induce plastic yielding.Unloading episodes were conducted periodically. The mainfindings pertain to the strains measured along the axial

direction, which is the direction subjected to the loadingunloading episodes. They are: (i) the average axial latticestrains for the different reflections are less distinct fromone another for increasing levels of biaxiality; and (ii) thechanges in the average axial lattice strains under removalof the axial load alone exhibit the same response regardlessof the level of biaxiality, as expected from linearity of theelastic behavior.

Analyses were performed using a simple model based onan assumption of uniform stress to aid in the interpretationof the diffraction data. The analyses indicate that one con-tribution to the observed trends in the experimental data

can be attributed to the changing proportions of the meanand deviatoric stresses with biaxiality. Because the volu-metric response is isotropic in this case, the differences inthe average axial strains for the various reflections are sup-pressed as the relative proportion of mean stress rises withincreased biaxiality. However, such simple models do notexplain the observed responses completely, as these requirea more complex description of grain interactions and inclu-sion of the inelastic behavior.

Acknowledgements

Support was provided by the US Office of Naval Re-search (ONR) under Contract N00014-06-1-0241 (forT.M. and P.R.D.). Neutron diffraction experiments wereperformed on a National Research Council (Canada) neu-tron diffractometer located at the NRU Reactor of AECL(Atomic Energy of Canada Limited).

References

[1] Cullity BD, Stock SR. Elements of X-ray diffraction. 3rd ed. PrenticeHall; 2001.

[2] Noyan IC, Cohen JB. Residual stress measurement by diffraction andinterpretation. Springer; 1987.

[3] Lu J, editor. Handbook of measurement of residual stresses. Society

of experimental mechanics. Fairmont Press; 1996.

0 0.25 0.5 0.75 10.5

1.5

2.5

3.5

4.5x 10

3

/

zz

Axialhkl

0 0.25 0.5 0.75 12

1

0

1

2

3x 10

3

/

zz

Hoop

hkl

{200}

{311}

{220}

{222}

typical uncertainty

Fig. 17. Axial and hoop lattice strains obtained from the experimental measurements at rzz= 370 MPa.

2 Note that, for the tests where the experimental data were missingbecause rzzwas not exactly equal to ryield, an interpolated value based on

a linear fit of the lattice strain for each {h k l} was used.



17/17

[4] Miller MP, Bernier JV, Park J-S, Kazimirov A. Experimentalmeasurement of lattice strain pole figures using synchrotron X-rays.Rev Sci Instrum 2005;76:113903.

[5] Pang JWL, Holden TA, Mason TE. In situ generation of intergran-ular strains in an Al7050 alloy. Acta Mater 1998;46:150318.

[6] Margulies L, Lorentzen T, Poulsen HF, Leffers T. Strain tensordevelopment in a single grain in the bulk of a polycrystal underloading. Acta Mater 2002;50(7):17719.

[7] Leinert U, Han T, Almer J, Dawson P, Leffers T, Margulies L, et al.Investigating the effect of grain interaction during plastic deformationof copper. Acta Mater 2004;52:44617.

[8] Krawitz AD. Introduction to diffraction in materials science andengineering. Wiley; 2001.

[9] Holden TA, Clarke AP, Holt RA. Neutron diffraction measurementsof intergranular strains in MONEL-400. Metall Mater Trans A1997;28:256575.

[10] Clausen B, Lorentzen T, Leffers T. Self-consistent modelling of theplastic deformation of F.C.C. polycrystals and its implications fordiffraction measurements of internal stresses. Acta Mater1998;46:308798.

[11] Pang J, Holden T, Wright JS, Mason T. The generation ofintergranular strains in 309H stainless steel under uniaxial loading.Acta Mater 2000;48:113140.

[12] Dawson PR, Boyce DE, Rogge RB. Issues in modeling heterogeneousdeformations in polycrystalline metals using multiscale approaches.Comput Model Eng Sci 2005;10(2):12341.

[13] Clausen B, Lorentzen B, Bourke MAM, Daymond MR. Latticestrain evolution during uniaxial tensile loading of stainless steel.Mater Sci Eng 1999;A259:1724.

[14] Dawson P, Boyce D, MacEwen S, Rogge R. Residual strains inHY100 polycrystals: comparisons of experiment and simulations.Metall Mater Trans A 2000;31:154355.

[15] Turner PA, Tome CN. A study of residual stresses in Zircaloy-2 withrod texture. Acta Mater 1994;42:404355.

[16] Cho J, Dye D, Conlon K, Daymond M, Reed R. Intergranular strainaccumulation in near-a titanium alloy during plastic deformation.Acta Mater 2002;50:484764.

[17] Gharghouri MA, Weatherly GC, Embury JD, Root J. Study of themechanical properties of Mg7.7 at.% Al by in situ neutrondiffraction. Philos Mag A 1999;79:167195.

[18] Rangaswamy P, Bourke M, Brown GKDW, Rogge R, Stout M,Tome C. A study of twinning in zirconium using neutron diffractionand polycrystalline modeling. Metall Mater Trans A 2002;33:75762.

[19] Allen AJ, Bourke MAM, Dawes S, Hutchings MT, Withers PJ. Theanalysis of internal strains measured by neutron diffraction in Al/SiCmetal matrix composites. Acta Metall Mater 1992;40:236173.

[20] Withers PJ, Clarke AP. A neutron diffraction study of loadpartitioning in continuous Ti/SiC composites. Acta Mater1998;46:658598.

[21] Oliver EC, Withers PJ, Daymond MR, Ueta S, Mori T. Neutron-diffraction study of stress-induced martensitic transformation inTRIP steel. Appl Phys A 2002;74(Suppl.):S11435.

[22] Mark AFL. Microstructural effects on the stability of retainedaustenite in transformation induced plasticity steels. PhD thesis,Queens University; 2007.

[23] Dunand DC, Mari D, Bourke MAM, Roberts JA. NiTi and NiTi-TiCcomposites. Part IV: Neutron diffraction study of twinning andshape-memory recovery. Metall Mater Trans A 1996;27:282036.

[24] Hutanu R, Clapham L, Rogge RB. Intergranular strain and texture insteel Luders bands. Acta Mater 2005;53:351724.

[25] Pang JWL, Rogge RB, Donaberger RL. Effects of grain size onintergranular strain evolution in Ni. Mater Sci Eng A 2006;437:215.

[26] Tomota Y, Lukas P, Neov D, Harjo S, Abe YR. In situ neutrondiffraction during tensile deformation of a ferritecementite steel.Acta Mater 2003;51:80517.

[27] Dawson PR, Boyce DE, Rogge RB. Correlation of diffraction peakbroadening to crystal strengthening in finite element simulations.Mater Sci Eng A 2005;399:1325.

[28] Imbeni V, Mehta A, Robertson SW, Duerig TW, Pelton A, Ritchie R.On the mechanical behavior of nitinol under multiaxial loadingconditions and in situ synchrotron X-ray diffraction. In: Shapememory and superelastic technologies conference. Menlo Park, CA:SMST Society Inc.; 2003. p. 22736.

[29] Mehta A, Gong X-Y, Imbeni V, Pelton AR, Ritchie R. Understand-ing the deformation and fracture of Nitinol endovascular stents usingin situ synchrotron X-ray microdiffraction. Adv Mater2007;19:11836.

[30] Gnaupel-Herold T, Liu D, Prask H. Biaxial fatigue failure ingas pipelines. Tech. rep. NIST Special Publication 1075, NISTCenter for Neutron Research Accomplishments and Oppor-tunities; 2007.

[31] Standard practice for strain-controlled axialtorsional fatigue testingwith thin-walled specimens. Annual Book of ASTM Standards; 1996.Designation: E 2207 -02.

[32] Gharghouri MA, Marin T, Dawson PR, Rogge RB. In situ neutrondiffraction study of the behavior of AL6XN stainless steel underbiaxial loading. In: Thompson C, Durr HA, Toney MF, Noh DY,editors. Materials in transition: insights from synchrotron andneutron sources, Mater. Res. Soc. Symp. Proc., vol. 1027E; 2007[Paper Number 1027-D01-08].

[33] Green DE, Neale KW, MacEwen SR, Makinde A, Perrin R.Experimental investigation of the biaxial behavior of an aluminumsheet. Int J Plasticity 2004;20:1677706.

[34] Smits A, Van Hemelrijck D, Philippidis TP, Cardon A. Design of acruciform specimen for biaxial testing of fibre reinforced compositelaminates. Compos Sci Technol 2006;66:96475.

[35] Makinde A, Thibodeau L, Neale KW. Development of an apparatusfor biaxial testing using cruciform specimens. Exp Mech1992;32:13244.

[36] Flugge W. Stresses in shells. New York: Springer-Verlag; 1960.[37] MacEwen S, Faber Jr J, Turner A. The use of time-of-flight neutron

diffraction to study grain interaction stresses. Acta Metall1983;31(5):65776.

[38] Allvac AL 6XN 6XN UNS N08367 UNS N08367 Specialty Steel.MatWeb Material Property Data. .

[39] Ritz H, Dawson PR, Rogge RB. Measuring the influence ofmagnesium on the elastic anisotropy of aluminum using in situneutron diffraction. J Neutron Res, in press.

[40] Kocks UF, Tome CN, Wenk H-R. Texture and anisotropy. Cam-bridge University Press; 1998.

[41] Raabe D, Roters F, Barlat F, Chen L-Q, editors. Continuum scalesimulation of engineering materials. WileyVCH; 2004.

[42] Voigt W. Uber die beziehung zwischen den beiden elastizitatskonstanten isotroper korper. Wied Ann 1889;38:57387.

[43] Reuss A. Berechnung der fliessgrenze von mischkristallen auf grundder plastizitatsbedingung fur einkristalle. Z Angew Math Mech1929;9:4958.

[44] Frank F. Orientation mapping. In: Kallend SJ, Gottstein G, editors.Proceedings of the eighth international conference on textures ofmaterials. Warrendale, PA: The Metallurgical Society; 1988. p. 313.

[45] Becker S, Panchanadeeswaran S. Crystal rotations represented asRodrigues vectors. Text Microstruct 1989;10:16794.

[46] Kumar A, Dawson PR. The simulation of texture evolution withfinite elements over orientation space. I: Development. ComputMethod Appl Mech Eng 1996;130:22746.

[47] Boyce DE. ODF/PF Function Set. .

[48] Dawson PR, Boyce DE, MacEwen S, Rogge RB. On the influence ofcrystal elastic moduli on computed lattice strains in AA-5182following plastic straining. Mater Sci Eng A 2001;313:12344.

http://www.matweb.com/http://anisotropy.mae.cornell.edu/onr/welcome.htmlhttp://anisotropy.mae.cornell.edu/onr/welcome.htmlhttp://anisotropy.mae.cornell.edu/onr/welcome.htmlhttp://anisotropy.mae.cornell.edu/onr/welcome.htmlhttp://www.matweb.com/

Marin ActaMat 2008

Documents