Tomography of small residual stresses

TL;DR

This paper investigates the inverse problem of reconstructing residual stresses in Man's model using tomographic data, focusing on theoretical uniqueness results for compressional and shear waves.

Contribution

It introduces a mathematical framework for residual stress tomography, proving uniqueness of solutions for both compressional and shear wave data under certain conditions.

Findings

Uniqueness results for the inverse problem in Man's residual stress model.

Equivalence of the inverse problem to the inversion of the longitudinal ray transform.

Extension to shear waves involving the mixed ray transform.

Abstract

In this paper we study the inverse problem of determining the residual stress in Man's model using tomographic data. Theoretically, the tomographic data is obtained at zero approximation of geometrical optics for Man's residual stress model. For compressional waves, the inverse problem is equivalent to the problem of inverting the longitudinal ray transform of a symmetric tensor field. For shear waves, the inverse problem, after the linearization, leads to another integral geometry operator which is called the mixed ray transform. Under some restrictions on coefficients, we are able to prove the uniqueness results in these two cases.