on the influence of crystal elastic moduli on computed lattice strains in aa-5182 following plastic...

Materials Science and Engineering A313 (2001) 123–144

On the influence of crystal elastic moduli on computed latticestrains in AA-5182 following plastic straining

Paul Dawson a,*, Donald Boyce a, Stuart MacEwen b, Ronald Rogge c

a 196 Rhodes Hall, Cornell Uni�ersity, Ithaca, NY 14853, USAb Alcan International Limited, Kingston, Ont., Canada K7L5L9

c National Research Council, Chalk Ri�er, Ont., Canada K0J1J0

Received 4 May 2000; received in revised form 28 December 2000

Abstract

Crystal lattice plane spacing is modified by the application of stress. The changes in spacing can be measured with neutrondiffraction and used to determine the elastic strains in loaded crystals. Using finite element methods, elastic strains can becomputed under loading that mimics the experiment. The quality of comparisons between the measured and computed strainsdepends strongly on accurate knowledge of parameters that quantify the single crystal elastic and plastic responses. For onealuminum alloy in particular, we have found that we can improve the match of lattice strains through careful choice of the singlecrystal elastic moduli. The parameters are selected on the basis of comparisons between the experimental results and a series ofsimulations in which the single crystal moduli were varied systematically. Good correspondence is obtained for a set of moduliwith higher single crystal anisotropy than those of pure aluminum. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Crystal elastic moduli; Computed lattice strains; AA-5182; Finite element; Neutron diffraction

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1. Introduction

The mechanical behaviors of polycrystalline materialsdepend on the properties of individual crystals as wellas the interactions of crystals with each other. Accurateknowledge of single crystal properties is essential to thesuccessful modeling of the mechanical response of poly-crystals. For predictive models to capture the observedbehaviors well, both reliable constitutive relations andaccurate determination of the parameters appearing inthem are needed. For some behaviors, the single crystalrelations are well accepted. For many other behaviors,however, relations that perform well over an adequaterange of imposed conditions have not yet been iden-tified. For example, the single crystal elastic relationsfor cubic metals are widely accepted, but the relationsfor the strain hardening of the crystal slip systems arenot. Even when the constitutive relations are not inquestion, the values of parameters may not be knownwith a high level of certainty.

Testing of single crystals is the obvious preference formeasuring properties of single crystals. However, foralloyed metals, it may not be feasible to obtain singlecrystals for mechanical testing with sufficient size andwith similar microstructure to those in a polycrystal.Nor is it advisable to assume that the single crystalparameters of the alloy are the same as those of thepure material, given the sensitivity of computed re-sponses to these parameters. Consequently, it becomesnecessary to determine them indirectly from measure-ments taken on polycrystals. This is made difficult bythe effects of grain interactions on the measured re-sponses of polycrystals. A range in single crystalparameter values may be obtained by invoking bounds,but even with the tightest bounds available, the rangemay be too large to enable precise simulation.

In this article, we present diffraction data taken fromtensile tests on AA-5182 in which elastic (lattice) strainswere measured in both the loaded and the unloadedstates. Finite element simulations in which each elementwas associated with an individual crystal were per-formed to replicate the experiments. We found that wecould not obtain a good match between the computed

* Corresponding author. Tel.: +1-607-2553466; fax: + 1-607-2559410.

E-mail address: [email protected] (P. Dawson).

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0921 -5093 (01 )00967 -4

P. Dawson et al. / Materials Science and Engineering A313 (2001) 123–144124

and measured lattice strains by using standard valuesfor the single crystal elastic moduli of pure aluminum(unalloyed aluminum of high purity), even if we homo-geneously scaled the moduli (to alter their values whilemaintaining constant proportionality between them).The lackluster agreement motivated a parametric studyin which we varied the degree of anisotropy of thesingle crystal moduli while holding the average direc-tional (Young’s) modulus constant. The best match tothe experimental data came with a set of single crystalmoduli of higher anisotropy than reported for the purematerial. While the moduli are different from those forpure aluminum, they lie well within the range of othercubic metals. Such a difference between the true moduliof the alloy and the corresponding moduli for the purematerial may indeed be real. However, there is legiti-mate concern over a false finding arising from numeri-cal artifacts that could result in an under-prediction ofanisotropy by the finite element model. To address thisconcern, the parametric study also included consider-ation of grain shape, sample size (numbers of crystals)and specimen initialization; further, the computed re-sponses were compared both to the lattice strain dataand to the peak intensity data. The conclusions reachedregarding the level of single crystal anisotropy for AA-5182 are independent of these factors.

Using single crystal moduli chosen from the paramet-ric study, the simulations of the tensile tests wererepeated using a more highly resolved specimen. Theresults of those simulations are presented, along withdetailed comparisons to the experimental data. Theresults are similar to those having the same moduli inthe parametric study. However, the larger numbers ofelements provide better statistics for averaging of thelattice strains, and consequently exhibit smoother histo-ries of evolving lattice strain and texture.

2. Background

Neutron diffraction is an effective experimental tech-nique for quantifying stresses at the level of individualcrystals in polycrystalline metals. Early development ofthe methodologies was reported by MacEwen, Faber,and Turner [1]. Measurements can be made either onthe externally unloaded specimens or, with the aid of anin situ load frame, on specimens in both externallyloaded and unloaded conditions. Use of the techniquehas since been reported for a number of applicationshaving macroscopically varying residual stress distribu-tions. Flower, MacEwen and Holden [2] reported onthe residual stress distribution in a thick stainless ringthat had been diametrically compressed to induce plas-tic deformation through the cross-section on loadingand to leave a condition of residual stress on unloading.Stress distributions in MONEL-400 and in Zircalloy-2

have been measured and compared with modeling pre-dictions by Holden, Clarke, and Holt [3] and byTurner, Christodoulou and Tome [4], respectively.

Recently several studies have been reported that fo-cus on stresses under loading conditions that aremacroscopically uniform, as is the case with the presentarticle. The intent is to better understand the develop-ment of stresses during plastic straining and to assessthe capability of certain modeling approaches to predictsuch behaviors. Clausen and coworkers [5–7] havemeasured the evolution of lattice strains in a number ofFCC materials (aluminum, copper and stainless steel)for several different sets of crystals corresponding todifferent diffraction peaks. They have compared thesedata to predictions made using a self-consistent schemeand contrasted the responses of the various peaks withregard to the degree to which each deviated from linearresponse. Pang, Holden and Mason [8,9] reported onthe generation of intergranular lattice strains in AA-7050 and in a high strength steel (350WT). Tensilespecimens were loaded in situ beyond the elastic limitwith diffraction measurements being taken during theloading and unloading. Subsequently, the residual lat-tice strains were measured for a suite of scatteringvector orientations relative to the tensile axis. Measuredstrains were compared with ones predicted using aself-consistent scheme. From the self-consistent model,they present a prediction of the magnitude of residuallattice strain as a function of the crystallographic orien-tation with respect to the specimen tensile axis. Whilethe experimental methodologies used in these investiga-tions are similar to that discussed in this article, themodeling methodology is significantly different fromthese prior works. Further, the focus of this work is onthe sensitivity of the lattice strains to the assumedmoduli values.

3. Experimental procedures and diffraction data

3.1. Strain measurement by neutron diffraction

Neutron diffraction measurements were made to de-termine lattice strains for a loading condition that ismacroscopically simple, namely uniaxial tension, asshown schematically in Fig. 1. Since neutrons penetratedeeply into aluminum, the measurements provide dataaveraged over a sampling volume located in the interiorof the specimen. To perform the measurement, a highlycollimated, monochromatic beam of neutrons providedby a nuclear reactor was directed toward the specimenas schematically depicted in Fig. 1. For a neutronwavelength, �, and an interplanar spacing, d, crystallo-graphic planes diffract the neutrons at an angle 2�

according to Bragg’s law [10].

P. Dawson et al. / Materials Science and Engineering A313 (2001) 123–144 125

�=2d sin �. (1)

In these experiments, the (115) lattice planes of agermanium monochromating crystal were used to selectneutrons of an approximate wavelength of 1.51 nm.The slit dimensions for the incident and diffractedneutron beams were chosen to maximize the number ofgrains for meaningful polycrystalline averaging, but toavoid instrumental aberrations. The slit width was 4.5mm for both for the incident and diffracted neutronbeams. Slit heights were selected to give the largestinstrumental gauge volume, while limiting background,for each scattering vector. These dimensions were 5 and20 mm for the axial and transverse specimen directions,respectively (axial and lateral directions are describedlater). All measurements were performed on the L3diffractometer located at the NRU research reactor,Chalk River Laboratories1, Ontario, Canada.

In a diffraction experiment, neutrons are diffractedover a distribution of angles — the diffraction peak —rather than at a single discrete angle. A distribution isobserved for several reasons, some related to the instru-ment and others to the sample. Instrumental effectsinclude the angular spread of the neutron beam paths,a small spread in neutron wavelength, and a mosaic

spread of the monochromating crystal. Sample-depen-dent sources include variations in the lattice planespacing within the sampled volume, disorder within thecrystals caused by dislocations and secondary phases,and finite size of the crystallites. The sample-dependentsources may or may not increase the width of thedistribution due to instrument resolution. A multiwiredetector was used to collect the count distribution anda Gaussian function with constant background wasfitted to the measured distribution [1,3]. Neutron count-ing intervals were sufficiently long to provide the de-sired level of uncertainty in the Gaussian parameters.

Three attributes of the distributions are of particularinterest — the change in the peak position (a measureof the average lattice plane spacing in the contributingcrystals); the change in the integrated peak intensity (ameasure of the texture evolution); and, the change inthe peak width (related to a number of factors asmentioned in the preceding paragraph). Of these, weare particularly interested in the average plane spacing(change in peak position). The lattice spacing within acrystal is altered by the application of stress. Latticestrains determined from changes in plane spacing areelastic strains. From the change in the average latticeplane spacing, �d, the normal component of latticestrain, �nn, parallel to the plane normal (which is coinci-dent with the scattering vector, as shown in Fig. 1) maybe computed [1] simply from

�nn=�dd0

(2)

where d0 is the reference plane spacing corresponding tothe unstressed condition. Subsequently, we refer tothese simply as the lattice strain components relative toparticular specimen directions (which are parallel to theassociated scattering vectors). Estimates of the experi-mental uncertainties associated with the average latticestrains are derived from the statistical quality of the fitsto the count distributions and are typically �0.0001.The uncertainties are shown with the data in the com-parisons of simulation to experiment presented in Sec-tions 5 and 6.

Each crystal has many families of lattice planes thatcan diffract neutrons. The planes examined in this workare given in Table 1, along with the nominal latticeplane spacings for AA-5182 powder. For a polycrys-talline material, only those crystals with a crystallo-graphic plane perpendicular to the scattering vector andthat satisfy Bragg’s law will diffract neutrons into thedetector. Thus, it is possible to selectively measurespecific lattice strain components in particular sets ofcrystals by choosing the appropriate combinations ofcrystallographic planes and scattering vector directionsrelative to the specimen orientation. For example, theaxial lattice strain for (400) planes refers to the normalcomponent of the elastic strain aligned with the speci-

Fig. 1. Neutron diffraction experiment schematic diagram. The insertdepicts a highly stylized image of diffracting crystals in the specimeninterior.

Table 1Diffraction plane powder data for AA-5182 (�=1.51 A� )

Plane spacing (A� )Family of planes {hkl} Angle (°)

63.6{220} 1.43{311} 1.22 76.4{222} 1.17 80.4{400} 96.41.01

1 The Chalk River Laboratories and NRU are owned and operatedby Atomic Energy of Canada, Ltd. The neutron scattering facilitiesare owned and operated by the National Research Council, Canada(see http://neutron.nrc.ca).


Fig. 2. Optical micrographs of AA-5182 taken from the RD-ND, TD-ND and RD-TD planes.Fig. 3. �100� and �111� pole figures for AA-5182. Scale is in multiple-of-random (MRD).


Fig. 4. Stress–strain curve for a typical specimen.

the average grain size specified above and the slitdimensions presented earlier, the instrumental gaugevolume contained on the order of 105–106 grains (lessfor the axial scattering vector measurements and morefor the transverse scattering vector measurements). Nosubsequent heat treatment was performed on the speci-mens prior to testing.

3.3. Loading sequence and measurement schedule

Specimens were stretched to approximately 15%strain at room temperature and at a nominal strain rateof 10−4 s−1 under displacement control using an ATSscrew-driven load frame. The loading was interruptedby unloading episodes at certain specified load levels tofacilitate diffraction measurements. During an unload-ing episode, the specimen was first unloaded to 95% ofthe specified load, and data for the four identifiedBragg peaks were obtained. Diffraction measurementswere repeated at a near zero load level2 and at the 95%reloaded level. The stress-strain curve for a typicalspecimen is indicated in Fig. 4. Labeled on the figureare the points in the unloading episodes at whichdiffraction measurements were made. The 95% loadlevel was chosen so as to prevent stress relaxation fromplastic deformation during the measurement process.However, the low crosshead speed allowed a smallamount of additional plastic deformation to occur asthe specimen load went from peak load to the 95%level. This accounts for the negative slope of the stress-strain curve between points of full load and the 95%levels, where diffraction measurements were taken.

A number of additional experiments were performedto assess the magnitude of stress relaxation over thedwell periods during which the diffraction measure-ments were made. These tests consisted of measuringdiffraction peaks at a number of times following un-loading to determine whether the average lattice strainsdiminish over the time frame of a typical unloadingepisode. No significant changes were detected, and con-sequently, stress relaxation was not viewed assignificant.

3.4. AA-5182 diffraction data

Three specimens were deformed to obtain data asso-ciated with the axial and transverse scattering vectors.In the first test, the scattering vector was aligned withthe specimen axial direction. In the other two tests, thescattering vector was aligned with either the TD-lateraldirection or the ND-lateral direction. Over the courseof each test, a total of eleven unloading-reloadingepisodes were performed. For each set of crystals, an

men axial direction in those crystals having (400) crys-tallographic planes perpendicular to the specimen axialdirection (i.e. the plane normals are parallel with theaxial direction to within a narrow solid angle of orien-tations about the scattering vector). Here, we measurestrains associated with three scattering vectors (oneaxial and two transverse with respect to the loadingaxis) and the families of crystal planes given in Table 1.In the remainder of the article, we will refer to the setof crystals with (hkl) planes orthogonal to the scatter-ing vector as (hkl) crystals.

3.2. Sample material and preparation

The material tested was an aluminum-magnesiumalloy (AA-5182), which has a nominal compositionconsisting of 95% Al, 4–5% Mg, and 0.35% Mn [11].Micrographs of the material for three planes are shownin Fig. 2. The fields of view were chosen to illustrategrain size, rather than constituent particle stringers. Themicrographs indicate that the grains that are fairlyequiaxed with an average size of less than 100 �m. Thecrystallographic texture of the plate stock was obtainedusing X-ray and neutron diffraction techniques inde-pendently, with the two providing equivalent character-izations. This alloy exhibited a mild cube texture, ascan be seen in the pole figures presented in Fig. 3.

Tensile specimens were machined from the platestock, with the specimen axis aligned with the platerolling direction (RD). The specimens were rectangularin cross-section, approximately 9 by 13 mm, and had agauge length of approximately 50 mm. The specimengauge length was chosen to guarantee sufficient lengthto eliminate end effects within the instrumental gaugevolume and to avoid interference of the neutron beamwith the loading apparatus. The short sides of thespecimens align with the plate normal direction (ND),and the long sides align with the plate transverse direc-tion (TD). Subsequently, in this article, specimen direc-tions will be referred to as either the axial direction, theND-lateral direction or the TD-lateral direction. Given

2 A small tensile load was retained (50 N or �2 MPa) to avoidintroducing any slack in the load train.


average lattice strain relative to the initial, near-zeroload level, lattice plane spacing was calculated from thecorresponding diffraction peak shifts. These averages,together with computed lattice strains, are shown forthe loaded and unloaded conditions of each episode forcrystal sets defined by the (220), (311), (222), and (400)lattice planes in Sections 5 and 6. In the case of theaxial direction scattering vector, the lattice strains re-ported are normal components of the strain tensor inthe specimen axial direction, and all averages are posi-tive under load (indicating extension); for both of thelateral scattering vector cases, the lattice strains arenormal components of the strain tensor in the specimenTD-lateral or ND-lateral direction, and all averages arenegative (indicating compression). The two lateral casesare similar, but not identical due to the initial texture inthe plate material. In the unloaded state followingplastic straining, some sets of crystals exhibit averagesthat are positive, and some exhibit averages that arenegative, for all of the scattering vectors. The peakintensities evolve over the course of the experiment as aconsequence of the imposed deformation. The mea-sured intensities are presented in Section 6 in conjunc-tion with the computed values.

4. Crystal equations and finite element methodology

4.1. Crystal elastic and plastic beha�iors

Crystal plasticity is a micromechanically-based ap-proach to modeling slip-dominated plastic flow in crys-talline solids. Plastic deformation in each crystal occursvia crystallographic slip within the crystal lattice, whichwith the material embedded in it, undergoes elasticstrains and rotations. Based on this description, thekinematics of crystal deformation is represented usingthe multiplicative decomposition of the deformationgradient F [12–15],

F=V*F� p and F� p=R*Fp (3)

where Fp is the purely plastic part of F, V* is thesymmetric left elastic stretch tensor, and R* is thelattice rotation. The deformation gradient F� p definesthe relaxed configuration B� obtained by unloading elas-tically, but without rotation, from the current configu-ration B to a stress free state. This configuration is usedto formulate the crystal constitutive Eq. (15). For thecase of small elastic strains, we write V*=I+�* with��*��1 and I the second-order identity tensor. Subse-quently, the kinematics can be expressed in rate form as

trD= tr�� * (4)

D �=�� *�+D� p+�*�W� p−W� p�*� (5)

W=W� p+�*�D� p−D� p�*� (6)

where D and W are the rate of deformation and spintensors, respectively, expressed in the current configura-tion B ; D� p and W� p are the plastic deformation rate andthe plastic spin tensors, respectively, written in therelaxed configuration B� ; D � and �*� are the deviatoriccomponents of D and �, respectively.

The crystal elasticity is assumed to be linearly an-isotropic. For the case of small elastic strains, it isexpressed as

�=L�* (7)

where L is the elasticity tensor and � is the Kirchhoffstress, which is related to �, the Cauchy stress, by�=��, �=det(I+�*). The specific form of L em-ployed here for cubic crystals is discussed in Section 5.

The viscoplastic flow rule is derived from crystallo-graphic slip and is expressed as

D� p=��

�� (s��m�)S and

W� p=R� *R*T+��

�� (s��m�)A (8)

where D� p and W� p are expressed in the configuration, B� ;s� and m� define the slip direction along and slip planenormal to the �-slip system at configuration B� ; thesubscripts S and A denote the symmetric and antisym-metric parts of the orientation tensor (s��m�). Theplastic shearing rate on the �-slip system, ��, is given bya power law form,

�� =�� 0�� (�)

�

�1/m

sgn(��) where ��=� �·(s��m�)S (9)

Here, �� is the resolved shear stress on the �-slipsystem, � is the slip system strength, �� 0 is a referenceshear rate, and m is the rate sensitivity of slip. The slipsystem strength is assumed to be the same for all slipsystems within a particular crystal (i.e. Taylor harden-ing), but is different from crystal to crystal due todifferences in the plastic deformations experienced bycrystals over the course of the deformation. The re-solved shear stress �� is plastic work rate conjugate to�� and is computed as the projection of � �, the devia-toric component of the Kirchhoff stress, onto the �-slipsystem. The slip system strength hardens with strainingaccording to a modified Voce law of the form

�� =h0� �s− �

�s−�0

�� (10)

where h0, �s and �0 are parameters specific to thematerial and �� =�� .

4.2. Finite element formulation

We restrict our attention to the isothermal, quasi-static, finite deformation of polycrystalline materials.Under these conditions, the field equations governing


the motion of the body B are given by the equilibriumequations plus appropriate surface traction (t� ) andvelocity (u) boundary conditions on the surfaceboundaries �B� and �Bu, respectively. Here, n is a unitnormal to the surface; body forces are neglected.

The formulation of the finite element equations isbased on the weak form of the equilibrium Eq. (16),�

B

� � · �� dV−�

B

p div � dV=�

�B�

t� · � dA (11)

where � is a weighting function for the velocity field u,� is the gradient operator, and � � and p are thedeviatoric and volumetric (pressure) components of themacroscopic Cauchy stress tensor �, i.e. �=� �−pI, p= −1/3 tr �. We note that the deformation rate iswritten in terms of the velocity as D=1/2(�u+ (�u)T).The constitutive behaviors are introduced separately forthe deviatoric and volumetric parts. First the elasticstrain rate �� * is written as a backward Euler differenceover the time interval, �t. Using this representation, thedeviatoric stress is written in term of the deviatoricdeformation rate by combining Eqs. (5)– (9). The resultis substituted into Eq. (11). The volumetric response isconstructed by combining Eqs. (4) and (7) with thedifference expression for the trace of the elastic strainrate, and then forming a residual over the domain B,�

B

(p+K�t

�trD+

K�

tr �*)q dV=0 (12)

where q is a weighting function for the pressure field p,�* is the elastic strain corresponding to the start of thetime step, and K is the elastic bulk modulus. The globalsystem of equations governing the motion of the poly-crystalline solid is obtained by summing the residuals ofEqs. (11) and (12) for all the elements comprising thediscretized domain and setting the total residual equalto zero. This procedure yields a nonlinear algebraicsystem of equations for the velocity, {U}, and pressure,{P} [16]. The formulation is three-dimensional, employ-ing eight-node, isoparametric, brick elements. Detailsare given in [16–19].

Polycrystal simulations often are quite resource in-tensive because of the large numbers of crystals re-quired to construct a truly representativethree-dimensional polycrystal. Consequently, parallelcomputing architectures are necessary for realizing sim-ulations with enough crystals to provide meaningfulstatistics. Parallel computing techniques are utilized atmany levels in the solution procedure employed to solve

the model equations summarized earlier [19]. The im-plementation of the formulation employed is a combi-nation of Fortran90 and the Message Passing Interface(MPI) standard.

4.3. Simulations of the experiments

A number of polycrystal finite element simulations ofthe AA-5182 were performed and compared with theexperimental data. The first set of simulations consti-tuted a parametric study spanning a range of values forthe single crystal elastic moduli. The parametric study isdescribed in greater detail in Section 5. These simula-tions were performed using a coarse mesh consisting of2000 finite elements. Subsequent to the parametricstudy, the experiments were simulated using mesheshaving many more elements (either 16 000 or 30 000)and specifying values for the elastic moduli correspond-ing to one of the sets of parameters from the parametricstudy. Those results are discussed later in Section 6.

In all of the simulations, each element of the mesh istreated as an individual crystal, and so is assignedinitial values for its own orientation, slip systemstrength, and elastic strain tensor. All of these statedescriptors are free to evolve during the course of thesimulation. The crystal lattice orientations were ob-tained by sampling the orientation distribution function(ODF) for the specimen material. The ODF was evalu-ated from the measured textures and expressed in termsof a set of harmonic coefficients [20]. A set of 100 000discrete orientations was generated randomly andweights were determined using the harmonic coeffi-cients. From these, a cumulative distribution functionwas created and sampled to construct a set of orienta-tions with unit weights. These orientations were as-signed to the elements in the mesh. The initial value ofslip system strength was taken to be constant across allelements, and the initial value of the elastic stretchtensor was taken as the identity (corresponding to zeroinitial lattice strain). There are twelve slip systems({111}�110�); the slip system parameters appearing inEqs. (9) and (10) are given in Table 2. These values andthe initial value of the strength were chosen so that thecomputed average axial stress agreed with the measuredaxial stress. There is some influence of the elastic mod-uli on this comparison, especially at specimen strains ofless than 2%. This point is discussed further in a latersection of the paper.

Boundary conditions were chosen to replicate theloading of crystals within the gauge section of thetensile specimen. One end of the mesh was constrainedin the axial direction while a specified axial velocity wasimposed on the other end; zero shear tractions wereimposed on both ends. Two adjacent lateral sides weretraction free, while on the other two a symmetry condi-tion was imposed. The velocity on the end was pro-

Table 2Slip system parameters for AA-5182

m (−) h0 (MPa)�� 0 (1/s) �0 (MPa) �s (MPa)

1551.0 0.02 42480


grammed to match the unloading episodes ofthe experiment. At each point at which neutrondiffraction measurements were made, the current orien-tation, hardness and elastic strain informationfor each element was written to a file. These rawsimulation results were subsequently post-processed toobtain quantities analogous to those obtained experi-mentally.

The first step in post-processing is to determinewhich subset of the orientations should be consideredactive when simulating a particular diffraction peak.Any element whose orientation (accounting for crystalsymmetries) is such that the designated crystal planenormal lies within a solid angle � of the scatteringvector is considered to be active in contributing to thesimulated diffraction peak. Experimentally, only thosecrystals that align with the scattering vector to a highdegree of resolution will contribute to the peak. Evenwith tens of thousands of elements, the simulationscannot produce the degree of resolution comparable tothe experiment, so typically we must specify a largerresolution angle, �, in the simulations than isset by the slit dimensions of the diffractometer. Theresolution angle should be as small as possible tomaximize the fidelity to the experiment, but we must atthe same time make it large enough to provide mean-ingful statistics. For the meshes with 2000 elements, wetypically used a value of 5°, which activatedbetween 5 and 107 elements; for the meshes with 16 000and 30 000 elements, we used a value of 2°, whichactivated between 10 and 135 elements and between 26and 239 elements, respectively. Notice that there is awide variation in the number of active elements for afixed angle of resolution. This is due both to the texturevariation and to certain crystal directions having ahigher multiplicity than others under crystal sym-metries.

In this work, we compare simulated lattice strainsand intensities to the experimental values. In bothcases, we first determine which elements are active forthe peak in question, as described above, and thenperform an appropriate average over that set of crystalsto obtain the simulated quantity. For the lattice strains,we use the elastic strain tensor, �*, to compute thestrain component in the direction of the specified planenormal for each element. We average over all elementsand record the result as the simulated lattice strain. Forthe intensities, we first divide the number of elementsactive at a given resolution by the total number ofelements to determine pa, the proportion of active ele-ments. Then we compare this to the proportion, p�,expected to be active for a uniform distribution (notethat p� accounts for the multiplicity of the particularcrystal direction). We record i=pa/p� as the simulatedintensity in multiples of random.

5. Parametric study

As background to the parametric study of the elasticmoduli, we note that the diffraction moduli (the ratioof change in macroscopic stress to change in averagelattice strain) depend on the grain interactions, and notjust the single crystal directional moduli for the respec-tive (hkl). Thus, the diffraction moduli depend implic-itly on the texture, as well. Due to this, it is not possibleto extract the directional moduli from diffraction datawithout the aid of a mathematical model. Models differwidely in the manner in which the grain interactions arehandled, as shown by de Wit [21] for untextured mate-rials. From a modeling perspective, let us arbitrarilycall grain interactions strong if compatibility is domi-nant so that all grains deform the same way. Con-versely, we call the interactions weak if compatibility ispassive, and the same stress acts on all grains. If thegrain interactions were strong, the diffraction moduliwould all equal the macroscopic modulus. If the inter-actions were weak, the diffraction moduli would equalthe single crystal directional moduli. Typically, neitherof these limits provides consistently good explanationsof observed responses. The parametric study presentedhere examines the influence of the elastic moduli on thecomputed changes in lattice strains for a number ofunloading episodes. We use a finite element formulationas our mathematical model. Two points of strength ofthe finite element model are in the quality of the graininteractions (each grain resides in a local neighborhoodof other crystals) and in the ease of including texture(including its evolution).

5.1. Single crystal elastic moduli

For a cubic material such as aluminum, three con-stants are required to determine the single crystal elasticbehavior. In a reference frame aligned with the cubicaxes, these three constants can be taken as scalingfactors for the off-diagonal entries, the deviatoric diag-onal entries and the volumetric entries of the straintensor. If we decompose the elastic strain tensor, �*,into its off-diagonal, deviatoric diagonal and sphericalparts, �off, �ddg and �sph, respectively, and denote thescaling constants by coff, cddg and csph, then the stresstensor, �, is related to the elastic strain tensor by [21]

�=coff�off+cddg�ddg+csph�sph. (13)

Note that if coff=cddg, then the single crystal iselastically isotropic, and 2G=coff=cddg, where G is theshear modulus. The volumetric response is inherentlyisotropic as it relates pressure changes to volumechanges, so 3K=csph, where K is the bulk modulus.The three constants are more commonly specified ascertain entries of the compliance or stiffness matrices,again in a reference frame aligned with the cubic axes,


Fig. 5. Values of the normalized spherical modulus csph correspond-ing to fixed Eavg. Superimposed are loci of constant rE and constantavg lying on the fixed Eavg surface.

However, only in the case that coff=cddg would theaverage shear modulus represent an actual isotropicbehavior. Instead, cshr provides a crude average of theanisotropic behavior that is independent of crystallo-graphic direction, and so is independent of the texture,as well. In general, the relationships between the an-isotropic single crystal moduli and their polycrystalaverages depend on the methodology used to averageproperties over the orientation distribution. Variousmethods are available, some of which provide bounds(a summary is given in [24]). Simple relationships typi-cally exist only for the case of a uniform texture.Isotropic behavior is realized either from isotropic sin-gle crystals (hypothetically) or from anisotropic singlecrystals and a uniform orientation distribution. Here,not only is the texture not uniform, it is evolving duringthe experiment, making the choice of an orientationindependent average attractive. Moreover, csph and cshr

are convenient for the purposes of the parametric studyin that the average behavior is easily fixed as the degreeof anisotropy is varied.

From Gavg and K, we can use standard formulaeto determine two average parameters used in theparametric study. The first is an average Young’s mod-ulus

Eavg=9KGavg

3K+Gavg

(16)

and the second is an average Poisson’s ratio

avg=3K−2Gavg

2(3K+Gavg)(17)

Since the experimental data consists of axial andlateral lattice strains, these two parameters are naturalchoices. The third parameter specifies the degree ofelastic anisotropy of the crystal. We chose to use theratio of Young’s modulus in the crystal �111� direc-tions to the Young’s modulus in the crystal �100�directions [24]. This ratio, called rE, can be computedfrom the single crystal constants as

rE=(1+2csph/cddg)(1+2csph/coff)

(18)

For a cubic crystal, the maximum and minimum valuesof Young’s modulus always occur in the �111� and�100� directions, with the maximum typically being in�111� directions. Most cubic materials have rE�1, butthere exist materials with rE�1. For pure aluminum,the commonly reported elastic moduli give rE=1.19,which is relatively close to unity (isotropic); in contrast,iron has rE=2.14.

In conducting the parametric study, we consider vari-ations in both the degree of anisotropy (rE) and in theaverage Poisson’s ratio (avg) under constant averageYoung’s modulus (Eavg). This choice was motivated bysystematic trends in the differences between the com-

than as coff, cddg and csph. For cubic materials [22–24],the three independent values appearing in the stiffnessmatrix are c11, c12 and c44

3 which are related to theconstants above as:

coff=c44

cddg=c11−c12

csph=c11+2c12

Commonly reported values for aluminum are 108.2,61.3 and 57.0 GPa for c11, c12 and c44, respectively [25].

5.2. Approach for �arying moduli

In conducting the parametric study, we chose to usea different set of three independent parameters to de-scribe the anisotropic single crystal elastic response.The three parameters consist of two average parametersand a third parameter that serves as a measure of thedegree of the anisotropy. To define the average parame-ters, we first construct two average moduli that areanalogous to those for an isotropic single crystal: one isthe bulk modulus

3Kavg=3K=csph (14)

the other is a shear modulus

2Gavg=cshr=12

(coff+cddg). (15)

As mentioned above, the volumetric response is in-herently isotropic, so the anisotropic and average mod-uli do not differ with respect to this parameter.

3 Here, c44 relates the shear stress to shear strain using the mechan-ics convention for shear strain (�ij= (ui,j+uj,i)/2).


Table 3Test Matrix for numerical parameter studya

(c11,c12,c44) =0.25 =0.30 =0.35

(94.23, 40.38, 53.85)rE=1.0 (112.35, 60.49, 51.85)(84.00, 28.00, 56.00)(79.94, 30.03, 62.10)rE=1.2 (90.47, 42.26, 59.49) (108.86, 62.24, 57.08)

(83.62, 45.69, 69.77)(72.57, 33.72, 73.15) (102.47, 65.43, 66.67)rE=1.7(69.44, 35.28, 77.83)rE=2.0 (80.69, 47.16, 74.16) (99.71, 66.81, 70.80)

a All values in GPa.

puted and the measured lattice strains for the variouscrystal sets when values for pure aluminum given earlierwere used. The values of the spherical modulus, csph,that give parametric combinations of moduli lying onthe fixed Eavg surface are shown graphically in Fig. 5 asa function of cddg and coff. The moduli have beennormalized by dividing each one by Eavg, which reducesall such surfaces to a single hyperbolic cylinder. Shownis that part of the surface that is bounded by thefollowing conditions: all moduli are positive (lie in thefirst octant); 0�avg�0.5; and rE�1.

Superimposed on the plot are two sets of intersectinglines that correspond to lines of constant avg and tolines of constant rE. The test matrix constructed byvarying the two parameters, avg and rE, is given inTable 3 and appears in Fig. 5 as the intersections ofthese sets of lines. Three values of avg were considered,and for each value of avg, four values of rE wereconsidered, thus providing twelve test problems. Thevalues of the anisotropy ratio rE (1.0, 1.2, 1.7 and 2.0)were chosen to cover a broad range in behaviors. Thecase of rE=1 corresponds to isotropic behavior; rE=1.2 was chosen to give a value corresponding to that forpure aluminum [24]; rE=1.7 and 2.0 were chosen toprovide higher degrees of anisotropy than given by thepure aluminum value. In preliminary simulations wealso considered values rE�1, but the results comparedpoorly with experiment and are not included here. Therange in values of average Poisson’s ratio is notas well described in the literature as is Young’s mod-ulus, which varies by 10–20% depending on thealloying elements [26]. Consequently, we consider awide range of avg based on several factors. First, thepure aluminum single crystal moduli produce a avg=0.348. Second, macroscopic data for aluminum alloysindicate values for of approximately 0.30 [27]to 0.35 [11]. Finally, values of Poisson’s ratio for metalsin general fall between 0.25 and 0.35. Thus, wechose to study the values 0.25, 0.30, and 0.35, butplaced higher importance on larger values because oftheir closer correspondence with data for both purealuminum and aluminum alloys. For comparison pur-poses, the values of an alternative measure of thedegree of anisotropy, the ratio of coff/cddg, are given inTable 4.

5.3. Trends from moduli �ariations

Recall that the data include lattice strain measure-ments for several sets of crystals (corresponding to the(400), (311), (220), and (222) diffraction planes) forthree scattering vector directions (the axial and the twolateral directions). For each combination of diffractionplane and scattering vector, lattice strains in the loadedand unloaded state are available from several unloadingepisodes spaced at intervals over the specimen totalplastic strain of about 15%. The assessment of thequality of the comparison between experiment andsimulation for the various moduli parameter sets ismade in light of this comprehensive body of data. Theresponses, both measured and computed, differ for thetwo lateral directions because the material is texturedinitially, but are sufficiently similar with respect to thetrends discussed here that there is no need to differenti-ate strongly between them. Rather, we primarily con-trast the differences between the axial and lateraldirections in relation to the influence that the differentmoduli have on the computed responses.

The data from the parametric study are shown inFigs. 6–11. Each figure contains four plots, one foreach crystal direction under consideration. Each plotshows the experimental data with the associated uncer-tainty (error bar) and four graphs of simulated data forthe values of rE used in the study. Figs. 6 and 7 showthe results for avg=0.35 in the axial and TD-lateraldirections, respectively. Similarly, Figs. 8 and 9 showresults for avg=0.30, and Figs. 10 and 11 show resultsfor avg=0.25.

We take the parameter set of (rE=1.2; avg=0.35) asa baseline, as it corresponds to the pure aluminummoduli. First considering results for the axial direction,

Table 4Correspondence of rE with coff/cddg

=0.30(coff/cddg) =0.35=0.25

1.0rE=1.0 1.01.01.221.23rE=1.2 1.24

1.88rE=1.7 1.83 1.80rE=2.0 2.28 2.21 2.15


Fig. 6. Lattice strains in the specimen axial direction for avg of 0.35 and various rE values. Experimental data are shown with error bars. In eachgraph, the upper family of curves corresponds to the loaded condition; the lower family corresponds to the unloaded condition.

Fig. 7. Lattice strains in the specimen TD-lateral direction for avg of 0.35 and various rE values. Experimental data are shown with error bars.In each graph, the lower family of curves corresponds to the loaded conditions; the upper family corresponds to the unloaded condition.

and comparing the computed strains with the measuredones, it is evident that under load the simulations givelattice strains in the (400) crystals that are too small inmagnitude and lattice strains in both the (220) and(222) crystals that are too large. In the unloaded condi-tion, the (400) crystals in the simulation are com-pressed, while the experiment indicates that thosecrystals are extended. The computed lattice strains in

the (220) and (222) crystals after unloading are toolarge in comparison to experiment. For the lateraldirections, the lattice strains computed for the (400)crystals are somewhat too small in magnitude underload and are slightly too large when unloaded. Thecomputed lattice strains in the (220) crystals are toohigh both under load and when unloaded. The com-puted magnitudes of the lattice strains in the (222)


crystals are much too large both under load and whenunloaded. Note that in general the lateral lattice strainsare smaller in magnitude than the axial lattice strainswhen under load (Poisson’s effect). The absolute valuesof the differences between measured and computedlattice strains, however, are comparable for the axialand lateral strain components.

By varying the value of rE with fixed avg, the com-parisons between measured and computed latticestrains generally improved as rE increased. First wediscuss the results for avg of 0.35 that are presented inFigs. 6 and 7 to consider variations from the baselinecase (rE=1.2; avg=0.35) by changing only rE. For theaxial direction under load, the lattice strains of the


Fig. 9. Lattice strains in the specimen TD-lateral direction for avg of 0.30 and various rE values. Experimental data are shown with error bars.In each graph, the lower family of curves corresponds to the loaded condition; the upper family corresponds to the unloaded condition.



Fig. 11. Lattice strains in the specimen TD-lateral direction for avg of 0.25 and various rE values. Experimental data are shown with error bars.In each graph, the lower family of curves corresponds to the loaded condition; the upper family corresponds to the unloaded condition.

(400) crystals are larger for larger values of rE, while inthe (220) and (222) crystals, lattice strains are smaller inmagnitude. At a value of rE of 1.7, the experiments andsimulations compare well. Further increase in the valueof rE produces mixed results in that some lattice strainscompare slightly better and others slightly worse. Simi-larly, for the axial direction when unloaded, the com-puted responses generally are in better agreement with

experiment for the higher values of rE. The discrepancyin sign of the residual lattice strains for the (400)crystals is rectified for rE of either 1.7 or 2.0, but rE of1.7 definitely gives better results. For the (311) crystals,the results both under load and when unloaded areinsensitive to the choice of moduli. In the lateral direc-tions, the lattice strains for the (400) crystals are pre-dicted best with a value of rE of 1.7. The (220) crystals


exhibit better correspondence for both the loaded andthe unloaded strains as rE increases. The strains in the(222) crystals are quite sensitive to rE, with greatlyimproved comparisons obtained for larger values of rE.For fixed avg, values of rE greater than the purealuminum value of 1.2 improve the comparisons forboth the axial and lateral lattice strains: the axialstrains compare within the initial uncertainty; differ-ences in the lateral lattice strains persist, but mainly forthe loaded condition.

The issue of the initial uncertainty has been intro-duced. In the simulations we initialized the latticestrains to be zero at the start of the tension test. Thismaterial has a prior processing history, of course, andthe lattice strains may not be zero (and most likely arenot). Having the differences between the lattice strainsfrom simulation and experiment remain constant withincreasing specimen strain is indicative of the responsesmatching to within this initial uncertainty. Attempts tofurther refine the comparisons may not be merited.

Varying avg with fixed rE had little impact on theaxial strains, regardless of the value of rE. The lack ofsensitivity of axial strains to avg was expected and waspart of the rationale for conducting the parametricstudy using Eavg, avg and rE. The lateral strains wereinfluenced by changing avg, however. Again, consider-ing variations from the baseline case of (rE=1.2; avg=0.35) there is noticeable improvement in thecomparisons between experiment and simulation for the(220) and (222) crystal sets under load for avg of 0.30instead of 0.35 as shown in Figs. 8 and 9. The latticestrains for the (400) and (311) crystals under load arefairly insensitive to decreasing the value of avg. For theunloaded state, the comparisons for all of crystal setsshow little change with variations in avg. The trendsobserved with varying avg occur smoothly as avg de-creases from 0.35 to 0.25, with the results for avg of0.25 given in Figs. 10 and 11.

The relative strength of texture components is givenby the intensities of diffraction peaks in the experi-ments. Changes in the texture that accompany theplastic strains associated with the specimen deformationare recorded as corresponding changes in the peakintensities. These can be compared with texture changescomputed in the simulations. From the perspective ofthe parametric study, the computed intensities are in-sensitive to the variations in the elastic moduli consid-ered in this study. This is to be expected since formonotonic deformations the reorientations of crystalsare strongly influenced by the slip system activity butonly weakly affected by the elastic behavior. Althoughthe intensity comparisons do not help differentiate be-tween elastic moduli, the comparisons do help establishthe overall correctness of the simulations. We defer afull discussion of the intensity comparisons until Sec-tion 6. Here we mention only that the simulations

replicated the intensity changes reasonably well, withthe exception of the (400) crystals aligned with the axialscattering vector. The texture component associatedwith these crystals grows too rapidly in the simulationsin comparison to experiment. Larger numbers of ele-ments improved the comparisons, as shown in Section6. In addition, a number of simulations were performedto test the sensitivity of this result, as well as to thelattice strain results, to the assumed grain shape, whichis cubic for the parametric study. With parameters of(rE=1.7; avg=0.25) the simulations were repeated us-ing meshes having element aspect ratios, and thus crys-tal aspect ratios, of 4, 8, 16, and 32. Allowing forflattened grains did improve the comparisons of theintensities in the axial case for the (222) and (400)crystals, although these intensities remained too high inthe computed results. The best results were obtained foraspect ratios of 4 and 8; for these cases the latticestrains and other texture intensities were not influencedappreciably by varying the grain shape, but the intensi-ties for (222) and (400) crystals increased more slowly.For larger aspect ratios, the lattice strain componentsin the ND-lateral case were altered in addition toaffecting the texture evolution. This could be attributedto poor element performance at such large aspectratios.

In summary, the general trends that were obtainedfrom the parametric study are that, relative to hand-book values of pure aluminum, larger values of rE andsmaller values of avg yield better comparison betweenexperiment and simulation over the full range of dif-fraction data (all of the combinations of crystal sets,scattering vectors, and load state). The preferred valuesfrom this study point to rE of 1.7 and avg of 0.3. Thevalues of the moduli for this combination are notgreatly different from the pure aluminum numbers,varying by 18% for c11, 29% for c12, and 22% for c44.This magnitude of variation is significant, but notdissimilar to the range observed in Young’s modulusfor various alloys of aluminum. It is important to notethat considerable improvement in the comparisons be-tween experiment and simulation also was obtained bychanging rE from 1.2 to 1.7 while leaving avg un-changed. In this case, the relative changes in the cubicmoduli were much smaller: 6% for c11, 5% for c12, and17% for c44. In either case, the impact of the strongerelastic anisotropy suggested by the results of parametricstudy is impressive.

6. Comparisons of computed and measured latticestrains

The complete loading history including all of theunloading episodes was simulated using two highlyresolved finite element meshes. The coarser of the two


meshes consists of 16 000 elements and the finer of thetwo has 30 000 elements. As with the smaller meshes usedin the parametric study, each element represents anindividual crystal and both meshes mimic the aspect ratioof the gauge section of the specimen.

The elastic moduli correspond to (rE=1.7; avg=0.30), which gave good overall comparisons in theparametric study. We consider several variations for thesaturation strength appearing in the slip system harden-ing equation ranging from 150 to 180 MPa (a value of155 MPa was used in the parametric study). The pointof comparison used in this regard is the specimen(plastic) strain at which a target load is reached to triggeran unloading episode. Larger values of the saturationstrength produce higher strain hardening rates, so targetloads are reached with less specimen strain. The experi-ments reached the final target load at between 15.9 and17.3% engineering strain depending on the test (the testwith the axial scattering vector showed the least whilethat with the TD-lateral scattering vector showed thegreatest). In the simulation using a saturation strengthof 170 MPa the target load is reached about 12% strain;using 155 MPa it is reached at about 14% strain. Forvalues below 150 MPa, the specimen showed signs ofinstability (localization), probably indicating that thesimple form of the hardening model is inadequate nearsaturation. We present results for a saturation strengthof 155 MPa as these results best allow comparison ofexperiment and simulation point by point for the unload-ing episodes. The computed and measured macroscopicstress-strain curves are shown in Fig. 12 for this value ofthe saturation strength. Note that the stress levels arereplicated well throughout the test. Since the strainhardening rate towards the end of the experiment is lowand the unloading episodes are triggered by reachingpreset load levels, the sample strains at which theunloading episodes occur deviate somewhat betweenexperiment and simulation. The differences, however,are not critical in comparing lattice (elastic) strains, ashaving a little more or less plastic strain occur in reaching

a given stress level is not as important as the stress levelitself.

The meshes distort with specimen deformation, whichis evident after 14% strain. This is shown for the 16 000element mesh in Fig. 13Figs. 14 and 15 after the loadingsequence has been completed and the externally appliedstress has been reduced to zero. Element colors indicatethe value of the lattice strain component associated withthe scattering vector of that plot. A prominent featureof each figure is the variation in the residual latticestrains that exists over the crystals that comprise thespecimen. Such variations are on the order of the elasticstrain at yielding. This point has been discussed ingreater detail for a high strength steel alloy [28], but giventhe attention in this article to the influence of the elasticmoduli on the overall behavior, it will not be discussedany further here. Also shown in each figure are sets ofelements corresponding to particular (hkl) reflections.

The averages of lattice strains are different for thevarious crystal sets, although the large variations inlattice strains make this difficult to discern from theplots. The averages are compared with experimentallydetermined values in Figs. 16–18 for three levels ofspecimen discretization: 2000, 16 000, and 30 000 ele-ments. In comparison to the results for the 2000 elementmesh used in the parametric study, the histories com-puted with more highly resolved specimens are muchsmoother, reflecting statistically superior samples pro-vided by the larger meshes.

Examining first the axial lattice strains Fig. 16, thesimulations are in good overall agreement with theexperiment. The measured lattice strains under load atthe end of the test range from less that 0.004 for the (311)and (220) crystals to approximately 0.005 for the (400)crystals. Interestingly, although the [222] direction hasthe largest elastic directional modulus, the average latticestrains of the (222) crystals are not the smallest. Thistrend is definitely captured in the simulations. Perhapsthe greater multiplicities of the (311) and (220) crystalsplay a role in this behavior. Overall, the residual latticestrains show correct trends with increasing specimenstrain. In the experiment, the (222) and (400) crystalsshow residual lattice strains that are tensile, while resid-ual strains in the (220) and (311) crystals are compres-sive. This is captured by the simulation, with theexception of the (220) crystals, which in the simulationexhibit residual lattice strains of approximately zero.

The comparisons between computed and simulatedaverage lattice strains for the ND-lateral and TD-lateral directions are shown in Figs. 17 and 18, respec-tively. First, we reiterate that the lateral strains are lessthan half of the axial strains, and the absolute values ofthe differences between experiment and simulationare about the same for the axial and lateral cases. Forthe ND-lateral scattering vector, the computed latticestrains under load are slightly too high in magnitude

Fig. 12. Comparison of computed and measured macroscopic axialstress.


Fig. 13. Simulated residual lattice strains associated with the axial scattering vector for the entire mesh and for the crystal subsets contributingto the labeled reflections.Fig. 14. Simulated residual lattice strains associated with the TD-lateral scattering vector for the entire mesh and for the crystal subsetscontributing to the labeled reflections.Fig. 15. Simulated residual lattice strains associated with the ND-lateral scattering vector for the entire mesh and for the crystal subsetscontributing to the labeled reflections.


for the (222) crystals, slightly too low for the (400)crystals, and quite close for the (311) and (220) crystals.For the TD-lateral case, the computed lattice strainsunder load are in reasonably good agreement for allpeaks. In the unloaded state, the computed magnitudesare slightly too large, but the signs of the lattice strains

for the different crystal sets are correct. For the ND-lateral scattering direction, both the measured and com-puted lattice strains in the (220) and (222) crystals arecompressive; in the (400) crystals they are tensile, andin the (311) crystals they are close to zero. For theTD-lateral scattering direction, the lattice strains in

Fig. 16. Lattice strains in the specimen axial direction computed using finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k)elements. Experimental data are shown with error bars. In each graph, the upper family of curves corresponds to the loaded condition; the lowerfamily corresponds to the unloaded condition.

Fig. 17. Lattice strains in the specimen ND-lateral direction computed using finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k)elements. Experimental data are shown with error bars. In each graph, the lower family of curves corresponds to the loaded condition; the upperfamily corresponds to the unloaded condition.


Fig. 18. Lattice strains in the specimen TD-lateral direction computed using finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k)elements. Experimental data are shown with error bars. In each graph, the lower family of curves corresponds to the loaded condition; the upperfamily corresponds to the unloaded condition.

both the (400) and the (311) crystals are tensile in bothexperiment and simulation. The measured lattice strainsfor both the (220) and (222) crystals are close to zero(perhaps slightly compressive), while the computed val-ues are compressive.

An interesting aspect of the results is that the com-puted lattice strains show a stronger dependence on thespecimen orientation (TD vs. ND direction) than do themeasured strains. However, if TD-lateral results wereoffset by a constant lattice strain (about 0.0002 for(220) crystals and 0.0001 for the (222) crystals) thestrains for the two lateral scattering vectors wouldalmost coincide. Further, the measured data does notshow such a large offset. This implies that there is anappreciable influence of the texture and the anisotropicbehavior on the response. From the parametric study,for fixed avg the offset between the two lateral casesdecreases with increasing rE. Thus, greater single crystalelastic anisotropy compensates for the directionalityintroduced by the texture. A degree of crystal elasticanisotropy greater than the pure aluminum value ofrE=1.2 is suggested by the diffraction moduli reducedfrom the unloading episodes for the Bragg peaks in theexperiment. All of these observations support the con-clusion that the elastic moduli for AA-5182 are moreanisotropic than those of pure aluminum, in addition todemonstrating an overall stiffer response.

Over the course of the experiment, some intensitiesrise, some fall, and others remain about the same. Themeasured peak intensities are presented in Figs. 19–21for the three scattering vector directions. Also shown in

the figures are the intensities computed for simulationsusing meshes with 2000, 16 000, and 30 000 elements.Note that the initial intensities are not all unity, butrange from about 0.5–2.0 based on the initial texture.With larger meshes, and thus more crystals, the histo-ries are smoother, and the correspondence between themeasured and simulated initial values generally im-proves. One texture component, the (400) crystals forthe ND-lateral scattering vector, initially apprarently istoo high, having a value of about 2.5 in the simulationsand 1.6 in the experimental data. This discrepancy wasintroduced in the representation of the ODF from themeasured textures, and was perpetuated through therandom generation of orientations from the ODF, asdiscussed in Section 4.3. All of the simulations have thissame discrepancy regardless of the number of elements(crystals) in the mesh. The simulations track the evolv-ing intensities from experiment reasonably well. In theaxial direction, the intensities for the (222) and (400)crystals both rise, while the intensity for the (220)crystals falls. This trend is captured in the simulation,although the simulated intensity change for axial (222)crystals is too large. With larger numbers of crystals thecomputed rate of change improves. Recall from Section5 that flattened crystal geometries also acted to slow therate of growth of this texture component. In the TD-lateral direction, the intensities of the (222) crystalsdecrease while those for the other crystals remain ap-proximately constant, both in the experiments and inthe simulations. The discrepancies in all remain nearlyunchanged throughout the experiment. In the ND-lat-


eral direction, the intensity changes for the (400) crys-tals in the simulations are more rapid than those ob-served in the experiment, which over the course of thespecimen straining compensates for the initially highintensity in the simulation. As the number of elementsincreases, the computed rate of change more closelyreplicates that measured. The other texture componentsare remarkably close throughout the experiments.

7. Discussion

The principal points that we are attempting to con-vey from the results of this combined experimental andmodeling program are the following:� Neutron diffraction experiments have provided ex-

tensive data for average lattice strains and peakintensities for plastically deformed tensile specimens.

Fig. 19. Texture intensities for the axial scattering vector and for finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k) elements.Experimental data are shown with error bars.

Fig. 20. Texture intensities for TD-lateral scattering vector and for finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k) elements.Experimental data are shown with error bars.


Fig. 21. Texture intensities for ND-lateral scattering vector and for finite element meshes having 2000 (2k), 16 000 (16k) and 30 000 (30k) elements.Experimental data are shown with error bars.

The data encompass one axial and two lateral scat-tering vectors, four different sets of crystallographicplanes, specimen strains up to about 15%, and quiteimportantly, both the loaded and unloadedconditions.

� For the simulations to match average lattice strains,a higher degree of single crystal elastic anisotropymust be invoked in the simulations than is com-monly reported for pure (unalloyed) aluminum. Theimprovement in comparisons between simulationand experiment occurs across the board as the elasticanisotropic ratio rE increases from a value of 1.2 forpure aluminum to about 1.7 for material used here(AA-5182).

� The elastic moduli of an alloy should be expected todiffer from those of the pure (unalloyed) element.Average constants such as Young’s modulus or Pois-son’s ratio can be determined in a straightforwardmanner by testing polycrystals, but the degree ofsingle crystal anisotropy can vary independently andis difficult to determine empirically. Although com-parison of simulation to experiment does not consti-tute conclusive proof, our results indicate that thesingle crystal anisotropy of AA-5182 is higher thanthat of pure aluminum.The implications of this are important for behaviors

that depend on the crystal stress state, such as recovery,recrystallization, fatigue, and fracture. The simulationsshow that inaccurate characterization of the single crys-tal anisotropy leads to substantial errors in the residuallattice strains, especially in the relative level of strain asa function of crystallographic orientation.

The findings clearly are dependent on the fidelity ofthe modeling. The finite element model is realistic in anumber of important ways. Elements constitute distinctgrains, and lattice orientations reflect the measuredtexture. Compatibility is strictly enforced everywhere,and equilibrium is satisfied in a weak sense based on aweighted residual. The material behavior includes bothanisotropic elasticity and plastic flow from crystallo-graphic slip with relations specific to face-centered cu-bic crystals. Of course, there are enhancements thatcould make the model better represent the real material.The representation of individual crystals could berefined so that each crystal has a more realistic geome-try (which here is brick like), is more finely discretized(using many finite elements rather than just one), andhas a shape that is determined from micrographs. Assummarized in Section 5, we did examine the influenceof flattened grains on the lattice strains as part of theparametric study. There is a tradeoff in terms of com-putational expense between discretizing the grains withmany elements and defining a sample with more grains.In some respects, both accomplish the same goal ofgiving the volume fraction of the material with a givenlattice orientation more degrees of freedom for itsdeformation. Clearly the topologies of the polycrystalare different for the two, and more work is needed toquantify the differences in computed lattice strains be-tween fewer highly-resolved crystals and more coarsely-resolved crystals. The equations that govern the singlecrystal plastic behavior are relatively simple. We use apower law form of the kinetics of slip at fixed state andan idealization of slip itself as a linear combination of


simple shearing modes on a limited number of slipsystems. The slip system strengths are the same for allslip systems within a particular crystal (but do varyfrom crystal to crystal); there is no attempt to allow theslip systems within a crystal to harden differently. Noris the role of constituent particles considered explicitly.We point out that the anisotropic trends in latticestrains that motivated the parametric study were ob-served throughout the test, so model improvements thatprimarily alter the strength and texture late in the testwould not be expected to change the nature of thefindings.

The initialization of the lattice strains remains anissue. The experiments give only an average latticestrain from the peak shift, but not the strain values ofspecific crystals. The lattice strains are distributed aboutthe mean indicated by the peak shift, and an initializa-tion process should sample from the distribution to givean accurate rendition of the starting strain state. How-ever, an examination of the scatter of lattice strainsabout the average has shown that there is a dependenceon orientation relative to the scattering vector for thosecrystals contributing to particular diffraction peaks[28]. One reason for stretching the specimens to strainsof more than 10% is to mitigate the influence of theinitial lattice strains. In this respect, less confidence isplaced on the first unloading episode than on ones laterin the test sequence.

8. Summary

A coordinated program of experiment and simula-tion was conducted to examine the (elastic) latticestrains of AA-5182 subjected to plastic deformation.Lattice strains were measured by neutron diffractionboth while the specimen was loaded and while it wasunloaded at a number of points characterized by in-creasing levels of plastic strain. A comprehensive dataset was obtained by resolving Bragg peaks for threescatting vector directions (giving three different straincomponents) and several families of crystals (deter-mined by a common crystallographic plane). Finiteelement simulations were conducted in which the speci-men was modeled as a polycrystal with each elementrepresenting an individual crystal and was subjected tothe same loading sequence as imposed in the experi-ments. To adequately match the data, a level of singlecrystal elastic anisotropy higher than that reported forpure aluminum single crystals was required. Based on aparametric study, values of the elastic moduli, whichrender a directional moduli ratio, rE, of 1.7 were foundto give superior comparisons of average lattice strainsover the full data set relative to the ratio of 1.2 that istypically reported for pure aluminum (the directionmoduli ratio is the ratio of the �111� Young’s modulus

to the �100� Young’s modulus). Using the more an-isotropic elastic moduli, the experiments were simulatedwith higher specimen resolution. Meshes with 16 000and 30 000 elements (crystals) showed very good com-parisons to the measured average lattice strain data.The larger meshes provided better statistical averages oflattice strains, reflected both in a closer correspondencebetween experiment and simulation and in smootherhistories of lattice strains. The comprehensive experi-mental data set and the finely detailed simulation re-sults, including excellent texture predictions, provide acompelling argument that the single crystal anisotropyof AA-5182 is higher than that of pure aluminum.

Acknowledgements

Support for this work has been provided by theOffice of Naval Research under contract NOOO 14-95-I-0314. Neutron diffraction experiments were per-formed on a National Research Council (Canada)neutron diffractometer located at the NRU Reactor ofAECL (Atomic Energy of Canada Limited). Comput-ing resources were provided by the Cornell TheoryCenter. Harmonic coefficients for the ODF measuredby X-ray diffraction measurements were provided byDr Jean Savoie, Kingston Research and Development,Alcan International, Ltd.

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on the influence of crystal elastic moduli on computed lattice strains in aa-5182 following plastic...

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