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PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO-STRESS
DETERMINATION RELATED TO SURFACE LAYER GRADIENTS AND
ANISOTROPY
S.J. Skrzypek I, A. Baczmaiiski 2 ‘Faculty of A4etallurgy and
Materials Science “Faculty of Physics and Nuclear Techniques
University of Mining andMetallurgy, Al. Mickiewicza 30, 30-059
Krakbw - Poland
Abstract ‘The well known sin2p method for macro-stress
measurement elaborated for Bragg-
Brentano geometry has some disadvantage e.g. penetration depth
of X-ray beam varies during m.easurement. This is particularly
important in the case of coatings or surface layers where residual
stresses appear with large gradients, The new version of the sin’v
method, named g- sin’w (based on the grazing angle scattering
geometry) was applied for determination of the macro-residual
stresses in TiN coatings deposited on sintered WC and on sintered
high speed steel. These types of samples present a wide range of
residual stresses i.e. from large tensile to large compressive.
Using the g-sin2w method the stresses are determined for chosen
near surface layers for which effective penetration depth remains
almost constant during measurement. Anisotropic elastic constants
were used for calculations. The measured results were compared with
a model based on thermal shrinkage.
Introduction The most universal sir& diffraction method
enables measurement of the macro-stress
tensors and the elastic properties of polycrystalline materials.
If a known stress can be applied during measurement, diffraction
elastic constants (DEC) and elastic constants or mechanical
compliances can be measured [l-4 J.
The diffraction methods have several important features such as
a non-destructive and non-reference character, possibility of
stress analysis for multiphase and anisotropic materials. The
conventional well-known sin’ p method has some disadvantage i.e.
penetration depth of X- ray rad.iation varies during experiment
when the w angle is changing. This is particularly important for
thin coatings, films or surface layers where the residual stresses
have large gradients which can reach several hundreds MPa/p.m. Till
now the gradients of residual macro- stresses in surface layers
were measured using different wavelengths [5,6] or by gradual
removing of surface layers and following measurements [7].
In this work the g-sin’ly method based on the grazing incidence
angle X-ray diffraction (MD) geometry is applied to macro-residual
stresses (RS) measurement in TiN coating. Van Acker et al. [8] and
Quaeyhaegen and Knuyt [9] have introduced the first approaches to
this geometry in term of residual macro-stress measurement. Using
this method the non-destructive analysis of the residual stresses
for different penetration depth bellow the sample surface can be
performed. The problems concerning penetration depth, diffraction
intensity distribution,
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2001,Advances in X-ray Analysis,Vol.44
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This document was presented at the Denver X-ray Conference (DXC)
on Applications of X-ray Analysis. Sponsored by the International
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critical angle of total reflection for GD geometry were also
discussed by Huang [lo] and Goehner [ 111.
In the presented geometry the penetration depth can be easily
chosen by incidence angle (o: in Figs 1 and 2) and it is almost
constant during measurement for wide range of \I, and 8 angles
(Eq.3 and 4). This method was used for determination of the RS in
TiN coatings deposited on sintered WC and on high-speed steel
(HSS). These kinds of samples presented a wide range of residual
stresses i.e. from large tensile to large compressive.
Diffraction methods of lattice strain measurement
The conventional diffraction sin2t+o method for the
determination of residual stresses is based on measurement of the
interplanar spacing for various directions of the scattering vector
characterised by the q and I,V angles (Fig. 1). In the diffraction
experiment, the mean interplanar spacing fikl) averaged only for
the reflecting grains having the scattering vector normal to the
Qzkl) crystallographic planes is measured. Moreover, in the case of
surface measurement (by X-ray radiation) the absorption should be
taken into account in calculations.
The average value of the lattice strain fikl, in the Lj
direction (Fig. 1) is defined a.s:
< d(+7v)>(hkl) - d(h”,) < w4vb(hkl) = __ (1)
d”
where: &@kI) is the interplanar spacing for (hkZ] planes in
the stress-free material and the (h~~~ average is defined for
grains volume as in Eq.2.
E’ig.1. Geometry of g-sin2 vmethod. The (hkI) spacings
aremeasured along LS axis in I-, system and the stresses d, are
defined with respect to the S-sample system. The incident angle a
is fixed during measurement. The orientation of s8cattering vector
is characterised by the np and w angles. For the u.nit incident
(KO) and diffracted (Qvectors the scattering vector is equal to
[12]:
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2001,Advances in X-ray Analysis,Vol.44
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In good approximation, the measured (J&l) is averaged
according to:
(2)
0
where; the < . . >(hkIj average is calculated over all
volume of the reflecting grains in the beam path from surface to
the depth t for which G, = 95% of radiation is absorbed (see Eq.4
and CMlity ]:12]), ,u is the linear coefficient of absorption and
Z(X) is the function of path length vs. depth (x) and the a and 0
angles are defined in Fig. 1,
The conventional method for stress determination, called sin’ry
has been elaborated for symmetrical Bragg-Brentano geometry and can
be applied to “w” and “Y;’ goniometers. In both cases tbe
orientation of the scattering vector varies, however, the planes
indexes (hkl) are kept constant during dm measurement. It can be
shown, that different penetration depths of measurement versus the
tilt angle w appear for these geometries [ 13,141 and causes
problems in stress measurement in the case of stress gradients,
non-uniform microstructure or texture in the near surface volume.
The importance of stress gradients in surface layers was recognised
by Perry [6] and Kra.uz and Ganev [5]. This problem can be solved
using the new geometry i.e. a new version of the sin’vmethod based
on grazing angle scattering geometry.
The new method, called g-sin2 vI/ [ 13,14 J is characterised by
small and constant incident angle (cr- in Fig. 1 and 2) and by
different lengths and orientations of the scattering vectors (Fig.
1). In contrast to the conventional sin” y method, the measurements
are performed for different fhkl) planes usiig appropriate values
of 8 fikl) angles. The interplanar spacing
-
Using some simplifications, the depth (t) for which fraction G,
(fraction of total intensity of the X-ray beam) is absorbed is
called effective depth of penetration (EDP) which depends on the
difI?action geometry and can be calculated from formulas:
a) for the sin”ty and psin’vmethods : t = - Zn(~ - Gx) cos( y)
sin( t9)
2P ,
b) for the cvsin’ty and p/c+sir?vmethods : t = - ln(l - Gx) I I
’
’ sin(Q+ y) + sin(O- y) I
(4)
c) for the g-sin2ry method : t= - ln(l - Gx)
i
I I P- sin a ’ sin(2 y f a) I
where, for every method the appropriate relations between angles
0, w and a were considered [ 14,151. The difference between
penetration depths vs. sin’ty for different geometries are
presented in Fig2 The main advantage of the g-sin’w method is
almost constant penetration depth for fixed small angle a (a = 1 -
10 degrees) and large range of I+V angle (Eq.3). Moreover, the
required penetration depth of X-ray radiation can be easily chosen
by changing the a angle and the stresses can be determined for
different thickness of layers under the sample surface (Eq.4c and
Fig.2).
Fig.2. The penetration depth t vs. sin2ry 20 .
calculated from Eq.4 for G,=O. 95 and for 18- --.:;::- 16-
.-.* different geometries. Absorption of Cu
*. -* 5 . sin’ y 14-
*. --* Kcli rladiation in TiN material (J = 561
*‘. *.
cm-‘) ‘was considered. In the case of the -y2- o-sin UC’\. -*..
* \ . . \
g-sin’ v, method calculations were SIO- *. .
- 8- \ .
. . . performed for two different a angles i.e.
. a=6 \
1 and 6 degrees. 6- \
4- g-sin’y ‘, 2- a=1
_-----w--w---- 0' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
sin’ w
The g-sin’p method is based on the non-standard geometry of
measurement for which the s;:cl(q, v/)>~kl, interplanar spacing
in directions defined by the q and I,V angles are measured for
different hkl reflections, The crucial point of the work is correct
interpretation of the eixperirnental data using properly calculated
(or measured) elastic difYraction constants and layer thickness.
The influence of crystal anisotropy on interpretation of the stress
measurement will be considered here.
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Determination of residual stresses from diffraction measurement
For a general stress state and for an quasi-isotropic sample, the
average lattice strain in the L3
direction is equal to [l-3];
-C &‘(qo, w / >(hkJ) = SI (hkl)(Lrfl + C& + & + 2
s.2 (hkl)(oi, cm2 p + O$~ sin’ v, f ~7:~ sin 44 sin’ w 1
I I (5) + 2 s.2 (hkUc& cos’ y + - s2 (hkl)( oi3 cos 9 +
&sin cp) sin 2ry
2 where: sl(hkl) and s.$hkt) are the diffraction elastic
constants for a quasi-isotropic
polycrystal and d, macrostresses are defined with respect to the
S system (Fig. 1.). Another definition of the DECs may be
introduced for a textured sample [Z, 18,191;
< E’(P~ YV) +a/+) = Fij OL, r P,I v’) CT; Pa)
The FV coefficients are not the tensor components because they
relate the stresses dV expressed in S system with the strains
cE$3>@kl, defined along & axes of the L-frame. Obviously,
using the appropriate transformation, the FV elastic constants can
be easily calculated from the R, ones, i.e. :
Fij (atkl, 4, vv, f(i9~ =R,, (hkl, 9, ~9 f@rO~ mi Y nj ( W where
the y matrix transforms stresses from the S to the L system (Fig.
l), i.e, :
o- 1 I= y mi y nj cz i. and R,, are DECs for textured
polycrystal defined in the following section. IEn
For the biaxial stress state in a qasi-isotropic sample the well
known formula for the measured interplanar spacing can be
obtained:
< dtp, y) >(hkll = {sl @W(a;l f & + ; s2 (hW& cm2
P + dz2 sin2 v + (6)
t& sin 2~) sin2 I// P ihlli+d ,,,lli The s&kJl and s$hkQ
elastic constants depend on the single crystal constants,
grain-matrix
interaction and hkl reflection. In the case of textured material
the FiJ depends also on orientation distribution. Moreover, for
biaxial stress state and for isotropic material, Eq.6 proves the
linear character of the
-
< a(qj v) +ikkg = {$I (hkl)(a:l+ o&J + i s2 (hkl)(cri,
cm2 p + dJ2 sin2 q +
0i2 sin 293) sin’ I+v )a *+a *
where for cubic structure :
(7)
and a”=d ’ h2+k2+12 (hkl) d In the above equation the
recalculated lattice constants a* and c~, are used
instead of interplanar spacings $ fikl) and okl in the fitting
procedure. Of course the experimental flk!, depends on the used
reflection and orientation of the scattering vector because it is
directly recalculated from (hkg using appropriate crystallographic
relation (e.g. for cubic structure Eq.7). The same type of average,
i.e. through volume of difkacting crystallites, is used for the
recalculated lattice constants and the measured interplanar
spacings.
It should be stated that in the presented method the s&ikl)
and s$ikC) constants used for one ~(9, +~w vs. sin’ly graph depend
on the hkZ reflection. Moreover, in the case of textured material,
the RG constants which depends on the orientation distribution
function must be used instead of s#ikl) and s$%kq. Consequently,
the diffraction elastic constants depend on the I,V angle and ekl,
vs. sin’ly graphs are not linear.
For calculated diffraction elastic constants s@zkJ), s.$hkJl or
FY , the experimental lattice constant,s
-
where all the above quantities are defined with respect to the
L-&me. Finally, in the case of Voigt model the d&-action
elastic constants do not depend on the hkl
reflection for an quasi-isotropic sample and they are equal
to:
For cubic crystals the above elastic constants are calculated
and they are usually expressed by components of single crystal
compliance tensor sijk [3, 211:
‘5 = soh11 +2~,,22)+~osl,22s1212 and s = 2w212 (s1111
-s1122)
3s Ill1 - 3k22 -I- 4SIZI2 2 3 Sllll - 3JII22 + 4SIZl2
where: SO = siril- ~112~ - 2 ~1212 In the case of textured
material more general DEC i.e. Rij’ are defined (see Eq.5a,b)
[2].
The calculation of [cklijr] is based on the texture mnction of
all crystallites from the irradiated volume which contribute here
to the average:
Rg = [c’j’~~ where [Cfqkl/ = ~C$H Ci$)f(g)dg (12) E
In this equation the c+‘(g) single crystal stifI?nesses
(expressed in the L system) are integrated over the whole
orientation space E and f(91 is the orientation distribution
function characterising texture.
Reuss model [22] In this approach we assume that the local
gQ-stress is homogenous across the sample, i.e.,
u;l=d,. Using the sYkl ’ compliance tensor for single crystal,
we can write the following equations in the L system:
-5+33 = s’ 338 y (-J.. = s)33. (p 31 cl (13)
< &‘33>(hkl) = < sl33ij>(hklj ok’ where; the
average is calculated over all diffracting grains having scattering
vector
normal to the fikg plane i.e.:
where: 5 is the angle of rotation around scattering vector
perpendicular to the fhkl) planes.
The diffraction elastic constants depend on the hkl reflection
used. For the cubic slymmetry and isotropic sample they can be
expressed by:
SI = -‘WI >(hkl) and S2 = 2 (
-
where: I&= @?f+h2~+kzf)/(h2+~+?)2 and ~0 is given by Eq.
11.
Consequently, using the Reuss model the DECs in textured
material may be expressed as:
(17)
where: f(g) is given by Eq. 12 and r is the angle of rotation
around the scattering vector perpendicular to the @k() planes.
Experimental results The g-sin’w geometry and from five to eight
diffraction lines (i.e.: ( Ill>, (2001, (2201,
(3111, (2221, (400) and (4203) were used for determination of
the residual stresses in TiN coatings deposited on sintered WC
carbide (sample no.V30) and on sintered high speed steel (HSS,
sample no.T31). The coatings were produced by the CVD method in the
case of V30 sample and by PVD for the other one [23,25]. Cu Kar
radiation was used with a Philips di@ractometer (X-Pert MPD) and
CoKa with a Bruker (D8 Advance) diffractometer.
The diffraction patterns were recorded for cp = 0 and cp = 90.
For verification the additional measurements for cp = 180 degrees
were performed using the Bruker (D8 Advance) diffractometer and the
splitting of the curves were not observed. It proves that the non-
linearities in the sin2v are not caused by the shear stresses
(i.e., o13 = 023 = 0).
The profiles of dieaction peaks were corrected for absorption
and Lorentz-polarisation fa.ctors [3,12]. The asymmetric Q
diEaction geometry for grazing incidence angles larger than the
critical angle of total reflection (0.28 degree) was applied in
experiments. A grain size of TiN was small enough, -1-2 pm [25].
Therefore, according to Hart et aZ.[24], correction due to
refraction can be neglected.
The diffraction elastic constants (sI and s2 ) were calculated
from single crystal elastic constants (S~III= 2.17 lo3 GPa-‘,
s1z22= - 0.38 10” GPa-‘, s12j2= 1.49 10” GPa-’ [3]) using the Voigt
and Reuss approaches.
The macro residual stresses determined for different penetration
depths are presented in the Table I and 2. The experimental lattice
parameter (a(p, v)>ek~) (recalculated from measured
-
However, both approaches give approximately similar values of
the macrostresses (see Table 1 and 2). It should be stated that the
linear regression can be used only for the Voigt method because the
diffraction elastic constants (.sI and s2 or R,) do not depend on
the hkl reflection.
Talb.1. Residual biaxial stresses (&I, = d& in TiN
coatings deposited on WC carbide substrate (V30). The g-sin2v
method was used and the penetration depth was calculated using Eq.4
for C&=0.95 and ,u = 561 cm-’ for Cu Kai . The thickness of the
coating was 5 pm.
Grazing angle (a)
Wgl
1
3
6
Stress Stress Average Penetration (oL=dzz) (dll’J22) stress
depth
Wal NW (&=d,z) (9 Reuss method Voigt method NW u-4
15s2+37 1550136 1566 0.9
1242539 1194&38 1218 2.6
1107&43 1045&41 1076 4.8-5.2
The tensile biaxial stress state in the coating of the V30
sample arises due to sample cooling from the high temperature (1173
K) of the CVD process to room conditions (AT=900 K). This stress is
caused by the difference of thermal expansion coefficients for TiN
(CX~ = 9.35 x 10d) and WC carbide (a~=6 x lo&). The thermal
origin residual stress (&II = d22 = 1760 MPa) calculated by a
simple model is larger then the maximum value obtained on the upper
layer of the coating [ 131. By the same calculation for the PVD
process for TiN on HSS steel (CX~S =17x10”) for AT=500 K the
thermal origin residual stress was; dll = dam = -2251 MPa.
R.esults in Table 1 and 2 show a gradient of residual stresses
in first case caused by a relaxation process in the intermediate
layers between TiN and the substrate. For TiN deposited on HSS
steel by the PVD method little gradient was found. The measured
macro-residual stresses are quite close to the calculated thermal
case. The results from Tab.2 obtained with the D%Advance
diffractometer are additionally presented in Fig.3.
The biggest deviation of the measured points from calculated
ones (from linearity in case of the Voigt model and from
approximated points for the Reuss model) appeared for the lowest
Bragg angle diffraction lines i.e. for ( 1 1 1 } and (200). It is
their nature, but could be caused additionally by stacking faults
which are common for this type of crystal lattice which result in
shifts in the 0 angles.
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4.255- I {111)
4.220-c I 4.220 ! 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
4.260
4.255
4.250
4.245 G- ; 4.240
4.235
4.230
4.225
sin’ v sin’ \v
Fig.3. The (,,~o lattice parameters (refers to Tab.2, sample T3
1) calculated using Eq.8 are fitted to the experimental points for
different grazing incident angles : a) a = I" (t = 0.6 pm), b) a =
6” (t = 3.3 urn). For better visualisation the calculated p~,~
values are connected using continues line for Reuss method and
dashed line for Voigt method. The measurements were curried out for
cp = 0 (e) and 180 degrees (+)
Tab.2, Residual biaxial stresses (d~l = d22> in TiN coatings
deposited on high-speed steel su.bstrate (sample T3 1). The g-sin2
y method was used and the penetration depth was calculated using
Eq.4 for G,=O.95 and p = 561 cm-’ for Cu Kar and p = 837 cm-’ for
CoKa. The thickness of the coatings was 5.1 urn.
Grazing
i
Philips (X-Pert MPD) Bruker (D8 Advance) angle ,,d’ Reuss
method, oll = cr22 [MPa] Reuss method oll = oz2 [MPa]
kl (hCUKa1) @COKal) depth ,,t” textured q-isotropic textured
q-isotropic depth ,,t”
b-4 +-120 MPa +-120 MPa +-130 MPa +-130 MPa [Pm1 1 0.9 -3532
-3562 -3177 -3074 0.6
2 1.8 -3482 -3556
3 2.6 -3575 -3607 -3207 -3207 1.75 (T)
4 3.3-3.5 -3572 -3652
6 4.8-5.2 -3662 -3737 -3588 -3496 3.2-3.4(T)
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2001,Advances in X-ray Analysis,Vol.44
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Conclusions A significant gradient of the residual stresses in a
TiN coating deposited on WC carbide
(Table 2) was found. The decrease of the measured stress versus
sample depth probably results from relaxation processes in the
interface layers between TiN and the substrate. The effect of
stress heterogeneity was observed using the g-sir? y method based
on the grazing angle scattering geometry. The conventional sin’w
and m-sin2p methods are not valid for the study of stress
gradients,
For the g-sin” v geometry the penetration depth of the X-ray
beam is much smaller than in the conventional sin2v methods (Tab. 1
and 2, Fig.2). Using the new geometry the stresses are: determined
for a chosen volume below the surface which can be easily changed
by choice of incident beam angle (Fig.2). The advantage of this
geometry is a constant penetration depth during the experiment. The
methodology of experimental data treatment presented, enables real
non-reference and non-destructive stress measurements to be made,
on the grazing angle scattering geometry.
The correct diffraction elastic constants should be used for
proper interpretation of the experimental data. These constants
have to be calculated for different hkl reflections and various
sample orientations. It is readily seen in the Fig.3 that the
crystal anisotropy creates the rmnlinearities on the strain vs.
sin’w plot. From a theoretical point of view these nonlinearities
can be easily modeled using the Reuss approach.
The results provided in Tab.2, which were obtained in two
laboratories with two different diffractometers and wavelengths are
satisfactorily close to each other.
Acknowledgements; This work has been partially sponsored by
European Commission under the PECO Program
(grant CIPA 92-3032) and by the University of Mining &
Metallurgy fkom Cracow - Grant No. Il. 11.110.230. Authors
acknowledge MSc. K. Chrusciel for his contribution in
calculations.
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Copyright(c)JCPDS-International Centre for Diffraction Data
2001,Advances in X-ray Analysis,Vol.44
12Copyright(c)JCPDS-International Centre for Diffraction Data
2001,Advances in X-ray Analysis,Vol.44 145ISSN 1097-0002