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Direct inversion of the Longitudinal Ray Transform for 2D residual elastic strain fields

C. M. Wensrich, S. Holman, M. Courdurier, W. R. B. Lionheart, A. Polyalova, I. Svetov

TL;DR

This work develops two direct inversion methods for the 2D Longitudinal Ray Transform of elastic strain, by linking Bragg-edge imaging to a Helmholtz decomposition using an Airy stress potential. The solenoidal component of the strain is recovered with a tensor-filtered back projection, while the remaining potential portion is reconstructed via Hooke’s law or a finite-element elasticity solve, enabling full reconstruction of the 2D strain field. The approach is demonstrated on simulated data and validated with experimental neutron imaging data, showing robustness to noise and compatibility with existing equilibrium-based measurements. The results clarify how scalar filtered back projection relates to the trace of the solenoidal component and provide practical tools for residual-stress tomography in planar structures, with potential impact for additive manufacturing and material science applications.

Abstract

We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal Ray Transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection algorithm whereas the potential part can be recovered using either Hooke's law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar filtered back projection algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.

Direct inversion of the Longitudinal Ray Transform for 2D residual elastic strain fields

TL;DR

This work develops two direct inversion methods for the 2D Longitudinal Ray Transform of elastic strain, by linking Bragg-edge imaging to a Helmholtz decomposition using an Airy stress potential. The solenoidal component of the strain is recovered with a tensor-filtered back projection, while the remaining potential portion is reconstructed via Hooke’s law or a finite-element elasticity solve, enabling full reconstruction of the 2D strain field. The approach is demonstrated on simulated data and validated with experimental neutron imaging data, showing robustness to noise and compatibility with existing equilibrium-based measurements. The results clarify how scalar filtered back projection relates to the trace of the solenoidal component and provide practical tools for residual-stress tomography in planar structures, with potential impact for additive manufacturing and material science applications.

Abstract

We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal Ray Transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection algorithm whereas the potential part can be recovered using either Hooke's law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar filtered back projection algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.
Paper Structure (20 sections, 3 theorems, 52 equations, 9 figures)

This paper contains 20 sections, 3 theorems, 52 equations, 9 figures.

Table of Contents

  1. Introduction and context
  2. Notation and definitions
  3. Helmholtz decompositions and LRT inversion formulas
  4. Elasticity theory and residual stress
  5. Problem statement
  6. Helmholtz decomposition of strain in $\mathbb{R}^2$
  7. Isotropic strain and scalar Filtered Back Projection
  8. Recovery of $\epsilon$ from $^s\epsilon$
  9. Recovery of $\epsilon$ from compatibility
  10. Recovery of the potential component
  11. Recovery of $\epsilon$ from Hooke's law
  12. Numerical demonstration: Simulated data
  13. Strain fields
  14. Procedure
  15. Results
  16. ...and 5 more sections

Key Result

Lemma 1

Suppose that $\Omega$ contains the support of $f$. If $h_\Omega$ in decomp_bd is zero, then ${^s}f$ and $u$ in decomp are equal to the extension by zero of ${^s}f_\Omega$ and $u_\Omega$ to $\mathbb{R}^n$. Conversely, if ${^s}f$ and $u$ in decomp are supported in $\Omega$, then $h_\Omega = 0$.

Figures (9)

  • Figure 1: Geometry of the Longitudinal Ray Transform and Bragg-edge strain measurements.
  • Figure 2: A reconstruction of a synthetic strain field computed from an Airy stress field. (a) The original strain field. (b) A reconstruction of the solenoidal component of this field from a simulated LRT consisting of 200 equally spaced projections over 360$^\circ$. (c) The recovered potential component from elastic finite element modelling. (d) The reconstructed strain field formed by the sum of the solenoidal and potential components.
  • Figure 3: Two samples representing strain fields used to perform numerical demonstrations of the reconstruction algorithm. (a) A crushed steel ring containing a distributed eigen-strain field. (b) An offset ring and plug system containing a discrete eigen-strain field generated through mechanical interference.
  • Figure 4: A reconstruction of a synthetic strain field computed from an elasto-plastic finite element model of the crushed ring. (a) The original strain field. (b) A reconstructed of the solenoidal component of this field from a simulated LRT consisting of 200 equally spaced projections over 360$^\circ$. (c) The recovered potential component from elastic finite element modelling. (d) The reconstructed strain field formed by the sum of the solenoidal and potential components.
  • Figure 5: A reconstruction of a synthetic strain field computed from an linear-elastic finite element model of the offset ring and plug system. (a) The original strain field. (b) A reconstructed solenoidal component of this field from a simulated LRT consisting of 200 equally spaced projections over 360$^\circ$. (c) The recovered potential component from elastic finite element modelling. (d) The reconstructed strain field formed by the sum of the solenoidal and potential components.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof