Uniqueness of solutions in high-energy x-ray based `eigenstrain tomography' and other inverse eigenstrain problems: Counter examples and necessary conditions for well-posedness

TL;DR

This paper investigates the mathematical conditions for unique and well-posed solutions in eigenstrain tomography, demonstrating non-uniqueness in some cases and establishing minimal measurement requirements for accurate residual stress reconstruction.

Contribution

It provides explicit counterexamples of non-uniqueness and identifies minimal measurement conditions for unique reconstruction in eigenstrain tomography.

Findings

Counterexamples show non-uniqueness with single strain component measurements.

Full elastic strain tensor can be uniquely reconstructed from three specific measured components.

Residual stress fields can be generated by diagonal eigenstrain, but not all by isotropic eigenstrains.

Abstract

Eigenstrain tomography combines diffraction-based strain measurement with elasticity theory to reconstruct full three-dimensional residual stress fields within solids. Notwithstanding a number of recent examples, the uniqueness of such reconstructions has not yet been clearly established. In this paper, we examine the underlying inverse problem in detail and construct explicit counterexamples demonstrating non-uniqueness for a recent implementation of x-ray eigenstrain tomography involving reconstruction from a single measured component of strain. We follow on to explore minimum conditions for well-posedness and conclude that the full elastic strain tensor within an isotropic sample can be uniquely reconstructed from three measured components; specifically the three shear components, or the three diagonal components. We further prove two key results related to eigenstrain reconstruction in a general sense; 1. That any possible residual stress field can be generated by a diagonal eigenstrain and 2. That residual stress fields exist that cannot be generated by isotropic eigenstrains. Together, these findings establish rigorous minimum experimental and computational requirements for well-posed eigenstrain tomography techniques and inverse eigenstrain problems in general.