A Deconvolution Method for the Mapping of Residual Stresses by X-Ray Diffraction

Abstract

Background

Inherent averaging effects in X-ray diffraction measurement of residual stresses, related to the finite size of the irradiated area, lead to inaccurate measurements in the presence of high surface stress gradients.

Objective

This paper develops a reconstruction method which allows the mapping of heterogeneous residual stresses from X-ray diffraction averaged measurements.

Methods

The stress reconstruction is based on a deconvolution of the average XRD measurements. The combination of a fine measurement grid and the use of two collimators lead to an overdetermined linear system on the average stress measured experimentally whose inversion provides the values of the local stress field.

Results

First, the method is successfully assessed in a reference problem solved by FEM in which the local distribution and average datasets are known. Then, it is applied to the reconstruction of residual stress mapping from experimental XRD measurements of a specimen processed by repetitive corrugation and straightening. The reconstructed field is in agreement with numerical (local) results of the process.

Conclusion

The method developed in this work permits the reconstruction of accurate distributions of heterogeneous near-surface residual-stresses.

Appendices

Appendix 1

Robust Spline Smoothing of Noisy Data

In order to reduce the effect of noise in the inverse problem (14), smoothing splines are considered as a very efficient technique to construct a smooth estimate $\widetilde{\varvec{\Sigma }}^a$ of $\varvec{\Sigma }^a$ by minimization of a functional $\mathcal {G}$ that balances the fidelity to the data, through the residual sum-of-squares (RSS), and the smoothness of the estimate $\widetilde{\varvec{\Sigma }}^a$, through some penalty term $\mathcal {P}$ [18]:

$$\begin{aligned} \mathcal {G}\left( \widetilde{\varvec{\Sigma }}^a \right) = \left\| {\varvec{\Sigma }^a}-\widetilde{\varvec{\Sigma }}^a\right\| ^2 + s \mathcal {P}\left( \widetilde{\varvec{\Sigma }}^a\right) . \end{aligned}$$

(21)

In equation (21), $\left\| \cdot \right\|$ denotes the Euclidean norm and s is a real positive scalar that controls the degree of smoothing. The penalty term $\mathcal {P}$ is defined from the point-values of the p-th derivative of $\widetilde{\varvec{\Sigma }}^a$ at the grid points; in the case of the second-order derivative, it reads

$$\begin{aligned} \mathcal {P}\left( \widetilde{\varvec{\Sigma }}^a\right) = \left\| \mathbf {D} \widetilde{\varvec{\Sigma }}^a \right\| ^2. \end{aligned}$$

(22)

In the one-dimensional case, the differentiation matrix $\mathbf {D}$ simply reads

$$\begin{aligned} \mathbf {D} = \frac{1}{\Delta x^2}\begin{pmatrix} -1 &{} 1 &{} &{} \dots &{} 0\\ 1 &{} -2 &{} 1 &{} &{} \vdots \\ 0 &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} &{} 1 &{} -2 &{} 1 \\ 0 &{} \dots &{} &{} 1 &{} -1 \end{pmatrix}. \end{aligned}$$

(23)

The minimization of $\mathcal {G}$ leads to the expression of the smooth estimate $\widetilde{\varvec{\Sigma }}^a$ [18]

$$\begin{aligned} \widetilde{\varvec{\Sigma }}^a = \mathrm{IDCT} \left( \varvec{\Gamma }(s) \circ \mathrm{DCT}\left( {\varvec{\Sigma }}^a \right) \right) , \end{aligned}$$

(24)

where DCT and IDCT respectively refer to the discrete cosine transform and the inverse cosine transform, and $\circ$ denotes Hadamard product (pointwise product). In the one-dimensional case, $\varvec{\Gamma }(s)$ is a vector whose components are given by

$$\begin{aligned} \Gamma _i = \frac{1}{1+s\lambda _i^2}, \end{aligned}$$

(25)

where the parameter $\lambda _i$ is given by

$$\begin{aligned} \lambda _i = 2-2 \mathrm{cos}\left( \frac{(i-1)\pi }{N-2k_a} \right) . \end{aligned}$$

(26)

The extension of the smoother to the two-dimensional case is straightforward [18]. The optimal smoother parameter s that avoids over- or under-smoothing is then estimated using the method of generalized cross-validation (GCV) introduced by [22]. The smoothing of the experimental data is thus fully automatic and the smoothing parameter estimate provided by the GCV method is unique [18, 23].

Appendix 2

Principles of XRD Measurements for a Fast Mapping of the Bi-axial Surface Stress

The principle of XRD measurements is to determine the interatomic spacing d of a family of diffracting planes using Bragg’s law. If we denote by $d_0$ the free-stress interatomic lattice spacing, the strain is given by

$$\begin{aligned} \varepsilon _{\Phi \Psi } = \frac{d_{\Phi \Psi }-d_0}{d_0}, \end{aligned}$$

(27)

where $\Phi$ and $\Psi$ are the angles associated with the X-ray direction. In the case of an isotropic elastic material, it is straightforward to note that

$$\begin{aligned} \varepsilon _{\Phi \Psi }= & \ \frac{1+\nu }{E} \left( \sigma _{11}\text {cos}^2(\Phi )+ \sigma _{12}\text {sin}(2\Phi )+\sigma _{22}\text {sin}^2(\Phi )-\sigma _{33}\right) \text {sin}^2(\Psi ) \\&+ \frac{1+\nu }{E}\sigma _{33} - \frac{\nu }{E}\left( \sigma _{11}+\sigma _{22}+\sigma _{33} \right) \\&+ \frac{1+\nu }{E} \left( \sigma _{13}\text {cos}(\Phi )+ \sigma _{23}\text {sin}(\Phi )\right) \text {sin}(2\Psi ). \end{aligned}$$

(28)

In the particular case $\Phi =0^\circ$ and $\Psi =0^\circ$, we have $\varepsilon _{\Phi =0^\circ , \Psi =0^\circ } = \varepsilon _{33}$, and therefore it is readily seen from equation (28) that

$$\begin{aligned} \varepsilon _{33} = \varepsilon _{\Phi =0^\circ , \Psi =0^\circ } = - \frac{\nu }{E}\left( \sigma _{11}+\sigma _{22} \right) + \frac{\sigma _{33}}{E}. \end{aligned}$$

(29)

Since the XRD measurements are performed on the surface of normal $\varvec{e}_3$, the normal stress $\sigma _{33}$ is null. Using equations (27) and (29), the bi-axial stress $\sigma _h$ finally reads

$$\begin{aligned} \sigma _h = \sigma _{11}+\sigma _{22} = -\frac{E}{\nu } \frac{d_{\Phi =0^\circ , \Psi =0^\circ }-d_0}{d_0}. \end{aligned}$$

(30)

Consequently the determination of the interatomic spacing $d_{\Phi =0^\circ , \Psi =0^\circ }$ allows a fast mapping of the bi-axial stress $\sigma _h$ since only one X-ray direction is required (with $\Phi = \Psi =0^\circ$). In practice, the free-stress interatomic lattice spacing $d_0$ is also required; it is determined in several points of the specimen (by calculating the full stress tensor using for instance 13 angles $\Psi$).