Spatial convolution of a stress field analyzed by X-ray diffraction

X-ray stress analysis suffers from homogeneity limitations of the stress field in the analyzed volume. When this homogeneity is not fulfilled, it is possible to reduce the irradiated volume down to stress homogeneity achievement. New limitation however occurs : the diffracting sites become too few for stress homogenization. We show that the diffractometry analysis corresponds to a spatially convoluted stress field. The inverse convolution problem is posed. An example of regularization method is given.


II. Stress field convolution and impulse response
In this section, we propose an experimental procedure to estimate the impulse response of the diffractometer. We call ) , ( 11 y x σ the local stress at a given point (x,y) on the specimen surface in the stress reference frames ( ) .We assume that the stress measures . , ( , 11 11 σ σ plus some Gaussian noise. The most direct way to determine h(x,y) of an X-ray diffractometer would be to perform measurements of a stress field that follows a Dirac distribution δ(x,y). Since it is pratically impossible to obtain such a stress field, we consider instead a stress field that follows a Heaviside function. Therefore, we consider a EN10084 steel sample containing a grinded and an annealed zones. The annealing treatment relaxes the surface stresses whereas the grinding introduces a residual stress ) , (  fig. 2a). This variation occurs within a range smaller than 100 µm (100 µm is the depth at which grinding stresses are usually relaxed [5]). By the specimen design, the stress field does not vary along the (Oy) direction. The impulse response studied is only in the (Ox) direction and thus it corresponds to the single-dimensional convolution: The experimental procedure is as follows. Stress measurements are performed every 100 µm along the (Ox) axis fig 2a. A diaphragm of 2.6 mm diameter is used in the collimator and the irradiated disc has an estimated diameter of 4.2 mm. The irradiation time is set to 30 s. Stress measurements reported in fig. 2b are obtained by the sin² ψ method wich is carried out over 8 ψ incidences with opposite pairs of angles. fig. 2b clearly shows that the stress goes from 0 to -280 MPa over a range of 4000 µm which is much larger that the zone of zone affected by grinding (c.a. 100 µm). Next, we propose a geometrical model for the spacial convolution of the stress field. We formulate the following hypothesis.
-The intensity of the incidencial X-ray beam is homogeneous inside the irradiated zone. It has been shown [3] that the intensity distribution is indeed close to a uniform distribution in the case of a collimator diameter larger than 3 mm and for the same XRD apparatus as the one used in the present investigation.
-The irradiated zone is a disc which shape is weakly affected by the incidence ψ as it will be confirmed in next section. These two hypothesis allow us to describe the determination of the For each measure, σ * corresponds to a solution u of equation (3). By plotting each value of u versus x, we obtain by linear regression a straight line with a slope of b 1 − . The stress stage response ( fig. 2b) is in agreement with the measured b value which is 2.03 mm. We thus obtain a convolution disc with a diameter b 2 close to the one of the irradiated (cf. section II, 4.2 mm) The derivative of equation (3) gives the impulse response: The curve of the impulse response is presented fig. 2c. It is observed that with a good approximation, the measured stress is the average value of the stress field in the extent of the irradiated zone.

III) Incidence angles and impulse response symmetry.
In this section, the influence of the set of Ψ incidences on the stress convolution is evaluated. To this aim, we choose to bend a specimen with an embeded extremity ( fig. 3a) leading to a stress field that is spatially varying. A planar stress state is assumed which leads to: ( )  fig. 3b. Slightly different evolution of the stress measures (slopes) are found depending on the used incidences. For y > 0, the stress value is minored for the set III and majored for the set I. Opposite results are obtained for y < 0. The actual irradiated zone position seems to be shifted when the incidence angle is changed. As the local stress is a function of x , the stress measures are found to be modified when the number of positive incidences is not equal to the number of negative incidences. The shift in the stress measures increases with y because the local stress ) , ( 11 y x σ is also a function y . However, the relative difference between the measures of stress I and III with measure II is roughly constant and is only about 10%.

IV Measurement and deconvolution of a spatially-varying stress field
In this section, we illustrate a deconvolution procedure to obtain the local stress values from experimental stress measurements. Since deconvolution techniques are sensitive to the amount of noise in the experimental data and in order to validate with no ambiguity the proposed deconvolution procedure, we consider an experimental situation where the local stress field is known (e.g. from analytical models). We chose a rectangular specimen containing a trough hole in the middle. We focus on the stress field around the specimen hole when the specimen is deformed under a uniaxial compression. . The theoretical convoluted stress field is shown fig. 4c2 and a good agreement is found with the experimental stress measurements. It can be noted that the theoretical non-convoluted stress field is slightly different from the convoluted response. The maximum difference of 20 MPa is found at θ = π/2. This value constitutes an estimation of the error made when stress analyses are performed without considering the spatial convolution of the stress measures. Next, we perform a deconvolution procedure of the experimental data ( ) > < θ σ , 11 r * in order to obtain the local stress ( ) θ σ , 11 r . The irradiated zone used to perform the experimental stress measures exhibits a relatively narrow radial dimension ( d =1mm). Therefore, it is reasonable to consider that the convolution is only carried over θ ∈[−α;+α]. The convolution equation becomes: is also given for few c,ω c ( ) domains and compared with the theoretical local stress at the center of the irradiated zone fig. 5c. A large domain where the solution is stable can be clearly seen and a good agreement is obtained with the theoretical local stress field.

Conclusion
Stress measurements using XRD result from the spatial convolution of the local stress over the irradiated surface of the specimen. For the experimental setup used here, we show that this convolution is simply the average of the local stress field over the analysed surface and the impusional response of the diffractometer is directly in connection with the circular shape of the irradiated surface. The irradiated surface is found to become slightly assymmetrical with respect to the goniometric center when incidence angles are varied. We show that the local stress field can be obtained from the experimental measurements using a deconvolution procedure based on the Fourier transform of the convolution equation. A stable solution is found when a rationalization procedure is performed and when noise in the experimental data is limited.