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Journal of

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ISSN 0021-8898

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A model for calculating diffraction elastic constants

Thomas Gnaupel-Herold, Adam A. Creuziger and Mark Iadicola  ̈

J. Appl. Cryst. (2012). 45, 197–206

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J. Appl. Cryst. (2012). 45, 197–206 Thomas Gnaupel-Herold  ̈ et al. · Diffraction elastic constants
research papers

J. Appl. Cryst. (2012). 45, 197–206 doi:10.1107/S0021889812002221 197

Journal of

Applied

Crystallography

ISSN 0021-8898

Received 25 August 2011

Accepted 18 January 2012

# 2012 International Union of Crystallography

Printed in Singapore – all rights reserved

A model for calculating diffraction elastic constants

Thomas Gna ̈upel-Herold,a,b* Adam A. Creuzigerc and Mark Iadicolac

a

Materials Science and Engineering, University of Maryland, Building 090, Room 2135, College

Park, MD 20742, USA, b

Center for Neutron Research, NIST, 100 Bureau Drive, Stop 6102,

Gaithersburg, MD 20899-6102, USA, and c

Metallurgy Division, NIST, 100 Bureau Drive, Stop

8553, Gaithersburg, MD 20899-8553, USA. Correspondence e-mail: tg-h@nist.gov

A model, dubbed the inverse Kro ̈ner model, is proposed to calculate the

diffraction elastic constants from the elastic constants of a single crystal. It is

related to the classic Kro ̈ner model, and both are identified as bounds on the

diffraction elastic constants. Through the grain shape as controlling parameter,

the classic Kro ̈ner model is bound by the hkl-independent mechanical limit

given by the bulk elastic constants of the matrix, while the inverse Kro ̈ner model

approaches the Reuss limit.

1. Introduction

Diffraction elastic constants (DECs) provide a relationship

between elastic lattice strains and macroscopic stress. As such,

they form the basis of diffractive stress determination. The

models used for the calculation of DECs have undergone a

very long development from the Reuss model (Mo ̈ller &

Martin, 1939), the Reuss–Voigt average, as suggested by Hill

(Hill, 1952), the Kro ̈ner model (Bollenrath et al., 1967;

Behnken & Hauk, 1986), and, more recently, the use of

perturbation theory by de Wit (1997). First and foremost,

DECs depend on the Miller indices (hkl) of the reflection used

for measuring the lattice spacings. Secondly, from Eshelby’s

theory (Eshelby, 1961) it is known that the strain/stress

response of a single grain depends on the elastic properties of

its surroundings as well as its shape. Prominent examples

where these interactions come into play are composites of

hard particles that reinforce an elastically soft matrix. DECs

are orientation-selective averages over a subset of all grains,

and therefore these dependencies remain valid to a degree.

Thirdly, it is known from various experimental results (Barral

et al., 1987; Brakman & Penning, 1988; Do ̈lle et al., 1977, 1978;

Baczmanski et al., 1993; Do ̈lle, 1979; Hauk, 1997) that

preferred orientation of grains can have a profound effect on

the magnitude and orientation dependence of DECs. Consequently, one should expect that DECs themselves depend on

these factors. The Kro ̈ner theory is the only model that can

account for all of these effects, yet the literature shows that,

because of the relative simplicity of the calculations and

despite their shortcomings, the Reuss model and the Reuss–

Voigt average are used predominantly. For isotropic aggregates and multiphase polycrystals in which the elastic properties of the constituents are not too dissimilar, the Reuss–

Voigt average and the Kro ̈ner model yield results in acceptable to good agreement with measured DECs. Other models

such as the modified Voigt model (Murray & Noyan, 1999), de

Wit’s model and the geometric average (Baczmanski et al.,

1993; Moraviec, 1989; Matthies & Humbert, 1993) have not yet

been compared with experimental results on a wider scale.

There are relatively few comparative data for materials with

preferred orientation, partly because the techniques used to

compare experimental results vary widely. For example,

recalculated stresses are sometimes compared with a known

applied stress, thus reducing the quality measure of the DEC

model used to a single number comparison. A comparison of

angular distributions of measured and recalculated lattice

strains has considerably more detail to offer but it also has to

reproduce the magnitude of the applied stress. A more

appropriate comparison can be gleaned from measured DECs

themselves. However, measuring DECs requires a large

number of measurements because lattice strains have to be

measured for different tilt angles and azimuth angles ’, as

well as for several applied stresses. This type of relationship is

conveniently expressed as stress factors Fij. An alternative

expression for Fij is developed in the following.

2. Theoretical considerations

The stress factors (Do ̈lle et al., 1977, 1978; Do ̈lle, 1979; Barral

et al., 1987; Hauk, 1997; Koch et al., 2004; Ortner, 2006) are

defined as

"ð’; ; hklÞ 1⁄4 dð’; ; hklÞ $ d0ð’; ; hklÞ

d0ð’; ; hklÞ

1⁄4 @"ð’; ; hklÞ

@!ij

!ij 1⁄4 Fijð’; ; hklÞ!ij; ð1Þ

where "(’, , hkl) is the average lattice strain of all grains

oriented such that their lattice planes (hkl) contribute to the

diffracted intensity in the direction given by the azimuth ’ and

tilt angle . Note that the reference d spacing d0(’, , hkl)

may depend on the orientation as well, as is the case for

samples with residual stresses from plastic deformation. The

!ij parameters are the stresses that pertain to the macroscopic

aggregate. For isotropic materials without preferred orientation, Fij is related to the traditionally used DECs s1 and 1

2 s2

through (Hauk, 1997)

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Fij 1⁄4

s1 þ 1

2 s2 cos2

’ sin2 1

2

1

2 s2 sin 2’ sin2

 1

2

1

2 s2 cos ’ sin 2 

1

2

1

2 s2 sin 2’ sin2

 s1 þ 1

2 s2 sin2

’ sin2 1

2

1

2 s2 sin ’ sin 2 

1

2

1

2 s2 cos ’ sin 2 12

1

2 s2 sin ’ sin 2 s1 þ 12 s2 cos2

 

0

B@

1

CA:

ð2Þ

The coordinate system used in the following is shown in Fig. 1.

In the following, Kro ̈ner’s notation is adopted, in which

fourth-rank tensors are denoted by upper case letters if they

refer to properties of the overall composite (matrix), and

lower case letters are used for the orientation-dependent

properties of the grains. Also, this discussion extends to

composites of noncrystalline and diffracting crystalline materials (grains) with known elastic properties. The grains in the

composite are described by an ellipsoidal shape with an

orientation ! of the ellipsoid axes. Furthermore, they are

characterized by their stiffness cijkl(g) and compliance sijkl(g),

which depend on the orientation g of the crystal lattice with

respect to the specimen frame. A fourth-rank tensor dependent on the orientation g is expressed as

aijklðgÞ 1⁄4 g$1

im g$1jn g$1ko g$1lp amnop; ð3Þ

in which amnop is the property tensor in the crystal reference

frame, aijkl(g) is the tensor in the specimen system and gij is the

transformation matrix [using Bunge angles (’1; "; ’2)] for the

transformation from the specimen reference frame to the

crystal reference frame (Bunge, 1982).

gf’1"’2g 1⁄4

cos ’1 cos ’2 sin ’1 cos ’2 sin ’2 sin "

$ sin ’1 sin ’2 cos " þ cos ’1 sin ’2 cos "

$ cos ’1 sin ’2 $ sin ’1 sin ’2 cos ’2 sin "

$ sin ’1 cos ’2 cos " þ cos ’1 cos ’2 cos "

sin ’1 sin " $ cos ’1 sin " cos "

0

BB

BBB

BBBB

BB@

1

CC

CCC

CCCC

CCA

:

ð4Þ

Note that equation (3) uses the inverse of g because the

transformation is from the crystal frame to the specimen

frame. The overall elastic constants of the composite, Cijkl and

Sijkl, are considered as known. The strain and stress in a single

crystallite depend on the orientation and shape of the crystallite. The relationship between the strain "ij(g) and stress

!ij(g) of a single grain and the macroscopic stress !ij and strain

"ij according to Kro ̈ner (1958) and Kneer (1965) is expressed

through the transfer tensors p and q:

!ijðgÞ 1⁄4 pijklðg; !Þ "kl; ð5Þ

"ijðgÞ 1⁄4 qijklðg; !Þ !kl: ð6Þ

Appropriate averaging of equation (6) yields the Kro ̈ner

model. Because of the difficulties associated with the determination of the distribution of the grain ellipsoid-axis orientations ! (not to be confused with the crystallite orientation

distribution), it is assumed in the following that the ellipsoid

axes (a1, a2, a3) align with (RD, TD, ND) (defined in Fig. 1),

and average values for the ellipsoid axis aspect ratios will be

used, where appropriate. Through Hooke’s law one has the

relationship

pijklðgÞ 1⁄4 cijmnðgÞ qmnopðgÞ Copkl: ð7Þ

However, it will be shown in the following that it is advantageous to consider the inverse tensor p$1

ijklðgÞ instead, where the

superscript indicates the inverse fourth-rank tensor. It is

obvious that, in general, p$1

ijklðgÞ 61⁄4 qijklðgÞ which, when averaged over all orientations, was shown by Kro ̈ner (1958) to

represent bounds on the overall elastic constants of the

polycrystal in the form

qijkl < Sijkl < p$1

ijkl or qijkl > Sijkl > p$1ijkl; ð8Þ

where the inequalities indicate that qijkl and p$1

ijkl can be either

an upper or a lower bound, depending on the single-crystal

constants, grain shape and orientation distribution. Furthermore, it was shown by Kro ̈ner (1958) and Kneer (1965) that

spherical grains or ellipsoids with randomly oriented axes

averaged over all crystal orientations yield qijkl = p$1

ijkl. It must

be pointed out that, because the orientation average for DECs

is not complete and involves only a small subset of grain

orientations, the DECs are generally subject to a relationship

of the type given by equation (8). The often-used Kro ̈ner

model is therefore only one of two possible formulae, and one

can express a relationship between lattice strain and macroscopic stress entirely equivalent to equation (6) as

"ðgÞ 1⁄4 p$1

ijklðgÞ!kl; ð9Þ

which in the following is referred to as the inverse Kro ̈ner

model because the inverse of tensor p(g) is used. Kro ̈ner

(1958) derived expressions that define pijkl(g) and qijkl(g):

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198 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants J. Appl. Cryst. (2012). 45, 197–206

Figure 1

Specimen reference frame with measurement direction m||h, where h is

the direction normal to the lattice planes (hkl) that contribute to a

reflection in the measurement direction m given by the spherical polar

angles ’ and . The direction h is defined in the Cartesian crystal

reference frame (not shown) by spherical polar angles !B and #B. RD is

the rolling direction, TD is the transverse direction and ND is the

direction normal to the sheet plane. The base vectors of the Cartesian

system must be chosen in accordance with the system used in the

measurement of the single-crystal elastic constants.

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pijklðgÞ 1⁄4 Cijkl þ rijklðgÞ; qijðgÞ 1⁄4 Sijkl þ tijklðgÞ; ð10Þ

rijklðgÞ 1⁄4 cijklðgÞ $ Cijkl þ cijmnðgÞ umnklðgÞ; ð11Þ

tijklðgÞ 1⁄4 uijmnðgÞ Smnkl; ð12Þ

uijklðgÞ 1⁄4 v$1

ijmnðgÞ 1⁄2cmnklðgÞ $ Cmnkl(; ð13Þ

vijklðgÞ 1⁄4 cijklðgÞ $ Cijkl þ Cijmnwmnkl; ð14Þ

where cijklðgÞ is the stiffness tensor of the crystallite, Cijkl and

Sijkl are, respectively, the stiffness and compliance tensors of

the overall composite, and wijkl is the inverse Eshelby tensor

(Kinoshita & Mura, 1971; Mura, 1987; Gavazzi & Lagoudas,

1990). From equation (10), Kro ̈ner established the conditions

for calculating the overall elastic constants

rðgÞ 1⁄4 0 and tðgÞ 1⁄4 0; ð15Þ

where the overbars indicate the average over all crystal

orientations. Note that r(g) and t(g) represent different

physical quantities, where r(g) is determined in units of GPa

and t(g) in units of GPa$1

. Equations (15) yield two solutions,

C(r) and C(t)

, that are not necessarily equal. As indicated

before, both yield the same solution only if the grains are all

spherical or ellipsoids with randomly oriented axes. Similar to

the Reuss and Voigt bounds but not as straightforward, the

existence of two solutions, C(r) and C(t)

, is to be interpreted as

bounds on the overall elastic moduli. The existence of bounds

is rooted in the condition of continuity of displacement

(fulfilled for Voigt) and equilibrium of traction (fulfilled for

Reuss) at the grain–matrix interface for all grains. If the

polycrystal under consideration were to consist of needleshaped grains aligned in the same direction, then the elastic

moduli may be closer to the Voigt bound in the direction of

the longitudinal axis, while Reuss bounds may be closer for flat

platelet grains with their short axis aligned in the same

direction.

In the case of DECs, one must calculate partial orientation

averages in the sense that only those orientations contribute to

the average for which a given crystal direction h is parallel to a

given sample direction m. In this case, one has both

rðgÞ

mkh

61⁄4 0 and tðgÞ

mkh

61⁄4 0, with the consequence that

S þ tðgÞ

mkh

61⁄4 C þ rðgÞ

mkh h i$1

: ð16Þ

In fact, the whole basis of the Kro ̈ner model for DEC calculation is tðgÞ

mkh

61⁄4 0. From equation (16), together with

equations (11) and (12), it can be concluded that the only

general exception to the inequality (16) is when both c(g) = C

for all orientations g. In other words, the grains have to be

elastically isotropic (tungsten comes to mind). For all other

cases, DECs calculated using the Kro ̈ner model or the inverse

Kro ̈ner model [first and second parts of equation (10)] will be

different from each other, except for singular combinations of

grain shape and grain orientation (see discussion in x3.1

below).

Using a compact notation without indices, equation (9),

which is also the right-hand side of equation (16), can now be

written as

"ðgÞ 1⁄4 p$1 ðgÞ!

1⁄4 cðgÞ þ cðgÞ1⁄2 ( cðgÞ $ C þ Cw $11⁄2 ( cðgÞ $ C ! "$1!: ð17Þ

As shown previously (Barral et al., 1987; Brakman & Penning,

1988; Do ̈lle et al., 1977, 1978; Baczmanski et al., 1993; Do ̈lle,

1979; Hauk, 1997; Pina et al., 1997; Van Houtte & De Buyser,

1993), the quantity that is measurable by diffraction is that

which is in the direction of the z axis of an intermediate

reference frame. The z axis of this intermediate coordinate

system is parallel to both the direction given by the tilt and

azimuth angles ( , ’) in the specimen coordinate system and

the direction h [normal to the diffracting lattice planes (hkl)]

in the fixed crystal coordinate system.

"L

33 1⁄4 "ð’; "; hklÞ 1⁄4 p$1

33klðgL; !Þ !kl; ð18Þ

where the overbar signifies that the orientation average is

taken over rotations about h for grains with the direction h

parallel to ( , ’). The transformation matrix gL is given by

(Bunge, 1982)

gL 1⁄4 g2 $; !B;

%

2 ; $#B

n og1 ’ þ

%

2 ; ; 0

n o; ð19Þ

as the product of the transformation g1 from the specimen

coordinate system to the intermediate system and g2 as the

transformation from the intermediate system to the fixed

crystal coordinate system. The inverse of gL is used in equation

(3) to calculate all orientation-dependent tensors in equation

(17). The position of the variables in the curly brackets

determines which of the angles {’1, , ’2} they replace. The

angles !B and #B are tilt and azimuth angles of the direction h

in the Cartesian crystal coordinate system that is normal to the

lattice planes (hkl). The elastic property tensors are referenced in a Cartesian coordinate system which means that, in

the case of non-Cartesian unit cells, the real space direction

normal to the lattice planes (hkl), i.e. the strain ‘direction’ or

scattering vector direction, has to be calculated. With the

lattice parameters a, b, c, &, #, ', and using the transformation

given by Sands (1995), one obtains for !B and #B

!B 1⁄4 arccos

hz

ðh2

x þ h2y þ h2zÞ

1=2

" #; #B 1⁄4 arccos

hx

ðh2

x þ h2yÞ

1=2

" #:

ð20Þ

h 1⁄4

hx

hy

hz

0

B@

1

CA 1⁄4

h=a

$h=ða cot 'Þ þ k=ðb sin 'Þ

hbc sin & cos & cos '$cos #

V sin & sin ' þ kac sin # cos # cos '$cos &V sin # sin ' þ lab sin 'V

2

6

4

3

7

5;

ð21Þ

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J. Appl. Cryst. (2012). 45, 197–206 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants 199

electronic reprint
where V = abc(1 + 2 cos&cos#cos' $ cos2

& $ cos2# $

cos2

')

1/2. Equation (21) assumes that x||a, y||(a ) b) ) x and

z||(a ) b), where (x, y, z) are the Cartesian base vectors of the

transformed crystal lattice. The base vector orientation must

agree with that used in the measurement of the elastic property tensors, which requires confirmation especially in the case

of monoclinic and triclinic symmetry. The angle $ is the

rotation angle about h, and it runs from 0 to 2%. Generally, one

has to assume that some degree of preferred grain orientation

is present, which requires the use of averages weighed by the

orientation distribution function (ODF) f(gL

). In practical

terms, both Kro ̈ner-type models, the Reuss model and the

Voigt models can be written as follows (Hauk, 1997; modified

Voigt: Murray & Noyan, 1999):

Kro ̈ner:

Fkl 1⁄4 q33klðgLÞ 1⁄4

R

2%

0

S þ tðgLÞ # $ijklmimjfðgLÞ d$

R

2%

0

fðgLÞ d$

; ð22aÞ

Inverse Kro ̈ner:

Fkl 1⁄4 p$1

33klðgLÞ 1⁄4

R

2%

0

C þ rðgLÞ # $$1

ijklmi

mjfðgLÞ d$

R

2%

0

fðgLÞ d$

; ð22bÞ

Voigt:

Fkl 1⁄4 mi

mj c gL ' ( % & $1

ijkl; ð22cÞ

Modified Voigt:

Fkl 1⁄4 mi

mj

R

2%

0

cðgLÞfðgLÞ d$

R

2%

0

fðgLÞ d$

2

6

6

6

4

3

7

7

7

5

$1

ijkl

; ð22dÞ

Reuss:

Fkl 1⁄4

R

2%

0

sijklðgLÞmimjfðgLÞ d$

R

2%

0

fðgLÞ d$

: ð22eÞ

Here, mi are components of the measurement direction

m 1⁄4 ðcos ’ sin sin ’ sin cos Þ: ð23Þ

Alternatively, in equation (22c), the single-crystal constants

c(gL

) can be replaced by matrix constants Cijkl calculated using

the Kro ̈ner model, which would yield a better approximation

of the mechanical average. The integrations in equations

(22a), (22b), (22d) and (22e) are performed numerically such

that for each $ the matrix product (9) is calculated, and the

Euler angles f’1; "; ’2g are determined from the matrix

components in equation (4).

It should be emphasized that, regardless of the model used

in equations (22a)–(22e), the stress factors contain no implicit

statement about the stress or strain state of the grains themselves, but rather how the lattice strains change in response to

changes in the macroscopic stresses. Thus, any pre-existing

strains in the grains, be it from plastic anisotropy, elastic misfit

or thermal expansion, are immaterial to this calculation.

3. Examples

For a discussion of examples we will focus entirely on ferritic

low-alloy steel (&-iron). Because of its industrial importance

this material has the most detailed body of theoretical analysis

and published experimental data. This material will allow the

study of the effects of preferred orientation and grain shape

with good sensitivity, owing to the appreciable elastic anisotropy visible in measured lattice strain distributions, combined

with high strength (large magnitude of elastic strains).

3.1. Isotropic case (no preferred orientation)

Concentrating for now on spherical grains without

preferred orientation, we find that, in terms of the familiar

quantities s1 and 1

2s2, the inverse Kro ̈ner model falls between

the Kro ̈ner and Reuss models (Fig. 2).

However, unlike the intersecting DECs for the Reuss and

Voigt models, 1

2s2 and s1 do not intersect for either the inverse

Kro ̈ner model or the Kro ̈ner model, and one is always greater

than the other. One should note that, in the case of grains in

the form of aligned nonspherical ellipsoids, s1 and 1

2s2 become

dependent on (’, ), and the relationship between lattice

strain and macroscopic stress must be expressed by the stress

factors Fij in equation (1). Both models yield substantially

different results if the grains are oriented such that the

difference between the tensors [c(g) $ C + Cw] in equation

(17) is large, and this is amplified by extreme grain shapes as in

the case of the 100 reflection. This is shown in Fig. 3.

One must be careful not to assume that the DECs display

similar symmetries to the overall or matrix elastic constants. In

particular, the DECs for spherical grains (a1 = 1, a2 = 1, a3 = 1)

are not equal in both models, as predicted above in the

discussion of the inequality in equation (16). This could be

interpreted as placing upper and lower bounds on the DEC.

However, as stated originally by Kro ̈ner [equation (8)], the

tensors p and q may have interchangeable roles. In fact, the

intersecting curves in Fig. 3 show that this is the case for the

Kro ̈ner and inverse Kro ̈ner models, with the grain shape as the

controlling factor. The asymptotic behavior of both models is

particularly revealing: for grains in the form of very thin

plates, the Kro ̈ner model yields stress factors that for all hkl

converge to the value of the mechanical limit F11(0, 0) = $(/E.

Note that this value is closely related to the Voigt value but

numerically different, because the matrix elastic constants

were calculated using the Kro ̈ner model and not the Voigt

model. The opposite behavior is found for the inverse Kro ̈ner

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200 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants J. Appl. Cryst. (2012). 45, 197–206

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model: the hkl dependence becomes even more apparent and

the model reaches the limit of values calculated by the Reuss

model for each hkl. The stress factors in the mechanical limit

for all hkl are calculated using equation (2) with s1 = (/E and 1

2s2 = (1 + ()/E. Poisson’s ratio ( and Young’s modulus E are

calculated from the matrix elastic constants (Table 1).

Isotropic constants are usually sufficient in accuracy (SC

isotropic in Table 1), and the following formulae can be used

(Hauk, 1997):

E 1⁄4 ðC11 $ C12ÞðC11 þ 2C12Þ

C11 þ C12

and ( 1⁄4 C12

C11 þ C12

: ð24Þ

The argument presented for the asymptotic behavior of the

Kro ̈ner-type models is a numerical one, i.e. it was inferred

from the convergence of hkl at the extremes of elastic crystal

anisotropy (100 and 111 in the cubic case) that this is the case

for all hkl. This procedure was applied to other materials

including corundum, niobium, titanium and uranium, all of

which cover a wide range of crystal symmetries and elastic

anisotropies. All showed the same behavior of asymptotically

reaching the respective limits of the mechanical average

(Kro ̈ner) and Reuss (inverse Kro ̈ner).

3.2. Preferred orientation

A study of the literature also shows that the Fe(211)

reflection is very sensitive to the elastic effects of preferred

orientation in terms of the variability of the slope d(sin2 ).

This reflection is also one of those most frequently used in

residual stress measurements on ferritic steel. In the extreme,

a large tensile stress can produce d spacings that decrease with

 in the range sin2 = 0.2–0.4. This reflection is therefore well

suited to studying the quality of DEC model predictions.

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J. Appl. Cryst. (2012). 45, 197–206 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants 201

Table 1

Overall (matrix) elastic constants calculated using different models

(GPa).

SC denotes self-consistent; SC [50,50,1] and SC [3.2,2.8,1] refer to the grain

shape used.

Model C11 C22 C33 C12 C13 C23 C44 C55 C66

Voigt 294.9 293.1 297.1 114.0 110.1 111.8 86.8 85.1 90.0

Reuss 275.3 273.5 277.6 123.9 119.8 121.6 72.1 70.6 74.3

SC [50,50,1] 285.5 282.4 287.3 119.2 114.3 117.4 81.6 79.2 81.5

SC [3.2,2.8,1] 286.0 283.5 287.2 118.6 114.1 116.3 80.1 78.1 82.4

SC, isotropic 282.4 282.4 282.4 118.3 118.3 118.3 82.05 82.05 82.05

Figure 3

Calculated stress factors F11 for different grain shapes and reflections for an isotropic iron polycrystal in the direction m||ND (left) and m||RD (right). It is

assumed that the grain ellipsoid axes align such that a1||RD, a2||TD and a3||ND. R refers to the Reuss limit of the particular (hkl) next to it, V is the

isotropic Voigt limit, and ML refers to the mechanical limit F11(0, 0) = $(/E (left) and F11(0, 90) = 1/E (right), where ( and E are the bulk values of

Poisson’s ratio and Young’s modulus, respectively.

Figure 2

Diffraction elastic constants s1 and 1

2s2 for spherical grains in an isotropic iron polycrystal over the orientation parameter " = (h2

k2 + h2l

2 + k2

l

2

)/(h2 +

k2 + l

2

)

2

.

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The samples used here are HSLA50 sheet metal extracted

as uniaxial tensile samples (60 mm long ||RD, 5 mm wide ||TD,

0.7–1.0 mm thickness ||ND) from larger sheets of as-received

material and from a sample strained to 20% in equibiaxial

mode. HSLA50 is a high-strength low-alloy ferritic steel used

for automotive applications. It has the composition given in

Table 2.

The single-crystal elastic constants used throughout are for

pure iron: c1111 = 237, 141 and 116 GPa (Alexandrov &

Ryzhova, 1961). A second sample with the same strain history

was used to extract six small plates (5 ) 5 mm) to form a

sample of roughly cuboidal shape for neutron diffraction polefigure measurements of the 110, 200 and 211 reflections. This

mode of operation was made necessary by the relatively small

area of uniform strain on the biaxially expanded sheet sample.

Also, the neutron diffraction experiments, performed at NIST

(Brand et al., 1997), allowed the measurement of full pole

figures. The equibiaxially expanded sample was used in

subsequent X-ray measurements (211 reflection, Cr K&

radiation) under five different levels of applied stress. The asreceived sheet was used to extract five tensile samples, which

were stacked together in a common fixture for stress testing in

a neutron diffractometer (gauge volume 3 ) 3 ) 5 mm) for

the reflections 200, 310, 220, 211 and 222 at sample orientations (’, ) of (0, 0) and (0, 90). Owing to instrumental

restrictions, a more detailed scan in the tilt angle was not

feasible.

3.2.1. Calculations using the ODF. The pole-figure data

were used in the calculation of the ODF in MTEX (Hielscher

& Schaeben, 2008), for which the ND inverse pole figures are

shown in Fig. 4.

For the subsequent calculations of DECs and overall

(matrix) elastic constants, the ODF was read in textual form

on a 5 ) 5 ) 5* grid, then expanded from the 90 ) 90 ) 90*

range of cubic orthorhombic symmetry in the Bunge angles

(’1, !, ’2) to the full interval of 360 ) 180 ) 360* using

symmetry operators. From this table the ODF values for a

given orientation (’1, !, ’2) were obtained through trilinear

interpolation between neighboring values on the 5 ) 5 ) 5*

grid. The cumulative effect of the numerical uncertainty from

this interpolation was determined in two ways. Firstly, using

the thus-read ODF, pole figures were calculated for all 24

permutations (hkl, klh, ...) of the 211 reflection, and secondly,

stress factors were calculated for the same permutations.

Because each hkl permutation results in a different but

symmetry-equivalent fibre orientation m||h through Euler

space, a comparison between fibres allows the quantification

of the numerical accuracy of the method. The relative deviations among different permutations hkl for both pole figures

and stress factors were of the order of 2% without bias, which

was regarded as sufficiently small.

Equation (17) requires the rotation of the single-crystal

stiffness tensor c(g), for which the orientation matrix g has to

be calculated in each sample direction m = (’, ) using

equation (19). More specifically, the numerical integration in

equations (22) over discrete rotation angles $ requires the

calculation of gL from equation (19) for each $, and then use

of inverse trigonometric functions on the matrix elements in

equation (4) to obtain the Bunge angles (’1, !, ’2), together

with the ODF value for f(gL

) in equations (22).

3.2.2. Matrix elastic constants. Through equation (17) the

integrations in equations (22) also require the overall or

matrix elastic constants of the polycrystal. These were determined through both equations (15) using a Newton–Raphson

root-finding iteration. For comparison, the upper [Voigt,

equation (25a) below] and lower bounds [Reuss, equation

(25b) below] on the matrix elastic constants were also calculated, with the results shown in Table 1:

CV

ijkl 1⁄4

R

2%

0

R

2%

0

R

2%

0

cijklðgLÞ fðgLÞsin ! d’1 d! d’2

R

2%

0

R

2%

0

R

2%

0

fðgLÞsin ! d’1 d! d’2

; ð25aÞ

CR

ijkl 1⁄4

R

2%

0

R

2%

0

R

2%

0

sijklðgLÞ fðgLÞsin ! d’1 d! d’2

R

2%

0

R

2%

0

R

2%

0

fðgLÞsin ! d’1 d! d’2

2

6

6

6

4

3

7

7

7

5

$1

: ð25bÞ

Inspection of the values in Table 1 shows that, compared with

the Reuss/Voigt bounds in self-consistent (SC) estimates, the

effect of both grain shape and preferred orientation on the

overall elastic constants is small. Also, the SC estimates fall

approximately in the middle between the Voigt and Reuss

estimates. Therefore, the use of the Voigt/Reuss bounds as

matrix constants should show the full extent of the matrix

effect on the DECs (Fig. 5). It should be emphasized that the

calculation of the matrix constants is not per se a necessity

because one could use matrix elastic constants that were

measured in some way. An example where this may be the

more appropriate solution is a multiphase material where only

the preferred orientation of one phase is known or measurable.

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202 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants J. Appl. Cryst. (2012). 45, 197–206

Figure 4

Inverse pole figures for ND, with levels in steps of 1 ) multiples of

random density (m.r.d.), for the HSLA as-received sample and after 20%

equibiaxial deformation.

Table 2

Chemical composition of HSLA 50 in wt%; Fe provides the remainder.

C Mn P S Si Nb

0.05 0.44 0.009 0.01 0.03 0.023

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The average grain shape of [3.2,2.8,1] was estimated using

an optical intersect method for the as-received sample

(average grain shape [1.85,1.6,1]) with subsequent 20% biaxial

elongation in-plane and contraction in the normal direction

due to volume conservation. Electron-backscatter diffraction

measurements on the strained sample were not feasible

because of the high dislocation density.

In the two solutions to equation (15), the differences

between components CðrÞ

ij $ CðtÞij are less than 0.1, and for

practical purposes they are set to C(r) = CC(t)

, with the result

given in the table. As shown in Fig. 5, the effect of the matrix

elastic constants is weak, and it has a visible effect on the

diffraction stress factors only if the Voigt or Reuss bounds are

used. The small differences between the SC estimates of the

matrix elastic constants yield similarly small differences in the

DECs, which deviate from one another by less than the line

width of the SC graphs in Fig. 5.

3.2.3. Stress factors. In order to facilitate the comparison of

calculated stress factors with experimental data from a plastically deformed sample, it is important to separate completely

the effects of intergranular strains from the stress factor

measurement. Intergranular strains, also known as residual

strains of the second kind (type II), are the result of plastic

deformation, and they are caused by plastic anisotropy

between grains of different orientations. Intergranular strains

are primarily visible in the absence of other stresses, i.e. under

zero applied or residual stress (see Fig. 6). Thus, any effect of

intergranular strains on a stress factor measurement can be

completely removed by choosing the lattice strains or d

spacings at zero applied load as the reference state from

which, through successive application of applied stresses, the

linear elastic changes are measured for each sample direction

m = (’, ). The actual stress factor determination is simply an

application of Hooke’s law, here done through the slope of the

linear regression of lattice strain over applied stresses (0, 141,

249, 358 and 475 MPa) (Fig. 6, right). Note that the actual

sequence should start at the highest load, with the zero load

measurement as the last. This way, in the absence of creep, the

stress factor measurements are guaranteed to reflect only

linear elastic effects, as shown in Fig. 6, right.

Using the matrix elastic constants for SC [3.2,2.8,1] from

Table 1, the F11 stress factors are calculated and compared

with measurements in the ND–RD plane (’ = 0*) and in the

ND–TD plane (’ = 90*) (Fig. 7).

For the ND–RD plane, the results reflect reports in the

literature (e.g. Barral et al., 1987; Baczmanski et al., 1993) that

good agreement is observed using the Reuss model, while the

Kro ̈ner model generally captures less well the ‘snake line’

distribution of the stress factors. Of all the models, the inverse

Kro ̈ner model as introduced here shows the best agreement

for platelet-shaped grains with aspect ratios [50,50,1]. Using

the experimentally determined estimate of the average grain

shape [a1 = 3.2, a2 = 2.8, a3 = 1], the agreement is still better

than for the Kro ̈ner model. The modified Voigt model

(Murray & Noyan, 1999) yields results very similar to the

Reuss model. The sensitivity of the Kro ̈ner model to the grain

shape is counterintuitive – needle grains aligned parallel to

ND (not shown) would yield a better agreement than platelets

research papers

J. Appl. Cryst. (2012). 45, 197–206 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants 203

Figure 6

Left: d spacings measured for the Fe(211) reflection versus sin2 and for different applied stresses for the equibiaxially expanded sample. The specimen

tilt is in the plane ’ = 0 (ND–RD||stress). Right: strain (d $ d0)/d0 versus applied stress. The reference d spacing is the value measured at zero stress (left,

0 MPa). The slopes are the values of F11 at = 0*, = +25.2* and = +45*.

Figure 5

Effect of the matrix (overall) elastic constants on the stress factor F11

using the ODF of the equibiaxially expanded sample. LB stands for

matrix constants calculated using the Reuss model, UB stands for the

Voigt model and SC is the self-consistent estimate for the average grain

shape [3.2,2.8,1]. The values are listed in Table 1.

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[50,50,1]. Plate-like grains with their small dimension along

ND would be expected after biaxial stretching along RD and

TD. The Kro ̈ner model with [50,50,1] grains approaches the

mechanical limit (close to the Voigt case). In contrast, the

inverse Kro ̈ner model behaves correctly, i.e. showing better

agreement as the grain shape varies from low aspect ratios

[3.2,2.8,1] to thin plates [50,50,1] where, in turn, the inverse

Kro ̈ner model approaches the Reuss limit of homogeneous

stress. In the ND–TD plane at ’ = 90*, one finds a reversal of

the roles of the two models. The best agreement is now found

for the Kro ̈ner model using the measurement-based estimate

of the grain shape [3.2,2.8,1]. For flat grains ([50,50,1]), the

Kro ̈ner model approaches the mechanical limit, just as in the

ND–RD plane. Remarkably, the modified Voigt model shows

similarly good agreement with the measured stress factors.

The inverse Kro ̈ner model, which performed best at ’ = 0*

with [50,50,1] as grain-shape parameters, again approaches the

Reuss limit, and both show the worst agreement. Therefore,

one can surmise that this type of model agreement reflects the

role of qijkl and p$1

ijkl as bounds, as expressed in equation (8),

with the grain-shape parameters controlling the proximity to

the respective Reuss or mechanical limit. Even for spherical

grains, the two Kro ̈ner-type models do not start out the same,

but rather in close proximity to the Reuss/Voigt average (not

shown), from which, on increasing the grain aspect ratio, the

Kro ̈ner model approaches the mechanical limit and the

inverse Kro ̈ner model approaches the Reuss limit. This is the

result of the asymptotic behavior of the Eshelby tensor for a1,

a2 >> a3. Here, one has nearly homogeneous strain (grain and

surrounding matrix) in the directions of a1 (RD) and a2 (TD),

while the stresses in the a3 direction (ND) are zero (Mura,

1987, p. 82).

It should be pointed out that grain elongation from slip due

to plastic deformation depends on the orientation of the

crystal lattice of the grain. For any given orientation (’, )||h,

the grains contributing to diffraction and to the integral in

equation (20) will not have the same shape parameters

[a1,a2,a3], thus making the Eshelby tensor a variable as well.

This has been disregarded in this work because of the lack of

appropriate data, but it provides a possible explanation of the

fact that extreme ellipsoid-axes aspect ratios show the best

agreement between measured and calculated stress factors in

Fig. 7.

It is tempting to interpret the very good agreement of the

modified Voigt model in both tilt planes (ND–RD and ND–

TD) as an overall confirmation for the validity of its underresearch papers

204 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants J. Appl. Cryst. (2012). 45, 197–206

Figure 7

Comparison of measured and calculated stress factors F11 at ’ = 0* (top) and ’ = 90* (bottom) for the Kro ̈ner model (left), the inverse Kro ̈ner model

(right), the mechanical average (calculated from SC [3.2,2.8,1] in Table 1), the Reuss model and the modified Voigt model. The sample was previously

stretched to 20% equibiaxial strain. The direction of the applied stress was ’ = 0*, = 90*. The uncertainties (1!) for the measured F11, if not visible, are

smaller than the size of the symbols. Orthorhombic sample symmetry (implied by the biaxial straining) was assumed and the and $ branches were

averaged. The matrix constants (SC [3.2,2.8,1] in Table 1) needed in the Kro ̈ner-type models were calculated from the iron single-crystal elastic constants

and the ODF using equation (25a).

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lying assumption, i.e. homogeneous strain for a particular hkl.

This is not the case, as shown by the comparison for different

lattice planes (hkl) in Table 3, where large differences from

measured values are found for the 200 and 222 reflections.

Table 3 shows, through the quality measure )2

, that the

Kro ̈ner model performs best, followed by the mechanical

average which, despite its simplicity, yields roughly the same

average agreement as the inverse Kro ̈ner model. Owing to the

weak texture, the inclusion of the ODF into the calculation

does not provide improvement over the isotropic model. All

models evaluated here give similar values F11 for 3" near the

mechanical average of 3" = 0.6 (intersect region in Fig. 2),

which explains not only the good performance of the modified

Voigt model for the equibiaxially strained sample, but also the

general preference given in the literature to the Reuss model

for mixed-type reflections such as 211. The homogeneous

stress assumption of the Reuss model is clearly not justified,

but the Reuss strains (which would violate continuity of

displacement) it implies are small because, for the particular

grain orientations involved in these reflections (such as 211),

the differences |c[(hkl)||z] $ C| are small. For a different

reason (small Voigt stress), the same applies for the Voigt

model.

The results from Table 3 are applicable primarily to neutron

diffraction because of the directions ( = 90* is normally

inaccessible to X-ray diffraction) and hkl involved. Considering the low level of preferred orientation in the as-received

sample, strain measurements along the principal directions –

as is typically the case in neutron diffraction – appear rather

insensitive to texture. The stress factor–tilt angle dependence

(sin2

 ) is more typical for X-ray diffraction, with results

shown in Fig. 8. The results demonstrate that even weak

texture matters in X-ray diffraction.

The differences between the two directions RD [F11(’ =

0*)] and TD [F22(’ = 90*)] are noticeable, with the inverse

Kro ̈ner model again capturing both curves better than the

Kro ̈ner model which, in turn, provides the better agreement

for F11(’ = 90*) and F22(’ = 0*). In effect, this outcome is the

same as for the equibiaxially expanded sample (Fig. 7).

The shape of the lattice strain distributions in Figs. 7 and 8 is

determined by the texture, the grain shape and the elastic

anisotropy of the grains. Because only the tilt-angle dependence of Fij(211) was analyzed, it would be premature to

generalize the observed respective preferences for each

Kro ̈ner-type model. For example, the extreme values of crystal

anisotropy are represented by the 200 and 222 reflections

(Table 3), and for both reflections the Kro ̈ner model is clearly

the better choice. More measurements involving different

material and texture combinations will be necessary to

develop general criteria for the selection of a particular

Kro ̈ner-type model.

4. Conclusions

A model for the calculation of diffraction stress factors, here

dubbed the inverse Kro ̈ner model, has been developed and

compared with measured stress factors from a sample with

strong preferred orientation. In X-ray measurements of the

211 reflection, the inverse Kro ̈ner model yields the best

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J. Appl. Cryst. (2012). 45, 197–206 Thomas Gna ̈upel-Herold et al. & Diffraction elastic constants 205

Table 3

Stress factors F11 for an as-received sample of HSLA steel measured by

neutron diffraction (TPa$1

).

)2 is defined as # ðFmeas

11 $ Fcalc11 Þ

2

=ðFmeas

11 Þ

2 # $. Kro ̈ner (isotropic) refers to the

Kro ̈ner model with equiaxial grains and no preferred orientation. The

modified Voigt, Reuss, Kro ̈ner and inverse Kro ̈ner models use the ODF, with

these last two models using an average grain shape of [1.85,1.6,1].

hkl 3" (’, ) F11 1!

Mechanical

average

Modified

Voigt Reuss Kro ̈ner

Inverse

Kro ̈ner

Kro ̈ner

(isotropic)

200 0 (0, 0) $1.78 0.10 $1.38 $2.83 $2.83 $1.67 $2.38 $1.89

310 0.27 (0, 0) $1.63 0.10 $1.38 $1.95 $2.36 $1.59 $2.04 $1.66

220 0.75 (0, 0) $1.36 0.07 $1.38 $1.15 $1.46 $1.40 $1.39 $1.26

211 0.75 (0, 0) $1.32 0.04 $1.38 $1.16 $1.49 $1.40 $1.40 $1.26

222 1 (0, 0) $1.07 0.06 $1.38 $0.79 $0.79 $1.14 $0.98 $1.05

)2 0.16 0.49 0.63 0.01 0.19 0.01

200 0 (0, 90) 5.16 0.08 4.67 7.59 7.59 5.83 6.06 5.70

310 0.27 (0, 90) 4.87 0.05 4.67 5.67 6.49 5.33 5.55 5.25

220 0.75 (0, 90) 4.19 0.07 4.67 4.01 4.53 4.40 4.64 4.45

211 0.75 (0, 90) 4.28 0.08 4.67 4.02 4.53 4.45 4.58 4.45

222 1 (0, 90) 4.06 0.13 4.67 3.52 3.52 3.97 4.14 4.03

)2 0.05 0.27 0.36 0.03 0.07 0.02

Figure 8

Measured (X-ray diffraction) and calculated stress factors F11 (left, measured with !applied||RD) and F22 (right, measured with !applied||TD) for the 211

reflection of the as-received sample. The Kro ̈ner and inverse Kro ̈ner model use the average grain shape [1.8,1.6,1] and the matrix elastic constants

calculated from the ODF of the as-received state. The X-ray test regimen was the same as for the equibiaxially expanded sample.

electronic reprint
agreement with measured stress factors in the tilt plane that

contains the stress direction, while the Kro ̈ner model is the

best choice in the tilt plane perpendicular to the stress direction. The use of elongated flat ellipsoidal grain shapes –

congruent with experimental findings – is necessary to achieve

this level of agreement. For extreme grain shapes, the Kro ̈ner

model reaches the hkl-independent limit of the bulk elastic

constants, while the inverse Kro ̈ner model approaches the

Reuss limit. It is concluded that both models represent bounds

on the diffraction elastic constants, albeit narrower than the

Reuss/mechanical average limit, and with the grain shape as

the controlling parameter that moves the respective Kro ̈nertype model from a Reuss/mechanical limit average position

closer to the respective bound.

For a sample with weak texture, neutron diffraction

measurements of stress factors involving other hkl along the

principal sample directions showed a general preference for

the Kro ̈ner model with a negligible effect from the weak

texture. However, for intermediate tilt angles typical for X-ray

diffraction, even weak preferred orientation has a visible

effect and it should be included in stress factor calculations.

The experimental data presented here or available in the

literature are not sufficient to develop a general guideline for

which model to use under a given set of circumstances (grain

shape, texture). However, the large differences that are

expected for h00-type reflections for platelet grains indicate

the type of experiment necessary to gain this insight. Both

models should also give different results if a realistic grain

shape–crystal orientation correlation (which is a result of slip)

is used.

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