Single-axis high-energy X-ray diffraction tomography for elastic residual strain: uniqueness and stability of solutions in the presence of equilibrium constraints

Abstract

It is well-established that single-axis Transverse Ray Transform tomography data from high-energy X-ray diffraction is insufficient for general reconstruction of three-dimensional elastic strain. In this paper we show that, when combined with the constraint of mechanical equilibrium, this problem becomes uniquely solvable for isotropic elastic samples with known elastic constants and non-zero Poisson's ratio. Building on the reconstruction framework of Desai and Lionheart [N.M. Desai, W.R.B. Lionheart, An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data, Inverse Problems 32 (11) (2016) 115009], we formulate this constrained single-axis tomography problem in the Fourier domain and show that the resulting system is invertible everywhere outside a finite collection of measure zero characteristic planes. In the case of bounded samples, this is sufficient to establish uniqueness for reconstructions. We further derive conditional stability estimates that quantify the regularity requirements associated with reconstruction of the various strain components.