Residual
stress determination by least square fitting method
Z. Pala1, A. Sveshnikov2, N. Ganev1, K.
Kolaøík1
1Department of Solid State
Engineering, Faculty of Nuclear Sciences and Physical Engineering, Czech
Technical University in Prague, Trojanova 13, 120 00
Prague 2
2Department of Physics, Faculty of
Civil Engineering, Czech Technical University in Prague, Thákurova
7, 166 29 Prague 6
zdenek.pala@fjfi.cvut.cz
Results of
macroscopic residual stresses (RS) obtained from the X-ray diffraction tensometry are increasingly used for quality characterization
of various surfaces and thin layers. In some cases even the depth distributions
of RS are coveted as indispensable information about the processes of
inhomogeneous plastic and thermal deformations which lead to the final shape of
the studied object. Since the knowledge about the state of RS is highly
desirable, the reliability and the accuracy of the macroscopic residual stress
tensor components is an important topic in powder diffraction. The first concern,
i.e. the results’ reliability, is
above all else given by the correspondence between the state of the measured
object and the choice of experimental techniques and evaluation methods. The
term “state of the measured object” encompasses not only the state of RS, i.e.
biaxial or triaxial, but the object’s real structure;
particularly stress gradients, preferred orientation and grain size. Another
vital feature of the stress results’ reliability lies in the selection of the
appropriate elastic constants as they directly link measured deformations with
the stresses requested by the results’ users. It is palpable that elastic
constants are influenced by isotropy of the irradiated volume, mostly by the
presence of texture and the history of plastic deformation. Calculation of
elastic constants is a frequently tackled issue and the simple initial models
of Voigt and Reuss [1] have been substantially
modified over the last half century, the most advanced calculations consider
interactions between crystallite and its inhomogeneous
neighbourhood [2] and even presence of texture [3].
The
benchmarks of the suitability of standard methods in RS determination, such as
sin2ψ, Dölle-Hauk and one tilt, are
deviations from linearity of the so-called dφψ versus
sin2ψ plots. These plots depict dependence of interplanar lattice spacing of the chosen set of planes {hkl} in respect to the orientation of the diffraction
vector to the surface normal given by the tilt ψ and chosen azimuth
φ:
(1)
where d0 stands for strain-free
material, ½s2 and s1 are X-ray elastic
constants (XEC), the superscripts L
and S stand for laboratory and sample
coordinate system respectively. If d0
is known the desired six components of macroscopic stress tensor
σ11, σ22, σ33, σ13,
σ12, σ23 can be calculated from the measured εφψ in six independent directions
given by tilt ψ and azimuth φ.
The results’ accuracy can be substantially
increased by measuring εφψ
in more than six directions and by application of standard least-square fitting
procedure. Detailed algorithm can be found in [4] and will be explained at
length in the oral contribution. However, this is path is seldom followed and Dölle-Hauk method is used in vast majority of full stress
tensor determination. Following Dölle-Hauk method, dφψ
versus sin2ψ plots are obtained for three azimuths. The
selection of these azimuths itself is a topic for detailed analysis and recent
papers, see e.g. [5], tend to prefer azimuths 0°, 60°, and 120° instead of
commonly chosen 0°, 45°, and 90°. The fundamentals for this switch in measurement
stereotype lies in the putatively smaller results’ inaccuracies for the choice
of the 0°, 60°, 120° combination.
It has been
shown [6] that presence of pronounced texture and coarse grain in the measured
polycrystalline material represent an impediment in RS determination by means
of standard methods. Residual stress analysis in textured material is an
especially arduous task, since its manifestations are oscillations in the dφψ
versus sin2ψ plots. Moreover, the texture is responsible for
different elastic properties and hence the XEC are not the same as in
non-textured materials. Residual stress determination by means of least-square
fitting method is capable of taking texture into account if anisotropic XEC Rijhkl and X-ray stress factors Fij
are implemented into its algorithm [7, 8]. The equation (1) is then rewritten
as
. (2)
In order to
assess and compare standard Dölle-Hauk method and
residual stress determination by least-square fitting a ground sample of high
carbon stainless steel M300 was investigated by XRD which was known to exhibit
the so-called ψ splitting and texture was virtually nonexistent [9]. For
this purpose {211} diffractions of α-Fe were investigated with CrKα radiation (diffraction angle 2θ ≈
156°) and X’Pert PRO MPD Bragg-Brentano diffractometer
in ω-mode equipped with a proportional detector. The goniometer
was adjusted in reference to a strain-free reference specimen of α-Fe
powder. The differential ψ–method, when the azimuth is kept constant and
the tilt is changing, was employed. Measurements were taken on the grinding
direction φ = 0° and in further four azimuths defined by 45°, 60°, 90°,
and 120°. For each azimuth seventeen tilts defined by sin2ψ = 0;
0.1; 0.2; ...0.8 were measured.
The aim was
to obtain diffraction profiles of good and poor background to Imax ratio and subsequently to characterize the profiles
measured in direction φ, ψ by their centre of gravity. These data
were subjected to residual stress calculation by Dölle-Hauk
method and by least-square fitting using XEC calculated according to Eschelby-Kröner model. These procedures were imposed on two
sets of data; the first consisted from azimuths 0°, 45°, and 90° and the other
from 0°, 60°, and 120°. Even though the characterization of diffraction dublet by centre of gravity may seem obsolete in comparison
with profile fitting methods, the [10] clearly shows that such treatment is
suitable. Nevertheless, and only for comparison, the profiles were also fitted,
after CrKα2 stripping, by Pearson VII function and its maximum
then entered the residual stress calculations. The comparison of results’
accuracy is performed on the basis of standard deviations.
References
1. I.C. Noyan, J.B. Cohen, Residual Stress: Measurement by Diffraction and Interpretation. New York: Springer. 1987.
2. H. Dölle, J. Appl. Cryst.,12, (1979), 489-501.
3. P.-O. Renault, E. Le Bourhis, P. Villain, Ph. Goudeau, K. F. Badawi, D. Faurie, Appl. Phys. Lett., 83, (2003), 473.
4. R.A. Winholtz, J.B. Cohen, Aust. J. Phys. , 41, (1988), 189-199.
5. B. Ortner, Int. J. Mat. Res., 98, (2007), 87-90.
6. V. Hauk, Structural and Residual Stress Analysis by Nondestructive Methods. Elsevier. 1997.
7. B. Ortner, J. Appl. Cryst., 39, (2006), 401-409.
8. B. Ortner, Z. Metallkd., 96, (2005), 9.
9. Z. Pala, N. Ganev, Mat. Sci. Eng. A, 497, (2008), 200-205.
10. H. Zantopulos, C. Jatczak, Adv. X-ray Anal., 14, (1971), 360 – 376.
Acknowledgements.
The research was supported by the Project No 106/07/0805 of the Czech Science Foundation and by the Project MSM 6840770021 of the Ministry of Education, Youth and Sports of the Czech Republic.