Diffraction tomography of strain

Abstract

We consider whether it is possible to recover the three dimensional strain field tomographically from neutron and x-ray diffraction data for polycrystalline materials. We show that the distribution of strain transverse to a ray cannot be deduced from one diffraction pattern accumulated along that path, but that a certain moment of that data corresponds to the transverse ray transform of the strain tensor and so may be recovered by inverting that transform given sufficient data. We show that the whole strain tensor can be reconstructed from diffraction data measured using rotations about six directions that do not lie on a projective conic. In addition we give an inversion formula for complete data for the transverse ray transform. We also show that Bragg edge transmission data, which has been suggested for strain tomography with polychromatic data, cannot provide the strain distribution within the material but only the average along the ray path.

Reconstructing intragranular strain fields in polycrystalline materials from scanning 3DXRD data

“…where m is the measured angle of diffraction and r is the corresponding angle of diffraction expected for a relaxed reference state. The scalar measured strain, " " " m , is an average property of the region R, and as explained by Lionheart & Withers (2015), it exists in the direction perpendicular to the diffracting Miller planes.…”

“…It has also been suggested, in the case of powder diffraction, that the full strain tensor can be retrieved using filtered back projection with a sufficient number of measurement directions (Lionheart & Withers, 2015). Similar ideas could, perhaps, be applied to scanning 3DXRD, which measures discrete diffraction events rather than powder rings.…”

Bragg-edge elastic strain tomography forin situsystems from energy-resolved neutron transmission imaging

“…[18], and prior work by [19], clearly demonstrate the general strain tomography problem from this transform is ill-posed. For conservative strain fields the LRT is only sensitive to boundary deformations, implying multiple strain fields can project to the same set of strain images (solutions to the problem are not unique).…”

Energy Resolved Neutron Imaging for Strain Reconstruction Using the Finite Element Method

“…As it stands, the inverse problem is ill-posed, and reconstruction of the strain is not possible without imposing further conditions [12]. Despite this problem, the reconstruction of the strain field under further assumptions has proven possible [10], [13].…”

“…Inherent symmetry of the transform implies 180 • are sufficient; however, in practice, measurements are taken over an entire revolution, i.e., 360 • . Lionheart and Withers [12] demonstrated that the integral line LRT is a non-injective map from ε → l ε (a, ϑ ) and hence the strain field produced by any given set of projection is not unique [18]. As a consequence, it is not possible to reconstruct the strain distribution within a body in the general setting.…”