Well-posedness and trivial solutions to inverse eigenstrain problems

TL;DR

This work analyzes the inverse eigenstrain problem for residual-stress analysis, revealing fundamental ill-posedness due to a large null space where nonzero eigenstrains produce no stress. By formulating a range-null decomposition with a $C$-weighted inner product, the authors show that inverse solutions are determined up to unobservable gradient components, yielding a trivial solution $\epsilon^{*\perp_C} = -S:\sigma$ in general. The paper demonstrates practical utility by enforcing equilibrium to extract information from incomplete data, illustrated with axisymmetric theory and two experimental case studies: ancient Roman artifacts and additively manufactured Inconel, including a direct link to strain tomography via the Longitudinal Ray Transform. The Maxwell-potential framework serves as a constructive way to model divergence-free (solenoidal) eigenstrains, enabling constrained inverse problems and suggesting tomographic pathways to reconstruct residual stress from LRT data. Overall, the work clarifies what can be learned from eigenstrain analyses and when prior information or priors are necessary to obtain nontrivial insights.

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.

Figures (9)

• Figure 1: An object represented by the domain $\Omega$ with surface normal $n$.
• Figure 2: Roman bronze medical tools dating to 1-200CE Italy (from RomanToolsURL)
• Figure 3: Geometry of strain measurements on the KOWARI instrument at ANSTO. Through precise measurement of the diffraction angle to the detector, normal strain in the direction of $\kappa$ can be measured within the gauge volume.
• Figure 4: Eigenstrain solutions for the residual stress fields in the four Roman utensils compared against the measurements from KOWARI.
• Figure 5: Eigenstrain solutions for the Roman utensils computed alongside a variable isotropic $d_0$ that was assumed to be a function of radius.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of Theorem \ref{['NullFields']}