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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 61 (2013) 428–449

0022-50

http://d

Abbren Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

A two-scale methodology for determining the residual stressesin polycrystalline solids using high energy X-ray diffraction data

Kevin P. McNelis, Paul R. Dawson, Matthew P. Miller n

Sibley School of Mechanical and Aerospace Engineering, 194 Rhodes Hall, Cornell University, Ithaca, NY 14853, United States

a r t i c l e i n f o

Article history:

Received 24 June 2011

Received in revised form

6 September 2012

Accepted 21 September 2012Available online 3 October 2012

Keywords:

Residual stress distribution

X-ray diffraction

Finite elements

Polycrystalline material

96/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jmps.2012.09.015

viation: LSHR, Low Solvus High Refractory; S

esponding author. Tel.: þ1 607 255 0400; fa

ail addresses: kpm64@cornell.edu (K.P. McNe

a b s t r a c t

This paper presents a new method for ascertaining residual stress fields in engineer-

ing components. Diffraction data are employed with a finite element discretization to

determine the macroscopic (continuum) residual stress field over the workpiece

simultaneously with the crystal-scale distribution of elastic strains at each diffraction

measurement point. Stress equilibrium and traction free boundary conditions constrain

the solution at the continuum scale. The thousands of lattice strain measurements

made at each diffraction volume ensure that the stress solution is consistent with

crystal-scale elastic distortions. Finally, integrated over all orientations within each

aggregate, the crystal-scale stresses must match the continuum stress. The method was

demonstrated using a shrink fit specimen with a nickel-base superalloy disk fit over a

stainless steel shaft. High energy synchrotron X-ray diffraction experiments conducted

at the Advanced Photon Source, Argonne National Laboratory, provided nearly 1 million

lattice strain measurements. Using these data, the new methodology produced a stress

field that satisfied the macroscopic constraints and matched the crystal-scale distor-

tions at each diffraction volume as manifest by spherical harmonic expansions of the

lattice strain tensor over orientation space. The residual stress distribution exhibits the

general features of an axisymmetric analytical approximation, but also contains details

that likely arise from the pre-existing material state and fabrication of the shrink fit

assembly. Projections of the spherical harmonic expansions of the crystal-scale lattice

strain tensor matched the experimental lattice strain pole figures. Issues related to

diffraction volume spacing (spatial resolution) and specimen shadowing are also

discussed.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Accurate prediction of mechanical failure is fundamental to engineering design. Numerous approaches have beendeveloped to more accurately predict failure in engineering components. These approaches have demonstrated thatmechanical failures result from the net effect of the stresses applied to a component during service and the stress existingin a material absent any applied load—the residual stress (Dowling, 2007). With the strong stress dependence of processessuch as microcrack initiation, it follows then that the accuracy of life prediction and damage evolution models aredependent on a quantitative understanding of the residual stress present in an engineering component. Knowledge of the

ll rights reserved.

PF, Strain Pole Figure; XEC, X-ray Elastic Constants

x: þ1 607 255 1222.

lis), prd5@cornell.edu (P.R. Dawson), mpm4@cornell.edu (M.P. Miller).

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 429

underlying residual stress will improve any life prediction of any processed part (Webster and Ezeilo, 2001). To be usefulfor the design of high reliability precision machines such as aircraft engines, however, the resolution of the residual stressfield estimate must be on a par with the stresses predicted during the analysis of service conditions. In this paper wepresent a new method for determining a residual stress distribution over an entire component using lattice strainsdeduced from diffraction peak shift measurements made at discrete points. The resulting stress field is consistent with themechanical constraints acting on the component and with the underlying lattice distortions measured at the crystal level.

1.1. Techniques for determining residual stress

The importance of the role of residual stress in design and failure of engineering components has long been appreciated,but the methods to quantify residual stress have not been consistently satisfactory for use in design. It is for this reason thatadvances in residual stress analysis continue to be pursued even after decades of attention. New experimental capabilitiesand modeling frameworks often have served as the drivers for a renewed interest in advancing the state-of-the-art. Theevolution of residual stress techniques is well-described in the literature (Society of Experimental Mechanics, 1996; Noyanand Cohen, 1987; Hauk, 1997; Withers, 2007), with recent advances regularly summarized at residual stress summits(Residual Stress Summit, 2003, 2005, 2007). We present a brief overview here—focusing on diffraction methods as theprincipal source of data. We first note that stress cannot be measured directly. Instead, to estimate the stress, anotherphysical quantity related to the motion, such as displacement, strain, wave speed or vibrational frequency, is measured andstress is calculated by modeling the experiment using some variant of Hooke’s law (Society of Experimental Mechanics,1996). The quality of the ‘‘measured’’ residual stresses, therefore, depends on both the experimental uncertainty and theaccuracy of the model.

Techniques can be delineated into one of two categories—destructive and non-destructive measures. The former, as thename implies, has the typically unwanted result of destroying part of the component to determine the existing state ofstress. Common destructive techniques include the sectioning (Tebedge, 1973) and hole drilling methods (Rendler andVigness, 1966; Shajer, 1988), each of which consist of removing a portion of the body in order to measure the strain whichwas present. Non-destructive residual stress techniques can be more desirable because they do not inflict permanentdamage on a component under investigation. Diffraction methods employing X-rays or neutrons are particularly well-suited for measuring distortion in crystalline materials. Laboratory X-ray sources are used extensively for the quantifica-tion of near surface residual stress fields (Society of Experimental Mechanics, 1996; Hauk, 1997). The technique has beenused frequently to approximate the residual stresses present in coatings (Kirchlechner, 2010) and thin films (Welzel et al.,2005), where measurements at larger depths within the component are not necessary. Determination of stresses withinbulk samples requires a high energy, high flux, X-ray source such as a synchrotron or a neutron source. Large componentscan be interrogated in neutron transmission experiments with valuable strain information from a relative large volumeof crystals from deep within the component gleaned from the data (cf. Allen et al., 1985; Krawitz and Holden, 1990;Turkmen et al., 2003; Dawson et al., 2001; Lorentzen et al., 2002; Holden et al., 2000; Clausen et al., 1999; Brown et al.,2003; Fitzpatrick and Lodini, 2003; Carter and Bourke, 2000; Loge et al., 2004). Transmission experiments usingmonochromatic synchrotron X-rays are rapidly being developed for quantification of crystallographic texture (Websteret al., 2001; Lienert et al., 2001) as well as lattice strains (Miller et al., 2005; Korsunsky et al., 1998; Wanner and Dunand,2000; Behnken, 2000; Wang et al., 2001). The collection time for a typical X-ray experiment is significantly less than thatneeded for an experiment conducted using neutrons and monochromatic X-ray area detector technology also makes itpossible to collect data in many directions simultaneously. Energy dispersive X-ray methods employing polychromaticbeams also hold significant potential (Croft et al., 2002; Korsunsky et al., 2002) but have their own limitations related tocollecting complete strain data in many directions.

Individual diffraction signals, or peaks, provide data on changes in lattice plane spacing in crystals that are orientedrelative to the beam to satisfy Bragg’s law (Cullity and Stock, 2001). The lattice planes are used as strain gages–elasticstrain gages, in fact, since the distortion of the unit cell is an elastic process. If one is observing the distortion of a singlecrystal using diffraction, lattice strains from various families of lattice planes (specified by Miller indices, fhklg, forexample) can be measured in many sample directions to form the full elastic strain tensor. The single crystal elastic modulithen are employed in conjunction with Hooke’s law to calculate the stress. Interpretation of diffraction data from apolycrystalline sample is more involved. The diffraction peak contains information from more than one crystal orientation.Rather, every crystal within the sample containing a suitably oriented set of lattice planes contributes to the peak (Wenk,1985; Kocks, 1998). The state of the elastic strain over the sample as a function of the crystallographic orientation can beextracted from the suite of measurements via an inversion procedure (Bernier and Miller, 2006).

In the past, diffraction data have been employed in residual stress determination only in a more limited manner.The traditional sin2 c approach considers the lattice strains from one family of lattice planes (one fhklg) and constructsspecial moduli, known as X-ray Elastic Constants (XEC), for the calculation of stress using Hooke’s law (Hauk, 1997).Since the magnitude of the peak shift will vary with the choice of lattice plane, the XEC are calibrated from the strainsfor a particular fhklg under known macroscopic stress (Hauk, 1997; Noyan and Cohen, 1987). Such XEC have beentabulated according to fhklg and material. The sin2 c approach has been employed widely using laboratory X-ray sourcesto estimate near surface residual stress states where the character of the stress state (e.g. plane or uniaxial stress) may beassumed a priori.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449430

1.2. Motivation for a new method

The methodology presented within this paper was designed to fill specific needs within the discipline of residual stressdetermination as applied to mechanical design. Two principal objectives were pursued in developing the methodology.First, the continuum scale residual stress distribution should conform to the data to the best extent possible while notviolating physical constraints associated with equilibrium (locally and globally) or with the surface tractions (mainly freesurfaces for an unloaded component). Second, all the data from diffraction measurements should be employed, not justdata for one family of lattice planes (fhklg). That is, the value of the stress computed by interrogating grains sampled ina diffraction measurement should be based on a complete orientation average to the greatest extent possible rather thana pre-selected subset of crystals that does not span the full range of lattice orientations present in the sample. Each of theseobjectives is explained in more detail in the following paragraphs.

To be of most value, a residual stress determination methodology should define the state of stress at every point withinthe component—not just the locations where measurements took place. Such a comprehensive representation is crucialif residual stresses are to be utilized in mechanical design in a quantitative manner, as the ‘‘worst case’’ need not occurat points that are easily accessible to experimental probes. Current diffraction-based residual stress measurementmethodologies calculate stress point by point from experimental measurements. As such, they neither quantify the fullstress tensor field over an entire component nor require that the field be mechanically consistent. For instance, mostdiffraction-based residual stress results consist of a single component of stress at one (or at most a handful of) discretepoint(s) within a component. The stress values are computed from the underlying lattice strains without consideration ofthe state of stress at spatially adjacent points. The most obvious limitations to such stresses calculated individually are thelack of any check on the satisfaction of stress equilibrium and the consistency of the underlying stress field withthe boundary conditions of the sample (most often the presence of free surfaces). To be a competent design tool,a methodology should take raw experimental data measured at discrete points within a component and impose a set ofmechanical constraints (stress equilibrium and satisfaction of boundary conditions) to produce an estimation of the stateof stress at every point.

Current diffraction methods permit making thousands of peak measurements that span broad combinations of fhklgsand sample directions, with each measurement providing one component of the lattice strain averaged over the diffractingcrystals. Using high energy synchrotron X-rays or neutron beams, it is possible to take these measurements in the interiorof bulk samples and from the data to first construct lattice strain pole figures (SPFs) and to then compute the orientation-dependent stress tensor (Miller et al., 2005; Schuren et al., 2011; Bernier and Miller, 2006). We have shown that the stressdistributions in bulk samples evaluated using this approach are consistent with both the macroscopic loading conditionsand the crystal-scale stresses predicted using crystal-based FEM simulations (Miller et al., 2008). There are several reasonsto use these diffraction methods over the more limited XEC approach in residual stress determination. First, the elasticanisotropy of the single crystals enters directly. This has important implications in computing the average stress tensorfrom the elastic strain distribution. Second, the crystallographic texture is included explicitly in the interpretation of themeasured average lattice strains, as will be shown later. Finally, the fraction of the total number of grains in a sample thatare interrogated is greatly expanded over the XEC approach. The net impact is that the data available for determininga residual stress distribution give a much more complete picture of the elastic strain distribution across the crystals thateffectively define the continuum (average) stress at points within the component.

1.3. Overview of the present approach

This paper describes a novel methodology to quantify residual stress distributions in engineering components. Theprincipal goals of the basic framework are to determine a continuum stress distribution that explicitly accounts for thephysical constraints of equilibrium and to effectively utilize the full array of diffraction data. For the latter, we will takeadvantage of crystallographically dependent lattice strain data to sample strains in many directions in grains of manyorientations, not just one or two strain components in a small sampling of grains associated with special reflections. Forthe former, we will allow for spatially varying stress fields without any a priori geometric assumptions that could precludeapplication of the method to general components. We open with a high-level overview of the methodology. This isfollowed in subsequent sections of the paper by more detailed explanations of its mathematical elements and of itsnumerical implementation. Finally, we demonstrate its use with determination of the residual stress distribution arisingfrom an interference fit.

The basic elements of the methodology are depicted schematically in Fig. 1. The goal is to minimize a mathematicalresidual, Rs, which quantifies the difference between a continuum stress distribution over the component and a collectionof volume averaged, crystal-scale stresses corresponding to diffraction measurements. The expressions for the continuumstress, Rh, and the crystal-average stresses, Rd, proceed along the left and right sides of Fig. 1, respectively. The continuumstress distribution is represented with finite element interpolation functions over the volume of the component. Thecrystal-average stresses are the volume-averaged crystal stresses at a discrete number of points within the component.Namely, these points are the centroids of the experimental diffraction volumes. For each diffraction volume, this stress iscomputed from a texture-weighted integration of the crystal stress distribution, which itself is computed using Hooke’slaw for single crystals and the crystal-scale elastic strain distribution, which is represented using a spherical harmonic

Fig. 1. Schematic of the two-scale method for determining the residual stress in engineering components. The method considers the material response at

both the macro- and crystal-scales. The method is set up as an objective function with explicit constraints, relating the continuum stress distribution to

the diffraction volume average crystal stress. Notation is defined in Section 2 of the text.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 431

expansion over the fundamental region of orientation space. The nodal point values of the continuum stress distributionand the sets of harmonic coefficients of the crystal-scale strain distributions (one set for each diffraction volume)constitute the solution variables sought in minimizing the residual.

The experimental data enter the methodology as constraints on the solution, labeled as ‘Data Constraints’ in Fig. 1.Specifically these data are diffraction measurements that provide the projection of the crystal-scale elastic strains inspecific directions of crystals lying along designated crystallographic fibers. As mentioned previously, one advantage ofusing high energy X-rays and area detectors is the ability to collect large numbers of peak shift measurements at eachmaterial point. These data contain information from many crystal orientations and many strain directions. The solutionalso is constrained to require that the continuum stress field is self-equilibrating over the body. This condition, togetherwith the free surface constraints, appears as ‘Equilibrium Constraints’ in Fig. 1.

The optimization simultaneously determines both the values of the continuum stress tensor at the nodal points ofthe discretized part and the harmonic expansion coefficients describing the orientation dependent crystal strains ineach diffraction volume. The method is unique in that it enforces information regarding both the crystal and continuumresponses of the component in finding optimal stress distributions collectively on both the crystal-scale and global-scale(continuum) stress distributions.

In residual stress analysis, stresses at different size scales are often designated as being of different types (Hauk, 1997).Type I residual stress refers to a stress averaged over a volume containing many grains (usually several thousand). Type IIresidual stress considers stress as a function of each crystal in the aggregate. In the method we have outlined, the continuumstress at the nodal points of the part is a type I residual stress. Meanwhile, the spherical harmonic coefficients that define thecrystal stress in each diffraction volume is similar in nature to type II residual stress. They differ somewhat in that the crystalstress is averaged for all grains in that diffraction volume which have a common crystallographic orientation.

2. Mathematical statement of the methodology

The methodology depicted in Fig. 1 is now presented in greater detail. This entails representing the stress distributionswith approximating functions, expanding the lattice strain distribution function in terms of a spherical harmonic seriesover orientation space, and evaluating the integrals that arise in the residual and the constraints placed on the solution.

2.1. Residual for the stress distributions

To determine the stress distributions, we define a weighted residual

Rs ¼

ZVFijðS

hij�S

dijÞ dV ð1Þ

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449432

which quantifies the difference between the continuum stress, RhðxÞ, and a distribution defined by the volume-averaged

crystal-scale stresses, RdðxÞ, over the body, V. Here, U is a set of tensor weighting functions common to finite element

approximations (Finlayson, 1972; Zienkiewicz, 1997). To evaluate the residual, trial (approximating) functions must beintroduced to represent the continuum and crystal-average stress distributions.

2.1.1. Representation of the continuum stress over the body

The continuum stress distribution is constructed using finite element interpolation functions. The body is discretizedwith a mesh. Over each element of the mesh the stress is represented as a polynomial function. The complete stressdistribution is the union of all the elemental representations

ShijðxÞ ¼

[elements

l ¼ 1

½NðxÞ�fSijgl ð2Þ

Here we use piecewise continuous (C0) interpolation functions, ½NðxÞ�, as is common in structural mechanics. Theinterpolation functions map the components of the continuum stress at the nodes of the finite element mesh, fSijg, to anypoint within each element.

It should be emphasized that the use of finite elements to represent the stress will not be to solve the field equations forelasticity, but rather to define a continuous stress distribution at the continuum scale that we can relate to the crystal-average stress states at common points over the body. Every element in the mesh is occupied by, and centered on, exactlyone diffraction volume centroid. The continuum stress at the nodal points of each element, as well as the volume-averagedcrystal-scale stress at the element centroid, is mapped to the integration points in that element where the residual isapplied.

2.1.2. Representation of the crystal-scale stress over a diffraction volume

Diffraction volumes are the regions within a component formed by the intersection of the region illuminated by theX-ray beam and that region with crystals that can possibly produce diffracted intensity that will intersect the detector.In this work, a diffraction volume typically holds many crystals (thousands or more). The volume-averaged crystal stress isthe mean value over this population of grains for all the components of the stress tensor. The first step in determining thevolume-averaged crystal stress is to represent the crystal-scale strain distribution, EijðrÞ over a diffraction volume. This isaccomplished with a harmonic expansions of the form

EijðrÞ ¼Xm

k ¼ 0

wkijH

kðrÞ ð3Þ

Here HkðrÞ is a set of k harmonic basis functions (or modes), and wk

ij are the coefficients of each harmonic mode. The latticestrains and the harmonic series are functions of the crystallographic orientation, parameterized here using the Rodriguesvector

r¼ tanx2

� �n ð4Þ

Here n is the rotation axis and x is the rotation angle for a particular crystal orientation. The crystal-scale strain distributionprovides an estimate of the most likely strain tensor for a specified crystal orientation within a given diffraction volume.For a general distribution we use piecewise polynomials, again in the form of finite element interpolation functions,to represent the strain over the cubic fundamental region in orientation space (Kumar and Dawson, 1996). Havingdefined the crystal strain distribution function, we can likewise determine the stress as a function of orientation usingHooke’s law

sijðrÞ ¼ CijpqEpqðrÞ ¼ Cijpq

Xn

k ¼ 1

wkpqHkðrÞ ð5Þ

Here rðrÞ is the crystal stress distribution function and C is the single crystal elastic modulus. The use of sphericalharmonics to represent orientation dependent quantities has been explored previously (for instance in QuantitativeTexture Analysis Wenk, 1985; Bunge, 1982). The expansion of strain and stress components in a series of sphericalharmonic functions is analogous to representing a function using a Fourier series with modes and coefficients. Thespherical harmonic modes in this case arise from the symmetry group of the crystal (Boyce, 2004). The coefficients dictatethe weighting of each mode in the distribution. The series converges to the exact distribution as the number of termsbecomes infinite. Truncation of the series can serve as high frequency noise suppression. Because the modes are knownfunctions that have been evaluated at the nodal points in the finite element mesh of the fundamental region of orientationspace, the degrees of freedom in specifying a general distribution are simply the weights.

The major benefit of using the spherical harmonic expansion for representing a field such as the crystal-scale straindistribution is computational efficiency. Because of the large amount of data which are collected experimentally,any technique which reduces the degrees of freedom in the problem, while not sacrificing an appreciable amount interms of the accuracy in the solution, is of great benefit. We have found that 23 harmonic modes are sufficient for

Fig. 2. Example of three of the 23 spherical harmonic modes (defined as HkðrÞ in Eq. (3)) used to approximate the orientation dependent crystal stress. Mode 2

(a), Mode 7 (b), and Mode 23 (c) in the cubic fundamental region of orientation space parameterized using the Rodrigues vector defined in Eq. (4).

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 433

representing orientation dependent quantities in the cubic fundamental region (Wenk, 1985; Altman, 1986; Miller et al.,2008). This reduces the necessary degrees of freedom for specifying a lattice strain state by over an order of magnitudecompared to the field itself. Provided in Fig. 2 are several harmonic modes evaluated over the cubic fundamental region inorientation space.

Given a stress distribution over orientation space, we may now determine the volume-average crystal stress for thatdiffraction volume. This averaging is performed as an integration over the cubic fundamental region, weighted by thevalues of the orientation distribution function (ODF), AðrÞ

Sdij ¼

ZOfr

AðrÞsijðrÞ dO ð6Þ

Here Rd is the volume-average crystal stress, AðrÞ is the ODF, and Ofr defines the volume of the cubic fundamental region inorientation space. In weighting by AðrÞ we can interpret Rd as the volume weighted average stress.

2.2. Constraints applied on the solution

There are a number of explicit constraints which are placed on a solution for the residual provided in Eq. (1), andexpressed schematically in Fig. 1. First, the stress distribution at the crystal scale will produce elastic strains within thecrystals. It is the components of the crystal elastic strains that are measured by diffraction and that ultimately dictate thesolution. To this, the conditions of equilibrium are appended, along with the requirement for traction-free conditions onfree surfaces.

2.2.1. Data constraint

Our data are the lattice strains measured at each diffraction volume. Any residual stress solution (and underlyingresidual strain state) must be consistent with the SPF data. Each diffraction peak is associated with a particular family oflattice planes (fhklg). The crystals contributing to the fhklg peak all share a common orientation of their fhklg planes up to arotation about a single axis, and thus lie along a crystallographic fiber in orientation space. If we designate this axis as asample direction, s, then the condition satisfied by all crystals lying on a common fiber is

cðrÞ ¼ 7s ð7Þ

where cðrÞ is the normal vector of a designated crystallographic plane for a crystal with orientation, r, and its symmetricequivalents. A crystallographic fiber defined by Eq. (7) is indicated by UcJs.

The measured lattice strain, which is the normal strain in the direction of cðrÞ ¼ 7s, is an average over all crystals lyingon the fiber. Using Eq. (7), each normal lattice strain, EcJs, now can be written in terms of the sample direction, s, and thespherical harmonic expansion

EcJs ¼

ZUcJs

AðrÞsisjEijðrÞ dU¼ZUcJs

AðrÞsisj

Xn

k ¼ 1

wkijH

kðrÞ dU ð8Þ

We enforce equation (8) for every lattice strain value available from the diffraction measurements.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449434

2.2.2. Equilibrium constraint

A second constraint is that the continuum stress distribution must be self-equilibrating for a body that is not subjectedto external loads, as should be the case for residual stress distributions. This constraint is enforced globally by means of aweighted residual, Re

Re ¼

ZVciS

hji,j dV ¼ 0 ð9Þ

where body forces are neglected. The vector weighting functions for the residual are defined as ci, and are chosen to be ofthe form ci ¼Fij,j.

2.2.3. Free surface constraint

Residual stress distributions exist in the absence of external loading. To be consistent with this condition, thecontinuum stress distribution must produce traction-free boundaries. This constraint is stated as a weighted residual, Rt

Rt ¼

ZSciðT

hi �T iÞ dS¼ 0 ð10Þ

using the same vector weights, ci, as employed in Eq. (9). Here Th is the traction vector consistent with Rh at the surface S

of the volume V and T is the imposed external traction at S, which is identically zero on a free surface.For thin geometries in which a plane stress assumption is valid, the traction-free condition may be replaced by

requiring that the out-of-plane stress components are identically zero everywhere over the body. This assumption satisfiesthe free surface constraint and reduces the number of unknown variables. We will exploit this simplification in theexample presented later, but point out here that the methodology is fully capable of dealing with three dimensionalcomponent geometries.

3. Numerical implementation of the methodology

The mathematical formulation laid out in Section 2 is solved using finite element procedures. Basically, the interpola-tion functions for the trial functions (for the stress) are also used as weights, the body is discretized with elements, andintegrations are performed numerically using Gaussian quadrature. Key to the efficiency of the procedure is the systematicuse of matrix representation, allowing the effective use of optimized linear algebra routines.

3.1. Discretization of the residual function

The residual from Eq. (1) is evaluated following standard finite element practices. The continuum stress is mapped overthe body using the finite element interpolation functions, as indicated in Eq. (2)

Rs ¼

ZVFijðS

hij�S

dijÞ dV ¼

Xelements

i ¼ 1

ZVeFijðS

hij�S

dijÞ dV ð11Þ

A Galerkin approach is taken whereby the weighting functions are the same as the shape functions

U¼ ½NðxÞ�fFg ð12Þ

At the elemental levelZVeFijðS

hij�S

dijÞ dV ¼ ½Mi

�fSg�fFig ð13Þ

where

½Mi� ¼

ZVe½N�T ½N� dV ð14Þ

fFig ¼

ZVe½N�T fSd

g dV ð15Þ

Here ½Mi� and fFi

g the matrices associated with the ith element, and we have adopted vector notation for the stress tensor.The integral is evaluated by mapping to a parent configuration using isoparametric elements for performing numericalquadrature. The complete objective function is obtained by summing the elemental contributions over all elements

Rs ¼

ZVFijðS

hij�S

dijÞ dV ¼ ½M�fSg�fFg ð16Þ

½M� ¼Xelements

i ¼ 1

½Mi� ð17Þ

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 435

fFg ¼Xelements

i ¼ 1

fFig ð18Þ

It now becomes necessary to express RdðxÞ in terms of the set of harmonic coefficients as per Eqs. (5) and (6). This

procedure is presented here step-by-step. The crystal-scale elastic strain in the sample frame (fEg) at each crystalorientation is first transformed to the crystal reference frame (f~Eg) by use of the orientation matrix, ½R�. This can be done ineither matrix or vector (Voigt) notation for the lattice strain (Ting, 1996)

f~Eg ¼ ½R�½E�½R�T ¼ ½Rn�fEg ð19Þ

The crystal-scale stress in the crystal reference frame (f ~sg) is determined from f~Eg using values of the single crystal elasticmoduli expressed in the crystal frame (½ ~C �) and then returned to the sample frame with the orientation matrix, giving

fsg ¼ ½Rn��1f ~sg ¼ ½Rn

��1½ ~C �f~Eg ¼ ½Rn��1½ ~C �½Rn

�fEg ð20Þ

Finally, an average state of stress, fSdg, for all crystals comprising the diffraction volume is obtained by integrating the

crystal stresses, weighted by the ODF, over the fundamental region of orientation space. For one component of fSdg this

can be written as

Sdi ¼ fAg

T

ZOfr

½NrðrÞ�T ½Nr

ðrÞ� dOfSdi g ¼ fAg

T ½L2�fSdi g ð21Þ

where the ODF and the crystal stress distribution have been represented over orientation space with piece-wisepolynomial interpolation functions of the form

AðrÞ ¼ ½NrðrÞ�fAg and sðrÞ ¼ ½Nr

ðrÞ�fSdi g ð22Þ

where ½NrðrÞ� are interpolation functions over orientation space and fAg and fSd

i g are the nodal point values of A and fsg,respectively. Here, ½L2� is the L2 inner product matrix (Belyschko et al., 2000). To compute the L2 inner product wediscretize the fundamental region of orientation space using finite element interpolation functions (Kumar and Dawson,1996). Eqs. (20) and (21) are combined into a single matrix operator that takes a set of spherical harmonic coefficients andproduces the volume-averaged crystal stress for each measurement

Sdi ¼ fAg

T ½L2�½C�½H�fwig ¼ ½Gi�fwig ð23Þ

Here, ½C� holds the elasticity matrix in the sample frame for each node of the mesh over the fundamental region and theharmonic expansion functions have been represented over orientation space with the finite element interpolation

HkðrÞ ¼ ½Nr

ðrÞ�fHkg ð24Þ

allowing the nodal point values, fHkg, to be used to form

½H� ¼ ½fH1gT fH2

gT fH3gT � � � fHk

gT � ð25Þ

The expressions for the components of the volume-averaged crystal stress, Sdi , in Eq. (23) are concatenated to form the

complete stress vector, fSdg, and the harmonic stiffness matrix, ½G�. The expression for the volume-averaged crystal stress

has now been represented in terms of a single matrix operation that isolates the spherical harmonic coefficients, Eq. (23).The vector matrix fFi

g is built assuming constant stress elements, whereby the value of the average crystal-scale stress,fSdg is assigned to all integration points in the element

fFig ¼

ZVe½N�TRd dV ¼ ½Qi

�½G�fwg ð26Þ

Here ½Q � is a finite element assembly matrix, whose functionality is to assemble the forcing vector from the integration ofRd over an element of the continuum mesh.

3.2. Discretization of the constraints

3.2.1. Discretization of the data constraint

The number of data constraints corresponds to the number of independent lattice strain measurements and ispotentially very large. Consequently, it is important to write Eq. (8) compactly and to evaluate it efficiently. We firstconstruct the constraint equation for a single lattice strain value and then expand the notation to incorporate all the data.An average value of the normal lattice strain, EcJs, along a crystallographic fiber, UcJs, as given in Eq. (8), is written in matrixform using finite element interpolation over the fundamental region as

EcJs ¼

ZUcJs

AðrÞfsngT fEðrÞg dU¼ fsngT fEf g ð27Þ

where

fsngT ¼ ½s1s1 s2s2 s3s3 2s2s3 2s3s1 2s1s2� ð28Þ

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449436

and one component of fEf g is

Efi ¼ fAg

T

ZUcJs

½Nr�T ½Nr� dUfEd

i g ¼ ff gTfEd

i g ð29Þ

Here, the crystal-scale elastic strains are written using finite element interpolation as: Esami ðrÞ ¼ ½N

rðrÞ�fEd

i g where fEdi g are

the nodal point values of Esami ðrÞ. Eq. (27) is expanded to include all the data constraints by introduction a projection

matrix, ½Gn�. Using this matrix, the data constraints over all p diffraction volumes are written as a single vector

fEcJsg ¼ ½Gn�½H�fwg ¼ ½D�fwg ð30Þ

where

½Gn� ¼

ðsn1ff gT Þ1 ðsn2ff g

T Þ1 ðsn3ff gT Þ1 ðsn4ff g

T Þ1 ðsn5ff gT Þ1 ðsn6ff g

T Þ1

ðsn1ff gT Þ2 ðsn2ff g

T Þ2 ðsn3ff gT Þ2 ðsn4ff g

T Þ2 ðsn5ff gT Þ2 ðsn6ff g

T Þ2

^ ^ ^ ^ ^ ^

ðsn1ff gT Þp ðsn2ff g

T Þp ðsn3ff gT Þp ðsn4ff g

T Þp ðsn5ff gT Þp ðsn6ff g

T Þp

266664

377775 ð31Þ

For the current method, the first 23 harmonic modes are used based on previous work (Wenk, 1985; Altman, 1986; Milleret al., 2008). Incorporating the modes of the discrete harmonics into ½Gn

� results in a modified projection matrix that maybe applied directly to the harmonic coefficients to approximate the values of the lattice strains from the diffraction data.This modified projection matrix is termed ½D�.

3.2.2. Discretization of the equilibrium constraint

The global framework can be completed by imposing the constraint of equilibrium of the continuum stress over thepart. This is achieved by penalizing solutions when the divergence of the continuum stress is non-zero over the part. Webegin by modifying our expression for the equilibrium condition in Eq. (9) so that it is in our finite element framework.Z

VciS

hij,j dV ¼

Xelements

i ¼ 1

ZVeciS

hij,j dV ¼

Xelements

i ¼ 1

ZVefCgT ½Nc�

T ½Bs�fSg dV ð32Þ

Here fCg are the weights, ½Nc� are the weighting functions, and ½Bs� are the shape function derivatives. The weightingfunctions are chosen to span the same space as div ðSh

Þ, thus we use ½Bs� ¼ ½Nc�. The construct is thus in a Galerkin form.The expression is first defined at the elemental level.Z

VeciS

hij,j dV ¼ fFgT ½Ki

�fSg ð33Þ

½Ki� ¼

ZVe½Bs�

T ½Bs� dV ð34Þ

We may then define the equilibrium condition over the entire component.ZVci,jS

hij dV ¼ ½K�fSg ¼ 0 ð35Þ

½K� ¼Xelements

i ¼ 1

½Ki� ð36Þ

3.2.3. Discretization of the free surface constraint

For fully three-dimensional component geometries, the free surface constraint would be enforced by constraining athree-dimensional representation of the continuum stress distribution. For the demonstration application presented later,the component geometry is that of a relatively thin disk. Consequently, we supersede the free surface constraint with aplane stress constraint in which we restrict both the continuum stress distribution and the degrees of freedom, fwg, used torepresent the crystal-scale stress distribution. The plane stress constraint can be written by appropriately filtering theharmonic stiffness matrix, ½G�. For the objective function, the filtering removes any out-of-plane components in Rd so thatonly the in-plane components of the stress tensor result when operating on the spherical harmonic coefficients. Thus,Eq. (26) becomes

fFig ¼

ZVe½N�TRd dV ¼ ½Qi

�½Gxy�fwg ð37Þ

where ½Gxy� corresponds to ½G� modified to constrain the resultant to allow non-zero components only in the x�y plane. Asecond filter, the complement to ½Gxy�, is termed ½Gz�. For plane stress conditions, ½Gz� identifies coefficients in the harmonicexpansion that will contribute to out-of-plane components of Rd. For plane stress, the solution should obey

½Gz�fwg ¼ f0g ð38Þ

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 437

This expression may finally be incorporated into the form of the global representation presented in Eq. (13), resulting ina single system of equations expressing the residual between the continuum stress and the volume-averaged crystalstresses. A weighting parameter, l, is applied to the constraint, so as to penalize solutions which are not in a state ofequilibrium.

ð½M�þl½K�ÞfSg ¼ ½Q �½Gxy�fwg ð39Þ

3.3. Assembly of system of equations

We may assemble these conditions to satisfy the global residual in Eq. (1). A single system of equations can bedeveloped which simultaneously solves for the harmonic coefficients describing the crystal response in each diffractionvolume, as well as the 2-D components of the continuum stress at the nodal points of the finite element mesh of the disk.This is accomplished by assembling the statement for the residual from Eq. (39) as well as the constraints on theexperimental lattice SPF data from Eq. (38) and the plane stress constraint from Eq. (30)

ð½M�þl½K�Þ �½Q �½Gxy�

0 ½D�

0 g½Gz�

264

375 fSg

fwg

( )¼

f0g

fEcJsg

f0g

264

375 ð40Þ

The assembly is now complete in that the formulation has the ability to simultaneously satisfy the global residual forboth the continuum stress and spherical harmonic coefficients. Another weighting parameter, g, has been introduced topenalize solutions which are not in a state of plane stress, in much the same way l penalizes solutions which are not in astate of equilibrium. In the example presented in Section 4, we iterated on the magnitude of the multipliers to determinethe values that would give the largest acceptable deviation from the fully satisfied constraint. For example, the value of lwas chosen so that no out-of-plane stress component was larger than 15 MPa.

4. Demonstration example

For the validation of the new residual stress determination methodology proposed here, we would ideally possess afully calibrated test specimen, one with a fully determined macroscopic stress field with a quantified link to the underlyingcrystal stresses. However, the creation of such a specimen would necessitate a methodology much like we are proposinghere. Therefore we employed a shrink fit sample often used for benchmarking residual stress methods (Hsu et al., 1982;Blessing et al., 1984; Gnaupel-Herold et al., 2000; Residual Stress Summit, 2003). Unlike the thick aluminum samples usedin the neutron-based experiments, however, our sample was a thin Ni-base superalloy disk attached to a steel shaft by aninterference fit, as shown in Fig. 3. To avoid end effects, the steel shaft was longer than the thickness of the disk. For thedimensions shown in Fig. 3, a plane stress assumption was valid. Since the traction vector must vanish on the front andback faces (because of the free surface constraint), gradients in the stress state through the thickness should be small, if atall present. We therefore employed a plane stress assumption in lieu of explicit free surface constraint discussed inSection 3.2.3. This assumption is made on both the global, Rh, and crystal average, Rd, stress distributions. Experimentally,each diffraction volume extends over the entire thickness of the disk. Even if a stress gradient were to exist in the through-thickness direction, averaging over the thickness of the disk will result in all z-component stress terms being zero. Removalof the third dimension to the stress state not only makes the computation of a global solution simpler, but also aids in thedetermination of Rd by further constraining the system of equations. As will be described in more detail later in this paper,an advantage of the shrink fit sample is the existence of an analytical axisymmetric solution for an ideal case (Budynas andNesbitt, 2010).

4.1. Sample design

At room temperature, the disk inner diameter was smaller than the shaft outer diameter. A residual stress state isimposed on the assembly by heating the disk and allowing it to thermally expand. It is then placed over the shaft andallowed to cool and contract, producing an interference pressure at the contact surface between the disk and shaft. Thenominal thickness of the disk (1 mm) was chosen by considering the attenuation of 90 keV X-rays in nickel. The diffractionexperiments were only performed over one quarter of the disk. The measurement locations (diffraction volumes) on thedisk are shown in Fig. 4. The dimensions of the assembly are provided in Table 1.

The material chosen for the disk was a Low Solvus High Refractory (LSHR) nickel-base superalloy. We examined thematerial in its as-forged state. Single crystal elastic moduli of nickel-base superalloys have been shown to be largelycomposition insensitive (Herman et al., 1996). The values we employed for the LSHR were chosen from Herman et al.(1996) and are given in Table 2. LSHR has a face-centered cubic (FCC) crystal structure. The as-forged material has anaverage grain size of 3 mm, which makes the material ideal for the powder diffraction experiments (Miller et al., 2005).

The orientation distribution function (ODF) of the material, AðrÞ, was determined using the synchrotron data and thedata reduction software Material Analysis Using Diffraction (MAUD) (Lutterotti et al., 1999). The ODF is plotted over the

Fig. 3. Photo of the fully assembled shrink fit sample. The LSHR nickel disk is heated and placed around the internal steel shaft. Upon cooling, residual

stress is induced within both parts.

Fig. 4. (Left) Diagram of the location of each diffraction volume on the quarter disk. Diffraction measurements were made over five radial bands (a¼ 01,

22.51, 451, 67.51, and 901). Each radial array consisted of 21 diffraction volumes, shown here as square symbols. (Right) Finite element discretization

of the interference fit disk. Each element was centered on the centroid of one diffraction volume (DV) from the experiment. Orientation of the sample

coordinate system is also shown.

Table 1Value of the dimensions (with tolerances) for the inner and outer diameter of the LSHR

disk (ri and ro), the radius of the stainless steel shaft (rs) and the disk thickness, t, in the

interference fit assembly shown in Fig. 4.

ri (mm) ro (mm) rs (mm) t (mm)

6:31870:006 15:87570:127 6.350 1:00070:006

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449438

cubic fundamental region of orientation space in Fig. 5. Throughout this example, a mesh having 1512 tetrahedralelements with 254 independent nodal point values is used to discretize the fundamental region (Kumar and Dawson,1996). The Multiples of a Uniform Distribution (MUD) for any particular orientation in the material is observed to varyonly slightly about unity, indicative of a weak texture. This texture was homogeneous over the sample.

Table 2Values of single crystal elastic moduli for the LSHR nickel-base superalloy

used in the disk of the shrink fit assembly (Herman et al., 1996).

C11 (GPa) C12 (GPa) C44 (GPa)

245 155 125

Fig. 5. The orientation distribution function (ODF), with respect to the sample coordinate system, for the LSHR nickel-base superalloy, plotted over the

cubic fundamental region of orientation space. The units are Multiples of a Uniform Distribution (MUD). Crystal orientations are defined relative to the x,

y, z sample directions defined using the Rodrigues parameterization (Eq. (4)).

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 439

4.2. Lattice strain measurements

A general description of high energy diffraction experiments is presented in Appendix and details for constructingstrain pole figures can be found in Miller et al. (2005), Bernier et al. (2008), and Schuren et al. (2011). Specific informationabout the experiments conducted for illustrating the new residual stress determination capability is provided here. Thediffraction experiments were performed at the Advanced Photon Source (APS) at Argonne National Laboratory, at beamline1-ID-C. A schematic of the experimental setup shown in Fig. 6 depicts the configuration of the sample and detector as wellas sample positioning instrumentation with respect to the X-ray beam. The powder diffraction data were collected intransmission at an energy of 90.5 keV. A General Electric fast scan detector (which is made up of an array of 2048�2048,200 mm square pixels) was used to take diffraction images. A beam cross-section of 250� 250 mm was used in theexperiment. Considering the average grain size, this beam illuminated on the order of 106 grains in each diffractionvolume. The sample was repositioned within the holding clamp then rotated (o) to obtain optimal pole figure coverage.A ‘‘calibrant’’ sample containing a standard powder with known lattice parameter (cerium dioxide) was affixed to thedownstream face of the sample to continuously fix the experimental geometry (Bernier et al., 2008).

Measurements were taken along five different radii on the disk, specified by the angle above the x-axis, a (see Fig. 4). Eachradial band of measurements consisted of 21 different diffraction volumes, at identical radial positions for each samplingarray. The locations of the diffraction volumes were spaced radially in approximate equal stress increments based on ananalytical approximation for the shrink fit disk presented in Section 4.3.1. Each set of diffraction measurements interrogatesfour crystallographic families – the f111g, f200g, f220g and f311g – at 25–37 different specimen orientations (o angles).Furthermore, the Debye rings that each measurement produces are subdivided into 72 azimuthal ‘‘bins’’, each serving as anindependent measurement for each lattice plane family. This results in over 10,000 independent lattice strain measurementsbeing made at each diffraction volume and 800,000–1,000,000 measurements made over the entire sample. Figs. 7 and 8depict partial lattice strain pole figures (SPFs) as examples of the experimental data. Fig. 7 illustrates changes in the latticestrain data as a function of radial position at a single a. Fig. 8 shows how the SPFs change with a at a fixed radius.Strains from the f200g family of planes are depicted, but SPFs for the f111g, f220g and f311g also were part of the datasetcollected at each diffraction volume. The coverage variations seen in Figs. 7 and 8 arise due to the specimen configurationand X-ray shadowing.

The lattice strain pole figures (SPFs) are a convenient means for depicting the lattice strain measurements. Each pointon the pole figure (sphere) represents a particular direction relative to the sample frame coordinate system(origin of thesphere). As described in Appendix of this paper, the normal strain depicted at each point on an {hkl} strain pole figure is themanifestation of the average lattice spacing change associated with every {hkl} lattice plane whose normal coincides with

Fig. 6. Schematic of the experimental setup at the Advanced Photon Source (APS), beamline 1-ID-C. The experiments are conducted in transmission. The

translational stages are used to center each diffraction volume depicted in Fig. 4 on the o rotation axis. To interrogate a broad range of crystal

orientations within each diffraction volume, images were gathered over a range of o. The X, Y, Z laboratory coordinate system is shown. It coincides with

the sample coordinate system defined in Fig. 4 when o¼0.

Fig. 7. Strain pole figures associated with the f200g family of planes, at a¼901 at radial locations of (a) r¼6.50 mm, (b) r¼7.27 mm, (c) r¼8.41 mm,

(d) r¼10.35 mm, and (e) r¼15.05 mm. The x, y, z sample directions are shown. Gray regions signify sample directions where no experimental

measurements were made.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449440

the pole figure direction. The orientations of the crystals containing these planes lie along a crystallographic fiber inorientation space. The advantage of employing high energy synchrotron X-rays in this study is their penetration power.The high X-ray flux at APS sector 1 also enables the collection of large amounts of data quickly using high speed areadetectors. Making the roughly 10,000 lattice strain measurements at each diffraction volume took only a few minutes andall crystals through the thickness of the specimen were sampled, not just those on the sample surface. The large amount ofstrain data is crucial for understanding the stress field over the workpiece as presented in the following section. Thehandful of measurements made in a conventional residual stress experiment could not produce such fidelity.

Examining the SPF data reveals several qualitative trends. In Fig. 7 we see that the lattice strains are observed todecrease in magnitude as we move outward along a radius of the disk. This is consistent with the expected trends in ashrink fit disk. At a¼901, the x direction coincides with the hoop direction and the y direction corresponds to the radialdirection. The sign of the measured lattice strains in the x and y directions are consistent with this observation. The strainpatterns observed in Fig. 8—which depicts measurements at different values of a are also consistent with the strains onewould expect in a shrink fit assembly. Since the overall goal of a residual stress measurement is to reconstruct the elasticstrain tensor from a set of normal strain measurements at each diffraction volume, the range of strain directionsinterrogated is a very important experimental variable. In Figs. 7 and 8 this range is manifest as the coverage of data on thestrain pole figures. For instance, due to the transmission diffraction geometry in general, lattice strain data in the specimennormal direction (z) cannot be obtained; hence, gray regions indicating no data are seen in every SPF. The variation of

Fig. 8. Strain pole figures associated with the f200g family of planes, for diffraction volumes at r¼9.05 mm and (a) a¼ 01, (b) a¼ 22:51, (c) a¼451,

(d) a¼ 67:51, and (e) a¼ 901. The x, y, z sample directions are shown. Gray regions signify sample directions where no experimental measurements

were made.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 441

coverage seen in Fig. 8 as a progresses from 01 to 901 is a good example of the effects of X-ray ‘‘shadowing’’ by regions ofthe workpiece or positioning apparatus. In our case, the shaft of the shrink fit sample, which can be seen protruding fromthe rear of the sample in Fig. 6, blocked X-rays at some sample orientations (o) and limited the SPF coverage at a¼01. Thediffraction volume at a¼901 was above the shadow and as a result, more complete pole figure coverage was obtained.Shadowing can be a very significant problem for real components with some regions experiencing reduced pole figurecoverage. Certainly a lack of coverage in a critical area will affect the accuracy of the stress ‘‘measurement’’. However, thefact that our method is full field will enable an informed approximation over a broad set of spatial locations even withincomplete strain pole figure coverage at some points.

4.3. Residual stress results

Using the lattice strains as data constraints, the methodology described in Section 2 was employed to simultaneouslydetermine both the continuum stress distribution over the physical component and the orientation dependent crystalstress distribution over each diffraction volume defined as fSg and fwg, respectively in Eq. (40). We had 105 such diffractionvolumes over the quarter disk which correspond to 105 elements in the finite element mesh. This results in 303 nodalpoints in the global mesh. Each nodal point has associated with it the three components of Rh (since we are assumingplane stress conditions) resulting in 909 unknown values (fSg). Meanwhile, for each of the 105 diffraction volumes thespherical harmonic expansion is used to determine EijðrÞ. We employed 23 terms in the expansion resulting in 23 unknowncoefficients (fwg) for each of the 6 strain components. This results in over 14,000 unknowns to describe the crystalresponse. The fact that we have more than 1,000,000 lattice strain measurements makes it reasonable to solve a systemof equations with over 15,000 degrees of freedom. The solution is obtainable in a matter of minutes using a singleprocessor computer.

4.3.1. Continuum stress distribution

Distributions of the continuum stress components (Shxx, Sh

yy, and Shxy) as determined using the new methodology are

shown in Fig. 9. We chose the shrink fit specimen for its relatively simple geometry; analytic solution for the stress fieldscan be used for comparison. An idealized shrink fit sample can be analyzed by approximating the interference fit as aninternal pressure applied to the inner radius of the disk (Budynas and Nesbitt, 2010). The interference ‘‘pressure’’, p, isapproximated by

p¼drs

1

Eh

r2oþr2

s

r2o�r2

s

þnh

� �þ

1

Esð1�nsÞ

� ��1

ð41Þ

Here the initial, room temperature inner and outer radius of the disk, ri and ro as well as the shaft radius, rs, are given inTable 1 and d¼ rs�ri. Young’s modulus and Poisson’s ratio for the shaft and disk are given by, Es, ns and Eh, nh, respectivelyand the values employed in the calculations are given in Table 3. The stress components in cylindrical coordinates (r, a, z)

Fig. 9. The residual stress field as determined by the new methodology. Distributions of (a) Shxx , (b) Sh

yy , and (c) Shxy are shown for the quarter disk

assembly.

Table 3Value of isotropic elastic constants for the LSHR disk (Eh and nh) and stainless steel

shaft (Es and ns) in the interference fit assembly (Online Materials Information

Resource).

Eh (GPa) nh Es (GPa) ns

195 0.3 205 0.31

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449442

can be computed using p and the thick-walled pressure vessel equations

SrrðrÞ ¼r2

i p

r2o�r2

i

1�r2

o

r2

� �ð42Þ

SaaðrÞ ¼r2

i p

r2o�r2

i

1þr2

o

r2

� �ð43Þ

where r and a are depicted in Fig. 4. Using these equations, the dimensions given in Table 1 and the elastic constants givenin Table 3, Srr and Saa were computed over the shrink fit disk. Values as a function of disk radius are shown in Fig. 10.Fig. 11 depicts plots of Sxx, Syy and Sxy obtained by transforming from ðr,a,zÞ coordinates to ðx,y,zÞ. Comparing the analyticsolution to the measured residual stress fields shown in Fig. 9, we see that the general trends for each stress componentcompare reasonably well. The obvious difference is the general lack of symmetry in the actual residual stresses in Fig. 9 ascompared to Fig. 11. Material heterogeneity and the finite component tolerances experienced in manufacturing tend tobreak the symmetry of stress within real assemblies. Typical residual stress benchmarking studies employing the shrink fitgeometry impose axisymmetry by employing a-averaged diffraction data (cf. Gnaupel-Herold et al., 2000). In order to fullyunderstand the detailed variations of stress within our disk, however, we chose not to smooth out the asymmetry byaveraging with respect to a.

The differences in the stress magnitudes observed in (9) and (11) can be explained by considering the machiningtolerances given in Table 1 and the elastic constants employed in each analysis. For instance, the difference variable, d, canvary up to 20%, which will produce a comparable variation in the stress values computed using Eqs. (42) and (43). This 20%

Fig. 10. Values of the hoop, Saa , and radial, Srr , components of the stress as approximated using the analytic solutions in Eqs. (42) and (43) and the

elastic constants given in Table 3.

Fig. 11. The analytic solution for the (a) Sxx , (b) Syy , and (c) Sxy components of stress in the disk of the interference fit assembly. The principal stresses

provided in Eqs. (42) and (43) are mapped to the Cartesian coordinate system associated with the sample. Stresses are plotted in MPa, and are shown

only in a quadrant of the disk due to symmetry of the stress field.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 443

variation in stress magnitude will bring the stress scales in Fig. 11 close to those seen in Fig. 9. Therefore, we focus oncomparing trends. There is particularly strong agreement in the Sh

yy and Shxy trends. Both components show smooth

transitions between the location of tensile and compressive stresses on the disk. Both components also show a cleardecrease in magnitude of the stress as the disk radius increases. The Sh

xx component, while not transitioning as smoothlyfrom a tensile to compressive response, does appear to have the proper sign at key locations on the disk—namely beingtensile along the y-axis and compressive along the x-axis.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449444

4.3.2. Crystal-scale strain distribution

Examples of the spherical harmonic coefficients, which describe the distribution of the crystal-scale strain tensor overorientation space, are shown in Fig. 12 at various lattice strain measurement locations (diffraction volumes) within thedisk. The distribution of each underlying component of the crystal strain tensor at each diffraction volume is representedby the series of spherical harmonic functions defined in Eq. (3). Examples of several spherical harmonic basis functions(modes) in orientation space are depicted in Fig. 2. The harmonic coefficients, wk

ij, specify the contribution of eachharmonic mode to a component of the crystal-scale strain distribution at each diffraction volume. As prescribed in Eq. (40),these coefficients are part of the solution to the overall optimization; in this way, the underlying crystal distortionsconsistent with the diffraction measurements contribute to the continuum stresses shown in Fig. 9. Shown in Fig. 12are the harmonic coefficients for several diffraction volume locations. While harmonic coefficients were found for allsix components of the full strain tensor, only those associated with the ExxðrÞ and EyyðrÞ components at several diffractionvolume locations are presented.

Consistent with understanding derived from quantitative texture analysis, we extend the spherical harmonic series to23 terms (Bunge, 1982). As can be seen, however, the dominant modes (values greater than 1.0) in most cases are lowerthan the 20th. It is apparent that the coefficient of each harmonic mode is dependent on both the component of strain andthe location of the diffraction volume on the disk. The coefficients for consecutive diffraction volumes on the same radialband are observed to be similar for most cases. The exception is ExxðrÞ at a¼ 451. Meanwhile, for common radial locations,the harmonic coefficients are obviously dependent on a.

It is difficult to ‘‘see’’ the effects of the harmonic coefficients on the resulting stress distributions shown in Fig. 9.However, using the predicted crystal-scale strain distributions, as manifest by the wk

ij from the solution, we can projectlattice strains that can be compared to the experimental measurements. This operation is shown in Eq. (27) as part of thesolution methodology. As an example of the comparison, we did this for one diffraction volume location in Figs. 13 and 14.Here, the harmonically generated f200g SPFs are shown in comparison with the same radial location as those provided inFig. 8. Difference pole figures can be generated by directly subtracting the values. These are presented for each location inFig. 14. While larger values of error are seen for some outlying points, most values on the difference SPFs do not exceed71� 10�4, which is only slightly larger than the nominal resolution of lattice strain measurement technique (Schurenand Miller, 2011). The strong agreement indicates that the solution from the optimization is consistent with theexperimentally measured strain data (the data constraint as described in Section 3.2.1). As has been shown in quantitativetexture analysis, an ODF that matches the orientation pole figures is not necessarily the ‘‘correct’’ one, due to the inherentnon-uniqueness of the pole figure inversion process (Wenk, 1985; Barton et al., 2002; Bernier et al., 2006). Therefore,additional constraints are necessary. This has also been demonstrated for the inversion of SPFs for the crystal-scale strain

Fig. 12. The spherical harmonic coefficients (wkij in Eq. (3)) for the crystal-scale strain distribution, ExxðrÞ and EyyðrÞ in diffraction volumes located on the

(a) a¼ 01, (b) a¼ 451 and (c) a¼ 901 radial bands. For each a, coefficients at three radial locations are shown.

Fig. 13. The f200g experimental Strain Pole Figures on the (a) a¼ 01, (b) a¼ 67:51, and (e) a¼ 901 radial bands are shown for diffraction volumes centered

at r¼9.05 mm on the interference fit disk. The harmonically generated SPFs are shown for the same diffraction volume locations in (d)–(f).

Fig. 14. The f200g difference Strain Pole Figures for diffraction volumes centered at r¼9.05 mm on the (a) a¼01, (b) a¼67.51, and (c) a¼901 radial bands.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 445

tensor for a sample subjected to uniaxial straining (Bernier and Miller, 2006). The equilibrium and free surface conditionsprovide the added constraints in the case of the current framework.

5. Discussion

Traditional diffraction-based residual stress determination techniques are limited to a relatively small number of latticestrain measurements over an engineering component. In this work, we sought to understand how the availability ofseveral orders of magnitude more lattice strain measurements might improve our understanding of residual stress. Can weobtain improved stress field fidelity by making many more measurements? The answer is yes, but it required creating anentirely new framework for reducing and analyzing residual stress measurements. This framework is the contribution ofthis paper.

In previous work using high energy synchrotron X-rays, our group developed lattice strain pole figure (SPF)experiments using in situ loading conditions to understand how crystal stresses vary within a deforming polycrystallineaggregate by making thousands of individual lattice strain measurements over many {hkl}s and a broad range of strainingdirections. By using an accurate load cell, we tracked the macroscopic stress (R) during in situ loading of our test sample.The crystal stresses over all orientations within the aggregate must be equal to this uniaxial value, i.e. (Rd

¼R). In aresidual stress field, the macroscopic stress is unknown and it varies spatially (Rh

ðxÞ). In the new framework, we used SPF

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449446

data measured at many spatial points to determine the macroscopic stress. We require that the macroscopic stresscoincide with the crystal stresses integrated over all orientations within each continuum point on the component. This isexpressed in a weighted residual sense in Eq. (1). Unlike the uniaxial in situ test specimen, we do not know what thisintegrated value of crystal stress should be beforehand. However, we do know that the continuum stress from onediffraction volume should be in mechanical equilibrium with the stresses at all other diffraction volumes. We impose thiscondition as Eq. (9). We also know that the residual stress state should be consistent with all free boundaries (Eq. (10)).

The sampling density (location of diffraction volumes) must be prescribed on a case by case basis but our newformulation has the flexibility to accommodate arbitrary sampling densities. We chose a dense number of diffractionvolumes for the shrink fit disk in order to establish the method. The resulting residual stress distribution is depicted in Fig. 9.The general trends from the analytic pressure vessel approximation (Eqs. (41) – (43), Fig. 10) that are presented in Fig. 11 canbe detected by comparing to Fig. 9. However, there is significant additional detail in the experimental field. The fluctuationsrelative to the analytic approximation seem consistent with machining and fabrication irregularities. In fact the other threefourths of the disk would most likely have different local fluctuations. There will be applications in which the detail seen inFig. 9 will be important: around local stress concentrators, for instance. Other applications, however, may impose lessstringent spatial resolution requirements. The sampling density can be coarsened or elements within the samplinggrid can be combined in this case. The effects of X-ray shadowing (obstruction of either incident or diffractedX-rays) due to structural conflicts can also contribute to sampling irregularities. We encountered a very simple exampleof shadowing in our experiments; the extended length of the shaft produced the pole figure coverage differences we see inFig. 8. There is often limited X-ray accessibility to regions on an engineering component during a residual stress experiment.While our finite element formulation enables discretization flexibility, each element was linked to a diffraction volume. Weare currently implementing an effective ‘‘decoupling’’ of the diffraction volumes and finite element discretization usingmeshless finite element methods, which significantly improves the flexibility of defining irregular diffraction volumeconfigurations independent of the FEM discretization.

6. Summary/conclusions

This paper presented a novel, multiscale methodology for determining the residual stress field within a polycrystallineworkpiece from diffraction data. Being able to estimate the residual stress with the same fidelity as the stress calculatedin mechanical design analyses served as the motivation for the new method. The method employs a finite elementdiscretization of the workpiece for introducing the lattice strain data, which are taken over a broad expanse of theworkpiece, and for imposing constraints of mechanical equilibrium and the effects of traction free boundaries.The diffraction data consist of thousands of elastic lattice strain measurements from several families of lattice planes.The methodology produces the distribution of residual macroscopic stress over the workpiece along with the orientation-dependent distribution of crystal-scale elastic strains at each diffraction volume. The solution is the one that best satisfiesthe macroscopic constraints of equilibrium and traction boundary conditions while simultaneously matching the latticestrain data, which are directly linked to straining at the crystal scale. The finite element framework enables interpolation ofthe continuum stress components over the full domain of the workpiece—enabling the designer to understand the residualstress field at any location.

The methodology was implemented on a simple test case common to residual stress analyses—the two dimensionalstress state produced in a thin circular disk that is attached to a circular shaft by an interference fit. A disk composed ofLSHR (low solvus high refractory) nickel-base superalloy was interrogated using synchrotron X-rays at sector 1 of theAdvanced Photon Source. Approximately 10,000 diffraction measurements were made at each of 105 diffraction volumesdistributed over one-fourth of the disk. Macroscopic (continuum) stresses were computed over the quarter disk andspherical harmonics coefficients were determined for each crystal-scale lattice strain component at each diffractionvolume. While matching the general trends, the resulting continuum stress field depicted fine detail not evident in anaxisymmetric analytical approximation. Lattice strain pole figures recomputed from the crystal scale stress distributionsmatched the experimental measurements close to the resolution of the measurement method.

We employed high energy synchrotron X-rays for these measurements. High energy enabled penetration through theLSHR sample, which is not possible with lab source X-rays. The high flux possible at APS sector 1 coupled with high speedarea detectors enabled rapid collection of the diffraction data. The new multiscale methodology, however, is independentof the diffracting radiation as long as adequate strain pole figure coverage is attained. New spallation neutron sourcesshould be capable of producing strain pole figures with the necessary coverage in acceptable amounts of time. Theextended penetration of neutrons would enable interrogation of larger engineering components.

The shrink fit disk example explored a two-dimensional stress field. However, there is nothing inherent in theconceptual framework that limits it to 2D. Creating a three-dimensional stress field and obtaining diffraction data fromsubsurface regions does present an experimental challenge, however. Motivated by the soller slits used in making depthresolved lattice strain measurements using neutrons (Allen et al., 1985), slits developed for high energy X-rays have madeit possible to make subsurface lattice strain measurements (Nielsen et al., 2000). Measurements made on a shrink fitsample containing three-dimensional stress gradients is the subject of an upcoming article. A refined numericalimplementation of the method described here, which decouples the diffraction volume from the exact specification ofthe finite element formulation will be presented in another upcoming article.

K.P. McNelis et al. / J. Mech. Phys. Solids 61 (2013) 428–449 447

Acknowledgments

Dr. Ulrich Lienert, of the Advanced Photon Source (APS)—currently at Deutsches Elektronen-Synchrotron, is stronglyacknowledged for his assistance and expertise in the conduction of the diffraction experiments and the data collection.Dr. Jay Schuren of Cornell University is acknowledged for his work on the data reduction algorithms and inversion scheme.Dr. Jun Park of Cornell University is acknowledged for experimental data collected at the APS. Donald Boyce ofthe Deformation Processes Laboratory (DPLab) at Cornell is acknowledged for his assistance with the generation of thespherical harmonic functions. This work has been supported financially by the Air Force Office of Scientific Research(AFOSR) and program director Dr. David Stargell, under Grant no. FA9550-09-1-0642. The material of interest wasprovided by Dr. T.J. Turner at the Air Force Research Lab (AFRL). Use of the Advanced Photon Source, an Office of ScienceUser Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, wassupported by the U.S. DOE under Contract no. DE-AC02-06CH11357.

Appendix A. High energy X-ray diffraction

X-ray diffraction has been a robust method for probing crystal structure for a century. Many methods exist for usingdiffraction data to probe the micromechanical response of polycrystalline metallic alloys. For detailed references on X-raydiffraction see Als-Nielson and McMorrow (2001) and Cullity and Stock (2001). Our technique involves diffraction of highenergy X-rays to measure the spacing of the crystal lattice in select grains of a crystalline sample. Bragg’s law describes thegeometry of the diffracted X-ray with respect to the incoming beam.

nl¼ 2dc sin y ð44Þ

Here, l, is the X-ray wavelength, dc is the lattice spacing for the diffracting crystallographic lattice planes, and n is aninteger value. The Bragg angle, y, is half the angle between the incoming and diffracted beam.

In polycrystalline samples, if the diffraction volume is large compared to the grain size, cones of diffracting X-raysintersect the detector and appear as rings on the detector image plate (see bottom-middle of Fig. 1). A typical diffractionpattern for the material which is the focus of this work, the LSHR nickel-base superalloy, is presented in Fig. 15a. The imagecontains diffraction rings from both the LSHR as well as a second material, Cerium Dioxide (CeO2). The diffractionspectrum for the DZ bin, integrated azimuthally, is provided in Fig. 15b.

Crystal scale elastic strain within a sample corresponds to changes in the interplanar spacing of the crystal lattice andappear as deviation in the rings on the detector from concentric circles. In the diffraction spectrum, the effect of strainpresent in the material is manifest as a shift in the location of the diffraction peak, for each crystallographic plane, from somereference (unstrained) position. The projection of the elastic strain in the scattering direction, Ec , may be written as a functionof the reference and deformed spacing of the particular crystal planes or, by use of (44), the change in the Bragg angle.

EcðsÞ ¼dc�d0

c

d0c

¼sin y0

c

sin yc�1 ð45Þ

Here d0c and dc correspond to the reference and deformed planar spacing for a particular crystallographic plane, and y0

c and yc

to the Bragg angle for the unstrained and strained material, respectively. The normal strain, Ec , is actually the integratedresponse for the portion of the diffracting beam intersecting the detector over each DZ bin.

Fig. 15. (a) A typical diffraction pattern for the LSHR nickel-base superalloy with a CeO2 calibrant powder affixed to the sample. Each ring on the image

corresponds to diffraction from a different family of crystallographic planes. Integrating the bin azimuthally produces a plot of intensity vs. radial location

on the detector. (b) The diffraction spectrum corresponding to the DZ ‘‘bin’’ in (a) is shown.

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The lattice strains for each fhklgwithin each DZ bin on the detector can be depicted on a lattice Strain Pole Figure (SPFs).For details on the generation of SPFs see Miller et al. (2005). Each point on an SPF corresponds to a direction relative to thesample coordinate system. The lattice strain value associated with the peak shift described above is the average value ofthe component of normal strain for all crystals whose fhklg normal aligns with a particular sample direction. We havedescribed these crystal orientations as lying along a crystallographic fiber in orientation space.

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