X-RAY STRESS ANALYSIS BY USE OF AN AREA DETECTOR

The most commonly known techniques of X-ray residual stress analysis use diffractometers with conventional counters or position sensitive detectors. The sin2-method is often applied to X-ray stress determination, but requires various measurements at different sample positions. This technique was combined with an imaging plate, which is an X-ray digital area detector. The use of a two-dimensional detector like an imaging plate offers the possibility to get diffraction in one measurement from lattice planes lying on the whole diffraction cone. Thus the number of measurements for the calculation of the stress components can be significantly reduced. In addition, the use of an imaging plate has several advantages for the stress calculation even for materials having relatively large grains or a strong texture. Measurements on an electron beam hardened steel sample are described as an application example.


INTRODUCTION
The X-ray diffraction method for stress measurements has been successfully carded out for many years (Hauk and Macheraueh, 1984, for instance). The basis for the determination of residual and load stresses is the Debye-Scherrer-rnethod and Bragg's law. The Bragg angles of one or several reflections are measured with high precision in the back reflection region. The results permit the determination of changes in interplanar spacings of polyerystalline materials.
Conventional stress analysis is based on two commonly used techniques, the so called "sin2'and "qf'-methods (Macheraueh and MOiler, 1961;Kiimpfe and Michel, 1986), which require measurements at different tilting and rotating positions of the sample with the help of aor -goniometer (Hauk and Macheraueh, 1984;; a and b). In these eases, the interplanar spacings and consequently the shift of the peak can be determined with conventional counters or position sensitive detectors.
From the whole dffraction cone only a small part of lattice planes {hkl with a special orientation (W, p) is analysed. The use of a two-dimensional detector like an imaging plate (STOE & CIE GmbH, 1992) offers the possibility to get diffraction patterns in a single measurement from lattice planes hkl} having one W and all q orientations (non-tilted ease) or different W and different (q orientations (tilted ease). Thus, the number of measurements for the calculation of the stress components can be significantly reduced. Moreover, it is possible to investigate both textured materials and coarse-grained samples  and b).

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A. SCHUBERT ET AL.

CONVENTIONAL METHOD OF X-RAY STRESS ANALYSIS (ONE-DIMENSIONAL APPROACH)
In polycrystalline materials, the stress determination can be applied with the help of a one-dimensional detector if the grains are randomly oriented and sufficiently fine-grained so that the Debye-Scherrer cones consist of uniformly distributed diffractions spots. Under these conditions, it is possible to apply isotropic elasticity theory to convert determined strains into stresses. The residual strain eq can be determined from the change in lattice spacing Dtp where D is the unstressed lattice spacing. This strain may be expressed in terms of the strains in e/j the sample coordinate system by the tensor transformation (g33) g3k g31 Eld g231 Ell + g232 E'22 + g233 F-'33 (2) + 2g3 g33 El3 "!-2g32 g33 E23 "1-2g31 g32 12 where g3k and g31 are the direction cosines between L (measuring direction) and sample axes S k, S respectively (Hauk and Macherauch, 1984;. Primed tensor quantities refer to the laboratory system L i, and unprimed tensor quantities refer to the sample coordinate system S r The direction cosine matrix for this case is ga g g Since e, depends on the angles and q, there are two basic measuring strategies the sin2 and q-method (Macherauch and Mtiller, 1961;Kimpfe and Michel, 1986).
The measurement in various sample directions involves considerable time expediture and even the use of a position-sensitive detector can not always reduce the time taken to make such measurements: In the case of coarse-grained material only a small number of crystals is recorded by the one-dimensional detector. Then the assumption of isotropic elastic behaviour is no longer satisfied, since the reflecting single grains are elastically anisotropie. -Due to the elastic and plastic anisotropy resdting from texture, in most cases non-linear D vs. sin29t distributions or only a limited D range of sin2w are observed.
The measurement is selective in the sense that only those grains properly oriented to diffract contribute to the diffraction profile.
To overcome the described problems, an imaging plate would be helpful for stress analysis. Since the beginning of X-ray diffraction, films have been used but they have some disadvantages because of the long time needed for picture development, film shrinkage, and the unsuitability of film data for direct computer processing. At last the filmtechnqiues are now exchanged by modern two-dimensional detectors or imaging plates which allow the digitization of the whole diffraction pattern. X-RAY STRESS ANALYSIS BY AREA DETECTOR 55 3. ADVANTAGES OF THE IMAGING PLATE Laser stimulated fluorescence imaging plates have been in use since 1986 as two-dimensional area detectors. The large dynamic range, high sensitivity, and low background render them ideally suitable as area detectors in X-ray crystallography (STOE & CIE GmbH, 1992).
The experimental arrangement is shown in Figure 1. The use of an imaging plate as a two-dimensional detector has several advantages for stress analysis fast and improved data processing better stress analysis for coarse-grained material by intensity integration of sectors or the whole Debye-Scherrer cone or by intensity measurement at discrete orientations improved stress determination for textured material by intensity measurement at preferred orientations (texture components), which are directly visible on the imaging plate (Figure 2 The incident X-ray beam is diffracted into a cone with an aperture angle of 2 (180-20). An imaging plate perpendicular to the cone axis intercepts the cone in a circle if the specimen is unstressed (Figure 3). If stresses exist, an ellipse-like pattern occurs. The deviation of its shape from a circle is caused by stresses (strains), (Borgonovi and Gazzara, 1989;Yoshioka and Ohya, 1994). b) Figure 3 Geometry of the two-dimensional approach (a) tilted and (b) non-tilted case A point on the circle of the imaging plate corresponding to a diffracted beam can be found by the polar coordinates radius R and the azimuthal angle fl with respect to the laboratory system. Once the position of the detector and the incident beam is known, it is easy to derive the angular position 20 as well as the position of the diffraction peaks from the coordinates of the spot on the detector for various fl ( Figure   4). All the normals of the crystal lattice planes responsible for the diffraction ring describe a cone with an aperture angle of 180-20. A special normal is also characterized by the azimuthal angle 1. In the sample coordinate system the same normal is described by the azimuthal angle and tilt angle W. To establish the correspondence between the sample orientation and the twodimensional detector, two coordinate systems are used. Figure 3 shows the relative orientation of the laboratory and sample system. The sample system is defined by the S3-axis along the normal direction and the other two axes S, and S in the surface.
L is the laboratory system, with the L axis oriented in the direction of the normals of the reflecting crystal planes. Using the angles 90-0, fl, and , the transformation of the sample system into the laboratory system is given by three consecutive rotations. The rotation matrix is of the form g, gg0o_ os2 gaS3 g,,S2 Now, substituting g3k and g31 in (2), the fundamental equation of X-ray strain determination for the two-dimensional approach with an imaging plate is given. The difference between the one-dimensional approach (sin-t-method) and the two-dimensional approach (imaging plate) can be shown using stereographic projection.

Absorption
A further point that must be considered in the analysis is the presence of strain gradients with depth. In such cases, the D spacings, and thus the strain obtained by the X-ray beam will be the average of this gradient over the effective penetration depth x of X-ray.
The diffracted intensities are affected by absorption, which is directly related to the path length of the incident and diffracted beams in the specimen for a given geometry (Figures 6 and 7). s+=s+s* s" =s+sz S +/" k +1" z k */'= 1/(cosWo) + 1 (cos (W +/-2q) Consider the case where a beam of intensity 10 of unit cross section F is incident on a flat plate at an angle '0-The decrease of the intensity by a layer of thickness dz at a depth z below the surface is described by the absorption factor where / denotes the linear absorption coefficient and k is a trigonometric function determined by the diffraction geometry 1/(costo) + 1/(cos0 +/-21/) (K, B two constants), (Figure 8).

Exposure time and peak resolution
Since the strains e,, are less than 0,001 for elastic behaviour in most materials, the absolute value of the peak position, i.e. the peak maximum, is required for the accurate calculation of D,, from Bragg's law and an appropriate method of peak location must be used. The most important factor in X-ray stress measurements is that proper peak resolution must be achieved of all specimen orientation. The selection was based on obtained satisfactory fme-line peak resolution with X-ray intensities of at least 103 to 104 counts. Figure 9 shows {420} reflections of a copper sample measured with Cu-K radiation. For data evaluation we reduce the two-dimensional image to a one-dimensional diffraetogram for a given measuring time.
To determine the influence of the exposure time on the quality of measured Debye-Scherrer rings the imaging plate was exposed for several times. The results are shown in Figure 10. An optimum ratio between peak and background and a maximum intensity of about 1000 counts were determined for an exposure time of about 15 min. For the 20 peak location 3rd order curves were fitted through the data points using a least square fit. The results are compared with the stress state determined with a conventional diffractometer (Go,).
This two-dimensional approach looks very promising for the determination of residual stresses. For the electron beam hardened steel it was shown that this method using an imaging plate gave stress distributions which were in good agreement with those from stress measurements with a one-dimensional detector (Figure 12).
In the future, the absorption effects, the information depth and possible stress gradients have to be taken into account to improve accuracy of data evaluation.