On recovering intragranular strain fields from grain-averaged strains obtained by high-energy X-ray diffraction microscopy

TL;DR

This work investigates recovering the full intragranular strain field from grain-averaged strains obtained by high-energy X-ray diffraction microscopy. By formulating the problem as finding a boundary traction distribution that drives the measured grain-averaged strains under elasticity, it proves the inverse problem is either unsolvable or admits infinitely many solutions due to an infinite-dimensional kernel. A best-approximate reconstruction with regularization is introduced, and its behavior is analyzed under Saint-Venant-type principles, showing exponential decay of kernel-field influence away from the ends in tall specimens. Numerical tests and an AlON experiment demonstrate the approach, revealing meaningful central-region strain information despite nonuniqueness and highlighting practical limitations and potential extensions to enforce physical constraints directly in reconstruction.

Abstract

We address an unusual problem in the theory of elasticity motivated by the problem of reconstructing the strain field from partial information obtained using X-ray diffraction. Referred to as either high-energy X-ray diffraction microscopy~(HEDM) or three-dimensional X-ray diffraction microscopy~(3DXRD), these methods provide diffraction images that, once processed, commonly yield detailed grain structure of polycrystalline materials, as well as grain-averaged elastic strains. However, it is desirable to have the entire (point-wise) strain field. So we address the question of recovering the entire strain field from grain-averaged values in an elastic polycrystalline material. The key idea is that grain-averaged strains must be the result of a solution to the equations of elasticity and the overall imposed loads. In this light, the recovery problem becomes the following: find the boundary traction distribution that induces the measured grain-averaged strains under the equations of elasticity. We show that there are either zero or infinite solutions to this problem, and more specifically, that there exist an infinite number of kernel fields, or non-trivial solutions to the equations of elasticity that have zero overall boundary loads and zero grain-averaged strains. We define a best-approximate reconstruction to address this non-uniqueness. We then show that, consistent with Saint-Venant's principle, in experimentally relevant cylindrical specimens, the uncertainty due to non-uniqueness in recovered strain fields decays exponentially with distance from the ends of the interrogated volume. Thus, one can obtain useful information despite the non-uniqueness. We apply these results to a numerical example and experimental observations on a transparent aluminum oxynitride (AlON) sample.

Figures (7)

• Figure 1: High-energy X-ray diffraction provides the grain structure (middle) and the grain-averaged strains (upper right) of a small cylindrical region of the specimen (left). We seek the full strain field, or equivalently the traction distribution on the top and bottom of the gauge section which induces a strain field satisfying the equations of elasticity (lower right).
• Figure 2: Three examples of kernel tractions (zero applied force and zero grain-averaged strains) and their corresponding strain fields in a unitless 2D domain with size $1 \times 1.2$. The different examples correspond to kernel tractions inducing maximal uncertainty in the subdomain $\Omega_h$ of height $h$ (denoted by the dashed white boxes) for (a) $h=1.2$ (whole domain), (b) $h=1.0$, and (c) $h=0.8$, see Section \ref{['sec:saint_venants_principle']}. The markers correspond to those shown in Figure \ref{['fig:max_eigenvalues']}.
• Figure 3: Best-approximate reconstruction using synthetic data. (a) "Ground truth", (b) uniform loading initial guess, and (c) best-approximate reconstruction loads and axial strain distributions. (d) Grain-averaged strains used to solve the inverse problem. (e) Uniform loading and (f) best-approximate reconstruction error/difference between ground truth (viz., difference between (a) and (b); (a) and (c)). (g) $\varepsilon_{11}$, (h) $\varepsilon_{22}$, and (i) $\varepsilon_{33}$ strain components plotted along the vertical black line shown in (d) ($x_2 \in (0, H)$, $x_1=x_3=H/2$) for the four fields shown in (a--d). (j--l) Normalized error (normalized by the mean axial strain) between recovered and ground-truth strain fields for the normal strain components along the same line. In (g--l), changes in grains are highlighted by the alternating lightly shaded regions.
• Figure 4: Best-approximate reconstruction for experimental data in AlON. (a) Grain structure (inverse pole figure coloring) and (b) grain-averaged axial strains as measured from ff-HEDM analysis. (c) Best-approximate reconstruction. (d) Experimentally measured grain-averaged axial strains and (e) associated best-approximate intragranular strain field in a grain fully contained within the volume.
• Figure 5: Normalized worst-case error (maximum eigenvalue normalized by volume) in a subdomain of height $h$ versus the domain height $H$. Worst-case error as a function of (a) domain height for fixed subdomain heights, and (b) subdomain height for fixed domain heights. The shaded regions approximately indicate where numerical breakdown occurs, and the non-circular markers correspond to the fields in Figure \ref{['fig:null_traction']}(a--c).

Theorems & Definitions (3)

• Theorem 1
• proof
• Remark 2