Well-posedness and trivial solutions to inverse eigenstrain problems

TL;DR

This paper investigates the mathematical structure of inverse eigenstrain problems, revealing the existence of trivial solutions and discussing implications for residual stress analysis and potential tomographic reconstruction methods.

Contribution

It analyzes the well-posedness of inverse eigenstrain problems, identifies trivial solutions, and links eigenstrain analysis to strain tomography via the Longitudinal Ray Transform.

Findings

Existence of a trivial solution in all inverse eigenstrain problems

Null space structure explains non-uniqueness of solutions

Potential for residual stress reconstruction using LRT measurements

Abstract

We examine the well-posedness of inverse eigenstrain problems for residual stress analysis from the perspective of the non-uniqueness of solutions, structure of the corresponding null space and associated orthogonal range-null decompositions. Through this process we highlight the existence of a trivial solution to all inverse eigenstrain problems, with all other solutions differing from this trivial version by an unobservable null component. From one perspective, this implies that no new information can be gained though eigenstrain analysis, however we also highlight the utility of the eigenstrain framework for enforcing equilibrium while estimating residual stress from incomplete experimental data. Two examples based on measured experimental data are given; one axisymmetric system involving ancient Roman medical tools, and one more-general system involving an additively manufactured Inconel sample. We conclude by drawing a link between eigenstrain and reconstruction formulas related to strain tomography based on the Longitudinal Ray Transform (LRT). Through this link, we establish a potential means for tomographic reconstruction of residual stress from LRT measurements.