Model for Analysis of Biaxial and Triaxial Stresses by X-ray Diffraction Assuming Orthotropic Materials

Reprinted from
REGULAR PAPER
Model for Analysis of Biaxial and Triaxial Stresses
by X-ray Diffraction Assuming Orthotropic Materials
Edson M. Santos, Marcos T. D. Orlando, Milton S. R. Milta˜o,
Luis G. Martinez, Alvaro S. Alves, and Carlos A. Passos
Jpn. J. Appl. Phys. 49 (2010) 056601
# 2010 The Japan Society of Applied Physics

Person-to-person distribution (up to 10 persons) by the author only. Not permitted for publication for institutional repositories or on personal Web sites.
Model for Analysis of Biaxial and Triaxial Stresses
by X-ray Diffraction Assuming Orthotropic Materials
Edson M. Santos1;2Ã
, Marcos T. D. Orlando2
, Milton S. R. Miltaõ1
,
Luis G. Martinez3
, Alvaro S. Alves1;4
, and Carlos A. Passos2
1
Laborato´rio de Fı´sica de Materiais, Departamento de Fı´sica, Universidade Estadual de Feira de Santana,
Av. Transnordestina, s/n, Novo horizonte, Campus Universitrio, Feira de Santana-BA, 44036-900, Brazil
2
Universidade Federal do Espı´rito Santo, Av. Fernando Ferrari, 514, Goiabeiras, Vito´ria-ES, 29060-910, Brazil
3
Instituto de Pesquisas Energe´ticas e Nucleares, Av. Lineu Prestes 2242, Cidade Universita´ria, Saõ Paulo-SP, 05508-000, Brazil
4
Instituto de Fı´sica, Universidade Federal Fluminense,
Av. Gal. Milton Tavares de Souza, s/n, Campus da Praia Vermelha, Niteroí-RJ, 24210-346, Brazil
Received November 20, 2009; revised December 26, 2009; accepted January 4, 2010; published online May 20, 2010
In this work we aim to develop expressions for the calculation of biaxial and triaxial stresses in polycrystalline anisotropic materials, and to
determine their elastic constants using the theory of elasticity for continuum isochoric deformations; thus, we also derive a model to determine
residual stress. The constitutive relation between strain and stress in these models must be assumed to be orthotropic, obeying the generalized
Hooke’s law. One technique that can be applied with our models is that of X-ray diffraction, because the experimental conditions are similar to the
assumptions in the models, that is, it measures small deformations compared with the sample sizes and the magnitude of the tensions involved,
and is insufficient to change the volume (isochoric deformation). Therefore, from the equations obtained, it is possible to use the sin2
technique
for materials with texture or anisotropy by first characterizing the texture through the pole figures to determine possible angles that can be used
in the equation, and then determining the deformation for each diffraction peak with the angles obtained from the pole figures.
# 2010 The Japan Society of Applied Physics
DOI: 10.1143/JJAP.49.056601
1. Introduction
In materials science, the study of residual stress is of interest
for a wide range of technological applications, for instance,
thermal barriers and corrosion resistance on metallic
structures,1)
mechanical stress in superconductor magnets,2)
residual tension in thin films,3)
corrosion fatigue in alloys
widely used for the structural components of older aircraft,4)
and so forth.
Thus, the importance of developing models that enable
the investigation of residual stress is clear. In this work we
define the material stress as the internal stress existing in a
material when it is not submitted to external forces.5)
Thus,
we can classify the material stress as intrinsic and extrinsic.
. Intrinsic stress appears during the process of material
growth and generally originates from the defects
incorporated in the material structure.
. Extrinsic stress appears after the process of material
growth and in general its causes are the thermal and
mechanical effects on the materials.
From the experimental point of view, the residual stress
in materials can be determined by different methods and
techniques:
. Mechanical methods are linear and are based on
applying dissection and section methods to the sample.
Examples of dissection methods include the ring-core
method and the hole-drilling method; an example of a
section method is the material-removing method.6)
These mechanical methods are destructive.
. Nonlinear elastic methods involve the inelastic mod-
ification of the sample structure, examples include the
ultrasonic and magnetic techniques.7)
. Diffraction techniques are linear and based on the
diffraction phenomenon because of the wave incidence
over the sample by elastic spreading. Examples include
X-ray diffraction1,3)
and neutron diffraction.8,9)
These
methods are not destructive.
The stresses cannot be directly measured. To obtain the
stress we need to measure various properties of the material
such as the deformation.1,8,10)
The methods given above give
the deformation.
Stresses in a material originate from many sources, and in
general they can be divided into three categories11)
accord-
ing to their length scale.
. Type I stress, which acts at a scale of a large number
of grains (i.e., millimeters), is, by definition, independ-
ent of individual grain orientation and is known as
the macrostress.11)
It is homogeneous over very large
crystal domains of the materials.12)
. Type II stress (intergranular stress), which varies from
grain to grain, is homogeneous within a small number
of crystal domains of the material (a single grain or
phase).12)
. Type III stress, which originates from local defects and
fluctuates within a grain, is homogeneous within the
smallest crystal domains (over several atomic dis-
tances).12)
In the case of real materials, the actual stress state at a point
results from the superposition of stress types I, II, and III. The
stress is strongly dependent on the material’s anisotropy.
As is well known, a crystal is characterized as a periodic
array of its elements in space. For this reason a dependence
of the crystalline properties on the direction is generated,
known as anisotropy. Most natural and artificial solids
contain many crystallites, which can have different sizes,
forms and orientations. The crystallites are units of the
microscopic monocrystals of the material.
The preferential orientation of the crystallites inside a
material is called the texture.13)
The texture is an intrinsic
characteristic of metals, ceramics, polymers, and rocks and
it has a strong effect on the anisotropy of the physical
properties of a material.Ã
E-mail address: emascarenhassantos@gmail.com
Japanese Journal of Applied Physics 49 (2010) 056601 REGULAR PAPER
056601-1 # 2010 The Japan Society of Applied Physics

The effect of texture on residual stress is a well-known
and ongoing problem14)
and is still an open issue from the
theoretical and experimental points of view.
Some works have been reported on this theme, for
example:
. The determination of Poisson’s ratio in orthotropic
materials was investigated by Lempriere.15)
. The applicability of the destructive hole-drilling tech-
nique to the experimental determination of residual
stresses in relatively thin rectangular orthotropic
materials was investigated by Lake et al.16)
and Schajer
and Yang.17)
. The determination of stress in textured thin films using
X-ray diffraction was investigated by Clemens and
Bain.18)
. The determination of the elastic constants of a fiber-
textured gold film by combining synchrotron X-ray
diffraction and in situ tensile testing was investigated
by Faurie and coworkers.19,20)
. The analysis using the elastic orthotropic model was
applied by Mascia21)
to determine the elastic properties
of wood.
In this work we investigate the effect of texture on type I
stress (or macrostress) in a material. We develop a model
that can take into consideration the orthotropic anisotropy of
a material by assuming small deformations in a continuum
material. A material is called orthotropic when three
symmetric orthogonal axes with different elastic properties
exist. In this sense, our proposed model differs from the
earlier models as we consider orthotropic anisotropy as a
diffraction technique.
The assumption on which our model is based is that of
small deformations. This implies that the deformations are
isochoric, which means they do not change the material
volume, allowing us to apply continuum media elasticity
theory. This can take into consideration the constitutive
relation between stress and strain, i.e., the generalized
Hooke’s law, which we model.
Taking into consideration the fact that from the exper-
imental point of view the stress is usually measured by a
diffraction technique, our model can be applied to diffraction
techniques (X-rays and neutrons), because the experimental
conditions are consistent with those assumed in the model,
i.e., (a) small deformations in comparison with the sample
dimensions and (b) the stress involved is not sufficient to
change the material volume.
As is well known, in the sin2
diffraction technique the
presence of texture in a material causes a nonlinear relation,
with snak-like "hkl
X sin2
curves.14)
Although the use of
this technique has not been reported for textured materials,
note that attempts have been made to remedy the effects of
nonlinearity22,23)
by semi-empirical procedures such as the
Marion-Cohen technique.4)
In our model we demonstrate that, under appropriate
considerations, we can continue using the sin2
technique,
even for materials with texture. Therefore, we characterize
the texture by pole figures of the majority diffraction peaks
with the objective of determining the possible values of
that can be used in our model.
This work is structured as follows. In §2 we review linear
elasticity theory and the constitutive relation between stress
and strain for orthotropic materials. In §3 we develop
expressions for the main triaxial and biaxial stresses. In §4
we give a proof of the consistency of these equations by
comparing them with previously reported models for
isotropic materials, and then we apply our model for biaxial
stress using the experimental data obtained by Faurie
et al.20)
As a result, we can observe the magnitude of the
elastic constants determined by our model. In §5 we present
our conclusions.
2. Linear Elasticity Theory
2.1 General considerations
Linear elasticity is a theory used to study the behavior of
material bodies that are deformed when submitted to
external actions (forces caused by the contact with other
bodies, gravitational force acting on mass, etc.) then return
to their original form when the external action is removed,
with no permanent change in the material volume. Within
certain limits that depend on the material and temperature,
the applied stress is roughly proportional to the strain.
In classical linear elasticity theory it is assumed that
displacements and displacement gradients are sufficiently
small for no distinction to be made between Lagrangian and
Eulerian descriptions.24)
Furthermore, we assume that the
deformation processes are adiabatic (no heat loss or gain)
and isothermal (constant temperature).
Linear elasticity theory25)
in the cartesian coordinate
system ðx1; x2; x3Þ includes the following equations:
ij;j À
@2
ui
@t2
þ Fi ¼ 0; ð1Þ
where i; j ¼ 1; 2; 3 this is the equation of motion for the
physical system;
ij ¼ Ãijklkl; ð2Þ
where i; j; k; l ¼ 1; 2; 3 this is the generalized Hooke’s law;
and
2kl ¼ ul;k þ uk;l; ð3Þ
this is the Cauchy formula, which expresses the strain in
terms of displacement.
In eqs. (1)–(3), ij ¼ ji are the components of the
symmetric stress tensor, ij ¼ ji are the components of the
strain tensor, Ãijkl are the components of the fourth-rank
tensor of the elasticity moduli, ui are the components of the
displacement vector, Fi are the components of the vector of
bulk forces, is the constant density of the material, and t is
the time. The comma before of the subscript indicates
differentiation with respect to the spatial coordinate denoted
by the subscript; repeated letters in the subscripts indicate
summation over their possible values.25)
The tensor symmetry in ij and ij is a consequence of the
assumption that there is no resultant torque in the material.
Elasticity theory is used to accurately determine the stress,
strain, and the relationship between them within a material.
In this paper, we study the relationship between stress and
strain given by the generalized Hooke’s law. Our goal,
therefore, is to determine the constants of the elasticity
tensor for a material with orthotropic symmetry. From this,
we obtain a model for determining the residual stresses of an
orthotropic material.
Jpn. J. Appl. Phys. 49 (2010) 056601 E. M. Santos et al.

2.2 Volumetric deformation
In general, the deformation of a solid involves the
combination of a volume change and a change in the
sample’s form. Thus, for a given state of deformation, it is
necessary to determine the contributions of changes in
volume and shape. In this work, we assume a domain with
small deformations, so that only the shape of the sample will
be modified.
From the literature we know that the constraints on
volumetric models for an anisotropic linearly elastic solid
are that the solid must be:26)
rigidtropic (zero deformations),
isochoric (volume change equal to zero), and hydroisochoric
(isochoric with hydrostatic stress). Thus, here we assume
that the deformation experienced by the material is iso-
choric.
The volumetric deformation is the volume variation per
unit volume.27)
Á ¼
Vf À V0
V0
¼
ÁV
V0
; ð4Þ
where Vf and V0 are the final and initial volumes of
the volume element respectively. Thus, the deformation
corresponds to a change in the shape of the continuum
between an initial configuration (undeformed) and a sub-
sequent configuration (deformed), considering two reference
systems.24)
As is known, the volume variation is given by
ÁV ¼ l0
1ðl0
2 Â l0
3Þ À l1ðl2 Â l3Þ; ð5Þ
where l1, l2, and l3 are the sides of a rectangular
parallelepiped.
Using the geometric interpretation of the normal defor-
mation, and considering an infinitesimal volume element
with li dxi, the new lengths of the infinitesimal paralle-
lepiped can be written as
dx0
1 ¼ dx1ð1 þ xxÞ;
dx0
2 ¼ dx2ð1 þ yyÞ; ð6Þ
dx0
3 ¼ dx3ð1 þ zzÞ;
where xx, yy, and zz are the deformations in the x, y, and z
directions, respectively.
Thus, the change in volume becomes
dx0
1dx0
2dx0
3 À dx1dx2dx3
¼ dx1dx2dx3ð1 þ xxÞ Á ð1 þ yyÞ Á ð1 þ zzÞ À dx1dx2dx3:
For small deformations, their products are neglected28)
ð1 þ xxÞð1 þ yyÞð1 þ zzÞ ¼ 1 þ ðxx þ yy þ zzÞ: ð7Þ
Substituting these values in eq. (4) and considering the
isochoric assumption, we obtain
dx0
1dx0
2dx0
3 À dx1dx2dx3
dx1dx2dx3
¼ xx þ yy þ zz
¼ 0: ð8Þ
The volume variation per unit volume (the volumetric
deformation) at a point is the sum of the normal strains,
which is equal to zero for small deformations.
2.3 Generalized Hooke’s law
In the development of a macroscopic phenomenological
theory, the use of constitutive relations has a key role. They
allow us to characterize the state of macroscopic physical
systems experimentally. A constitutive relation is a relation-
ship between two physical quantities that is specific to a
material.
Constitutive relations are particular to each material and
are used to classify different materials according to their
behavior. The mechanical constitutive equations that classify
engineering materials, such as mechanical behavior, relate
tension with a parameter of the body’s movement, usually
the deformation or deformation rate. There are many other
types of constitutive equations such as (i) those that relate
stress with deformation and temperature, (ii) those that relate
stress with electric or magnetic fields, (ii) Ohm’s law,
(iv) the law governing the friction force, and (v) the law
governing linear elasticity (Hooke’s law).
In the case of linear elasticity theory, the constitutive
relation given by the generalized Hooke’s law eq. (2), has a
key role because it semi-empirically determines the behavior
of a solid material under existing tension.24)
Equation (2) is
represented in a tensorial form by
ij ¼ Ãijklkl: ð9Þ
As stress is a second-order tensor, its components can
be represented by a matrix where the diagonal components
ii are called the normal stresses and the non-diagonal
components are called the shear stresses or tangential
stresses. The non-diagonal components ij act in the
direction of the jth coordinate axis and on the plane whose
outward normal is parallel to the ith coordinate axis.24)
Then,
ij ¼
xx yx zx
xy yy zy
xz yz zz
0
B
@
1
C
A: ð10Þ
As we have seen earlier, the stress tensor is symmetric,
which means that ij ¼ ji.
Similarly, the strain tensor kl is also symmetric and of
second order with a matrix representation given by28)
ij ¼
@u
@x
@u
@y
@u
@z
@v
@x
@v
@y
@v
@z
@w
@x
@w
@y
@w
@z
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
: ð11Þ
Furthermore, the elasticity tensor, Ãijkl, is a fourth-order
tensor and contains 34
¼ 81 elements. It is worth noting that
this tensor is a characteristic of each material, which implies
that it is invariant under coordinate transformations, such
as that given by eq. (12). Its form and components, Ãijkl,
are the same for different observers and characterize the
material properties completely.29,30)
The orthogonal coordinate transformation is given by29)
x0
i ¼ aijxj; ð12Þ
where aij are the components of the orthogonal operator
A,29)
an element of Oð3Þ.
The set Oð3Þ of all orthogonal transformations, aij,30)
is
called the symmetry group of the material. In relation to this
set, the elastic properties are invariant and this symmetry
group provides the basis for the standard classification of the
various types of anisotropy in elastic media.29)

We exploit these symmetries to study the elastic proper-
ties of an orthotropic material.
Following Annin and Ostrosablin,25)
we introduced a
six-dimensional vector space with the usual scalar multi-
plication and, for the above equations, Hooke’s law, eq. (2),
takes the following form
i ¼ Ãijk; ð13Þ
with i; j ¼ 1; . . . ; 6.
The elasticity tensor in this vector space takes the
following form:
Ãij ¼
Ã1111 Ã1122 Ã1133
ffiffiffi
2
p
Ã1123
ffiffiffi
2
p
Ã1113
ffiffiffi
2
p
Ã1112
Ã2211 Ã2222 Ã2233
ffiffiffi
2
p
Ã2223
ffiffiffi
2
p
Ã2213
ffiffiffi
2
p
Ã2212
Ã3311 Ã3322 Ã3333
ffiffiffi
2
p
Ã3323
ffiffiffi
2
p
Ã3313
ffiffiffi
2
p
Ã3312
ffiffiffi
2
p
Ã2311
ffiffiffi
2
p
Ã2322
ffiffiffi
2
p
Ã2333 2Ã2323 2Ã2313 2Ã2312
ffiffiffi
2
p
Ã1311
ffiffiffi
2
p
Ã1322
ffiffiffi
2
p
Ã1333 2Ã1323 2Ã1313 2Ã1312
ffiffiffi
2
p
Ã1211
ffiffiffi
2
p
Ã1222
ffiffiffi
2
p
Ã1233 2Ã1223 2Ã1213 2Ã1212
0
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
A
: ð14Þ
As the strain and stress tensors are symmetric, the number
of elastic constants of the tensor Ãijkl, which represents the
material, is reduced from 81 to 21.2,28)
Thus, a general
anisotropic material with linear elasticity has 21 independent
constants.29,30)
Materials characterized by these tensors are
said to be triclinic or full anisotropic because their material
properties do not have planes of symmetry. The matrix
representation in Voigt notation, is given by:
Ãij ¼
11 12 13 14 15 16
22 23 24 25 26
33 34 35 36
Symet. 44 45 46
55 56
66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
: ð15Þ
Considering the orthogonal coordinate transformation
given by eq. (12), the number of independent components
of Ãij decreases from 21 to 18.31)
When a material exhibits symmetric operations (rotations,
reflections, or inversions) in its structure, the number of
independent constants, in the tensor of elasticity, eq. (15), is
also reduced. It turns out that material invariance places
constraints on the elasticity tensor Ã, and this provides a
natural way of classifying elastic media.29)
Our objective in
this work is to study these constraints in detail in the case of
orthotropic materials.
We consider only rigid symmetric operations such as
reflections and rotations for two points, ðx; y; zÞ and ðx0
; y0
; z0
Þ,
to explore the symmetries in the structure of an anisotropic
material.
When a material has reflection symmetry in one plane
through the origin, the number of independent components is
reduced from 18 to 13.29)
Such materials are called
monoclinic.
When a material has two planes of reflection symmetry
that are orthogonal to each other, the number of components
is reduced from 18 to 9.29)
Such materials are called
orthotropic.
The symmetry group for an orthotropic material has
eight elements,30)
in three-dimensional space, and is given
by29)
Gortho ¼
8

:
1 0 0
0 1 0
0 0 1
0
B
@
1
C
A;
À1 0 0
0 À1 0
0 0 À1
0
B
@
1
C
A;
1 0 0
0 1 0
0 0 À1
0
B
@
1
C
A;
1 0 0
0 À1 0
0 0 1
0
B
@
1
C
A;
À1 0 0
0 1 0
0 0 1
0
B
@
1
C
A; and their combinations
9
=
;
:
ð16Þ
Because of the symmetry group Gortho, eq. (15) has the
following form29)
(which means that it is written in relation
to its principal coordinate system32)
)
Ãij ¼
11 12 13 0 0 0
12 22 23 0 0 0
13 23 33 0 0 0
0 0 0 44 0 0
0 0 0 0 55 0
0 0 0 0 0 66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
: ð17Þ
This is the matrix for orthotropic materials, the object of
our study, which has only nine independent elastic coef-
ficients. Moreover, orthotropic materials have three mutually
orthogonal planes of elastic symmetry.
3. Expressions for Triaxial and Biaxial Stress States
for Orthotropic Materials
3.1 General considerations
To determine the components of the orthotropic elasticity
tensor, given by eq. (17), we use the elements of the
orthotropic symmetry group, eq. (16), with a phenomeno-
logical approach.
The elastic behavior of an orthotropic material is
characterized by nine independent constants. In phenom-
enological terms they are referred to as three longitudinal
moduli of elasticity or Young’s moduli ðEx; Ey; EzÞ, which
describe the tendency of an object to deform along an
axis when opposing forces are applied along that axis;
three transverse moduli of elasticity or shear moduli

ðGxy; Gyz; GzxÞ, which describe an object’s tendency to shear
(the deformation of a shape at a constant volume) when
acted on by opposing forces; and three Poisson coefficients
ðxy; yz; zxÞ, which describe the ratio of transverse contrac-
tion strain to longitudinal extension strain in the direction of
the stretching force.
Note that there is a connection between the stress and the
Poisson effect, because at the molecular level, the Poisson
effect is caused by small movements between molecules and
the stretching of molecular bonds within the material lattice
to accommodate the stress. When the bonds elongate in the
stress direction, they shorten in the other directions. This
behavior multiplied by millions of times throughout the
material lattice drives the phenomenon. Mathematically, the
Poisson coefficients are represented as follows:
ij ¼ À
jj
ii
; ð18Þ
where j represents the transverse strain and i represents the
longitudinal strain.
Let us find the inverse of Hooke’s law. Multiplying both
sides of eq. (2) by ÃÀ1
klij, we have
kl ¼ ÃÀ1
klijij; ð19Þ
where ÃÀ1
klij is the tensor of stiffness (compliance), repre-
sented using the same structure as that of the tensor of
elasticity:
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
¼
c11 c12 c13 0 0 0
c21 c22 c23 0 0 0
c31 c32 c33 0 0 0
0 0 0 c44 0 0
0 0 0 0 c55 0
0 0 0 0 0 c66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
: ð20Þ
To calculate the constants cij, we use data obtained
empirically by a phenomenological approach. To obtain
these experimental data, we apply uniaxial stress (defined
as a stress state for which only one component of the
principal stress is not zero) to the sample. Thus, we perform
a test using shear stress, which enables us to obtain the
relations between Young’s moduli and Poisson’s ratio, for
example, in the x direction. We then carry out the same
procedure in the y and z directions, following the same
reasoning, to determine the other Poisson coefficients, yx,
zy, and xz.
Using the stiffness tensor, the elastic behavior of an
orthotropic material is represented as follows:
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
¼
1
Ex
À
yx
Ey
À
zx
Ez
0 0 0
À
xy
Ex
1
Ey
À
zy
Ez
0 0 0
À
xz
Ex
À
yz
Ey
1
Ez
0 0 0
0 0 0
1
2Gyz
0 0
0 0 0 0
1
2Gzx
0
0 0 0 0 0
1
2Gxy
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
: ð21Þ
Considering the orthotropic elasticity tensor, given by eq. (17), for the stress-strain relationship in terms of the elastic
constants of the material we obtain the following equation given as the inverse matrix of the standard equation eq. (21)
[following15)
]:
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
¼
ðÀ1 þ yzzyÞEx
Á
À
ðyx þ yzzxÞEx
Á
À
ðyxzy þ zxÞEx
Á
0 0 0
À
ðxzzy þ xyÞEy
Á
ðÀ1 þ xzzxÞEy
Á
À
ðzy þ xyzxÞEy
Á
0 0 0
À
ðxz þ yzxyÞEz
Á
À
ðyz þ xzyxÞEz
Á
ðÀ1 þ xyyxÞEz
Á
0 0 0
0 0 0 2Gyz 0 0
0 0 0 0 2Gzx 0
0 0 0 0 0 2Gxy
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
xx
yy
zz
yz
zx
xy
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
; ð22Þ
where Á :¼ xzyxzy þ xzzx þ xyyx þ yzxyzx À 1 þ
yzzy. This is the matrix representation of the gener-
alized Hooke’s law in the Voigt notation for orthotropic
symmetry.
3.2 Strain, stress, and coordinate system
According to Lima,33)
to develop the basic equations relating
strain and stress it is necessary to write the stress in relation
to a coordinate system. The system chosen is the orthonor-
mal coordinate system coincident with the principal axes
of the sample, i.e., the deformation direction (P1 axis),
the transverse direction (P2 axis), and the normal direction
(P3 axis).
When a measurement is carried out, we use the laboratory
coordinate system, represented by the L1, L2, and L3 axes,
shown in Fig. 1.

The transformation of the coordinates between the two
reference systems is given by eq. (12).
The relationship between the strain tensors in the two
coordinate systems is given by
0
ij ¼ wikklwT
jl; ð23Þ
where ij is the strain tensor in the principal system Pi, 0
ij
is the strain tensor in the laboratory system Li, and w is
the matrix giving the coordinate transformation between Pi
and Li, obtained from
wik ¼ aijajk; ð24Þ
where
aij ¼
cos 0 À sin
0 1 0
sin 0 cos
0
B
@
1
C
A;
and
ajk ¼
cos sin 0
À sin cos 0
0 0 1
0
B
@
1
C
A:
To obtain the above two matrices we consider the
rotations of a rigid body in the xz and xy planes, respectively,
for the Euler angles and .
To carry out the phenomenological approach, we choose
one direction as the direction of the experimental measure-
ment (L3 direction) and assume that the shear stresses are
negligible. This approach gives the following relations
z0z0 ¼ cos2
sin2
xx þ sin 2 sin2
xy þ cos sin 2 xz
þ sin2
sin2
yy þ sin 2 sin yz þ cos2
zz;
x0x0 ¼ y0y0 ¼ 0;
y0z0 ¼ z0x0 ¼ x0y0 ¼ 0: ð25Þ
Taking into account the assumption of a small deforma-
tion, ij vanish for i 6¼ j. Thus, in this case, eq. (25) is
rewritten as
z0z0 ¼ xx cos2
sin2
þ yy sin2
sin2
þ zz cos2
: ð26Þ
The strain and stress tensors are always related to the main
axis in the phenomenological approach. Therefore, we will
study the states of the principal triaxial stress when there are
tensions in the x, y, and z directions and the states of the
principal biaxial stress when there are tensions in the x and y
directions.
3.3 Principal triaxial stress state
To calculate the state equation of principal triaxial stress, we
use eq. (21), taking into account the principal directions.
Then,
xx ¼
1
Ex
xx À
yx
Ey
yy À
zx
Ez
zz;
yy ¼ À
xy
Ex
xx þ
1
Ey
yy À
zy
Ez
zz; ð27Þ
zz ¼ À
xz
Ex
xx À
yz
Ey
yy þ
1
Ez
zz:
Substituting these theoretical equations in eq. (26), rep-
resenting the experimental situation, we have
z0z0 ¼ cos2
sin2 xx
Ex
À
yx
Ey
yy À
zx
Ez
zz

þ sin2
sin2

À
xy
Ex
xx þ
1
Ey
yy À
zy
Ez
zz

þ cos2
À
xz
Ex
xx À
yz
Ey
yy þ
1
Ez
zz

: ð28Þ
After appropriate manipulations, we obtain the following
result, which relates strain with stress for orthotropic
materials:
z0z0 ¼ sin2

xx
Ex
½ð1 þ xyÞ cos2
þ ðxz À xyÞŠ
þ
yy
Ey
½ð1 þ yxÞ sin2
þ ðyz À yxÞŠ
À
zz
Ez
ð1 þ zy sin2
þ zx cos2
Þ

À
xz
Ex
xx þ
yz
Ey
yy À
zz
Ez

; ð29Þ
where xy, xz, yx, yz, zy, and zx are the Poisson
coeﬃcients and Ex, Ey, and Ez are the Young’s moduli in
the x, y, and z directions, respectively.
Equation (29) is the key equation used in the
model proposed for the principal triaxial stress
state.
3.4 Principal biaxial stress state
We now set zz ¼ 0 in eq. (27) to consider superﬁcial
tension. We obtain
xx ¼
xx
Ex
À
yx
Ey
yy;
yy ¼ À
xy
Ex
xx þ
yy
Ey
; ð30Þ
zz ¼ À
xz
Ex
xx À
yz
Ey
yy:
Substituting these theoretical equations in eq. (26), rep-
resenting the experimental situation, we have
z0z0 ¼ cos2
sin2 xx
Ex
À
yx
Ey
yy

þ sin2
sin2
À
xy
Ex
xx þ
1
Ey
yy

þ cos2
À
xz
Ex
xx À
yz
Ey
yy

: ð31Þ
Fig. 1. Laboratory coordinate system (Li ) used for carrying out
measurements on the sample.33)

After manipulating this equation, we obtain:34)
z0z0 ¼ sin2

xx
Ex
½ð1 þ xyÞ cos2
þ ðxz À xyÞŠ
þ
yy
Ey
½ð1 þ yxÞ sin2
þ ðyz À yxÞŠ

À
xz
Ex
xx þ
yz
Ey
yy

; ð32Þ
where xy, xz, yx, and yz are the Poisson coefficients
and Ex and Ey are the Young’s moduli in the x
and y directions, respectively. This equation relates
the strain with the superficial stress for orthotropic
materials.
Equation (32) is the key equation used in the model
proposed for the principal biaxial stress state.
4. Analysis of Consistency of Orthotropic Equations
Equations (29) and (32) describe the behavior of mate-
rials with orthotropic anisotropy. Under isotropic conditions
they should also describe isotropic materials in accordance
with the previously reported models.10,35)
To show this
consistency we determine the states of triaxial and biaxial
stress for an isotropic material using the equations in
our model.
Furthermore, we demonstrate the consistency of
the model with previously reported experimental
results.20)
4.1 Principal triaxial stress state
In isotropic materials the Poisson coefficients, ij, and the
Young’s moduli, Ei, are equal in any direction, since the
transversal deformation in any direction and axial deforma-
tion are equal, that is,
xy ¼ xz ¼ yx ¼ yz ¼ zy ¼ zx ¼ ;
Ex ¼ Ey ¼ Ez ¼ E:
Substituting these into eq. (29), after some algebraic
manipulation, we obtain
z0z0 ¼
1 þ
E

sin2
ðxx cos2
þ yy sin2
À zzÞ
À

E
ðxx þ yy þ zzÞ þ
1 þ
E

zz; ð33Þ
which is in agreement with the result for isotropic
materials35)
in the principal triaxial stress state.
4.2 Principal biaxial stress state
Similarly, we have
xy ¼ xz ¼ yx ¼ yz ¼ ;
Ex ¼ Ey ¼ E:
Substituting these in eq. (32), after some algebraic
manipulation we obtain
z0z0 ¼
1 þ
E

sin2
ðxx cos2
þ yy sin2
Þ
À

E
ðxx þ yyÞ; ð34Þ
which is in agreement with the result for isotropic
materials10)
in the principal biaxial stress state.
4.3 Consistency with experimental results for the biaxial
stress state
It is necessary to measure the elastic constants in order to
verify the predictions of the model. The simplest way to
measure the constants is to apply a uniaxial stress to a
sample of the material and then to measure the strain
evolution z0z0 for different values of applied stress.
Generally, we apply a tensile or compressive stress using a
four-point bending device.12)
For this, we consider the work of Faurie et al., which
establishes the elastic constants of a fiber-textured thin gold
film with transverse isotropic symmetry by combining
synchrotron X-ray diffraction and in situ tensile testing.20)
In that work only the stresses xx and yy were considered,
thus, in this work let us consider, using our model, the case
of a principal biaxial stress state.19)
Observe that a material with transversely isotropic
anisotropy has five constants30)
that characterize its elastic
tensor. Assuming that the plane of symmetry is xy, we
have the characteristic constants: Ex, Ez, Gxy, xy, and xz.
We emphasize that the use of biaxial stress does not allow
the determination of the elastic constant Ez. Considering the
assumption of small shear stress, the elastic constant Gxy is
negligible.19)
Thus, to test the experimental coherence of our theo-
retical model, we use the experimental data of this study to
evaluate the predictions arising from our model in compar-
ison with those in Faurie et al.’s work.20)
To this end, we use
the inputs of xx and yy obtained by Faurie et al.20)
given in
Table II of their work.
It is important to note that in the case of a material with
texture, or an anisotropic material, it is necessary to obtain
the X-ray pole figure20)
for each plane family in order to
determine the angles that can be used in the equation in
the model.
Studying the experimental results in their article, we will
compare then with our proposal in order to observe the
experimental consistency.
To this end, from the work of Faurie et al.,20)
we consider
four experimental graphs (Figs. 7, 8, 10, and 11 in their
study). We then compare the equation extracted from each
graph with the theoretical equation arising from our model
to obtain the elastic constants, which are compared with
those of Faurie et al. In fact, we only required two figures
for this determination, as discussed below.
Let us start with Fig. 7 of Faurie et al., which shows
the strain evolution of as a function of the load F applied
to the film/substrate composite for each family of planes.
As shown in this graph, for each constant angle and
¼ 0, previously obtained from the pole figures, a load
that generated tensions xx and yy was applied on the
thin film.
Let us fit the equation of the principal biaxial stress state,
eq. (32), in our model to the experimental data in Fig. 7
of ref. 20.
0; ¼ ½sin2
þ ðsin2
À 1ÞxzŠ
xx
Ex
þ ½ðsin2
À 1Þyz À sin2
yxŠ
yy
Ey
: ð35Þ
Considering the transversely isotropic anisotropy of the
thin film under study, we have the conditions15)

xz ¼ yz ; Ex ¼ Ey;
xy ¼ yx ; Ez ¼ undetermined for a biaxial state; ð36Þ
Gxy; negligible by assumption;
where we have taken into account the use of the biaxial
stress state.
Substituting these conditions in eq. (35) we have
0; ¼ ½sin2
þ ðsin2
À 1ÞxzŠ
xx
E
þ ½ðsin2
À 1Þxz À sin2
xyŠ
yy
E
; ð37Þ
which is the equation in our model considering the data in
Fig. 7 of Faurie et al.’s work.
We now consider Fig. 8 in ref. 20, for the case of ¼ 0
eq. (32) is then written as
0; ¼ sin2 xx
Ex
ð1 þ xzÞ þ
yy
Ey
ðyz À yxÞ

À xx
xz
Ex
þ yy
yz
Ey

: ð38Þ
Using the conditions of transversely isotropic anisotropy,
eq. (36), we have
0; ¼ sin2 xx
E
ð1 þ xzÞ þ
yy
E
ðxz À xyÞ

À
xz
E
xx þ
xz
E
yy

; ð39Þ
which is the equation in our model considering the data in
Now consider Fig. 10 and Table II in the work of Faurie
et al. shown in Fig. 2 below.
This graph represents the experimental loads applied to
the sample with stresses in the xx and yy directions. For the
values of ¼ 0 and 90, the strain average according to the
theoretical equation in our model is written as:
0; þ 90;
2
¼ sin2

xx
2Ex
ð1 þ 2xz À xyÞ
þ
yy
2Ey
ð1 þ 2yz À yxÞ

À xx
xz
Ex
þ yy
yz
Ey

: ð40Þ
For transversely isotropic anisotropy, this equation is
written as
0; þ 90;
2
¼ sin2 ð1 þ 2xz À xyÞ
2E
ðxx þ yyÞ

À xx
xz
E
þ yy
xz
E

: ð41Þ
This is the equation in our model considering the data in
To study the graph in Fig. 11 in ref. 20, we deﬁne as the
magnitude of the slope of the graph ð0; þ 90; Þ=2 as a
function of xx þ yy. For the experimental data in ref. 20,
we obtain the following equation, from our model:
P ¼
1 þ 2xz À xy
2E

ðxx þ yyÞ: ð42Þ
To determine the elastic constants in ref. 20, we consider
the curves in Figs. 8 and 10.
Let us start with the curve in Fig. 8 of ref. 20. Using
the curve for the stress T5, by performing linear regression
[see Fig. 3 below] we obtain the following experimental
equation:
0; ¼ 2:27 Â 10À3
sin2
À 7:22 Â 10À4
: ð43Þ
The correlation coeﬃcient is R ¼ 0:954.
Comparing this experimental equation with eq. (39), after
algebraic manipulation we arrive at the following relation-
ship:
3:648 À xy ¼ 9:967xz: ð44Þ
Next, using the graph of T5 load in Fig. 10 in ref. 20 and
applying the least-squares method, we obtain
0; þ 90;
2
¼ 1:29 Â 10À3
sin2
À 7:49 Â 10À4
: ð45Þ
Comparing eq. (41) with eq. (45) we obtain
1 À xy ¼ 1:446xz: ð46Þ
Then, solving eqs. (44) and (46), we obtain
xz ¼ 0:311; ð47Þ
xy ¼ 0:551; ð48Þ
E ¼ 74:1 Â 109
Pa: ð49Þ
Fig. 2. ½ð0; þ 90; Þ=2ŠX sin2
graphic. Fig. 3. Graph of strain versus sin2
with correlation factor R.

In Table I we compare the values obtained by our model
with those obtained by Faurie et al. (given in Table III
in ref. 20).
We observe that the elastic constants obtained using our
model are consistent in terms of magnitude with the
coefficients obtained by Faurie et al. In addition, we
obtained a Poisson coefficient, xz ¼ 0:311, that does not
appear in the results of Faurie et al.
Furthermore, the results obtained for the elastic constants
using our model, are in agreement with eqs. (4), (7), and
(10) in the work of Lempriere15)
Ex; Ey 0; ð50Þ
ð1 À xyyxÞ 0; ð51Þ
jxyj
Ey
Ex
1=2
; ð52Þ
jyxj
Ex
Ey
1=2
; ð53Þ
À1 xy 1: ð54Þ
These results demonstrate the consistency of our model of
the biaxial stress state with experimental results.
5. Conclusions
In this work, we developed a model to study materials that
have orthotropic symmetry in their physical properties. The
proposed model was proven to be a generalization of the
model for isotropic materials.
It can be used for the calculation of stresses in biaxial and
triaxial polycrystalline materials with texture or anisotropy
using the sin2
technique with X-ray and neutron diffrac-
tions. It is first essential to study the texture of the material
through the pole figures in order to obtain the angles
that can be used in the model equation. Next, measure-
ments of the deformations for the possible and for
various crystallographic planes are performed in order
to obtain the deformations (L3
; ). Thus, this procedure
is different from the previous methodology used to
measure (hkl
; ).
To study the residual stress it is necessary to obtain the
elastic constants of the material, since the sin2
technique is
used. To characterize the material in relation to the residual
stress we can use the equations derived in this study. To
this end, it is important to make a series of independent
in situ measurements of the applied tensions in the main
directions, i.e., xx and yy, for the angles ¼ 0, ¼ 90,
and ¼ 0.
Acknowledgment
We would like to thank CNPq Grant 504578/2004-9 and
LNLS—Laborato´rio Nacional de Luz Sıńctron—XRD1
beam line. The authors also thank Dra. Ludmila Oliveira
H. Cavalcante (DEDU-UEFS) and the editors for the
suggestions that made the text more clear and readable.
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Table I. Comparison of the results of Faurie et al.’s work20)
with those obtained using our model.
Model Young’s modulus, E (GPa) Poisson’s ratio, Shear modulus
Faurie et al. E ¼ 75:7 Ez undetermined ¼ 0:517 — Gxy negligible
Ours E ¼ 74:1 Ez undetermined xy ¼ 0:551 xz ¼ 0:311 Gxy negligible

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