General reconstruction of elastic strain fields from their Longitudinal Ray Transform

TL;DR

The paper addresses reconstructing elastic strain fields from the Longitudinal Ray Transform (LRT) in both 2D and 3D, exploiting mechanical equilibrium to achieve full recovery in connected domains with a single boundary component. It combines Helmholtz decomposition, LRT inversion formulas for the solenoidal part, and a boundary-value formulation to recover the potential part from boundary displacements, proving a boundary-determination result up to rigid motions on each boundary component. The authors develop a practical algorithm and implement it in MATLAB, including a 2D simulation on a disk to demonstrate recovering $^{s}\epsilon$, $du$, and the full $\epsilon$ from simulated LRT data with boundary traction. This work enables energy-resolved neutron Bragg-edge strain imaging to recover full elastic strain fields in simple geometries and clarifies the role of boundary components and stability, paving the way for applications in non-destructive testing and materials characterization. The significance lies in providing a rigorous, implementable pathway to reconstruct full tensor-valued strain from projected measurements under general loading, expanding the scope of elastic tomography in neutron imaging.

Abstract

We develop an algorithm for reconstruction of elastic strain fields from their Longitudinal Ray Transform (LRT) in either two or three dimensions. In general, the LRT only determines the solenoidal part of a symmetric tensor field, but elastic strain fields additionally satisfy mechanical equilibrium, an extra condition that allows for full reconstruction in many cases. Our method provides full reconstruction for general elastic strain fields in connected objects whose boundary only contains one component, while previous results included other requirements such as no residual stress, or zero boundary traction. This allows for full reconstruction in energy resolved neutron transmission imaging for simple objects. Along the way, we prove that the LRT of a potential rank-2 tensor restricted to a bounded set determines the potential on the boundary of the set up to infinitesimal rigid motions on each component of the boundary. The method is demonstrated with numerical examples in two dimensions.

Figures (10)

• Figure 1: Bragg-edge imaging of strain at a pulsed neutron source. (a) Pulses of neutrons are generated at a source (usually a spallation target) before travelling a fixed distance down a guide to a sample and detector. High-energy (short wavelength) neutrons travel faster and arrive before low-energy (long wavelength) neutrons; the time-of-flight of a detected neutron is proportional to its wavelength. (b) Pixelated time-of-flight detectors now allow for rich-imaging where a 'radiograph' consists of a stack of images - one for each neutron wavelength in a spectrum. (c) At each pixel, Bragg-edges are formed at wavelengths defined by the spacings of various crystal planes within the sample. Relative change in the location of these edges provide a measure of strain averaged along the ray paths. e.g. (d) A projected strain image from a steel 'ring-and-plug' reference sample computed from relative shift in the (110) edge position. (adapted from data measured by Gregg et.al.gregg2018tomographic)
• Figure 2: Geometry of a ray passing from a source, through the sample $\Omega$, to a detector.
• Figure 3: Geometry of a ray passing through a sample $\Omega$ and intersecting the boundary at multiple points.
• Figure 4: A schematic of the numerical scheme to reconstruct $u$ on a discretised boundary.
• Figure 5: Singular value decomposition of $A$ for a non-convex 3-lobed shape. Top: Singular values of $A$ listed in decreasing order. Bottom: Zero-singular vectors of $A$.

Theorems & Definitions (6)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Lemma 1
• proof : Proof of Theorem \ref{['BdryThm']}