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

The paper analyzes the uniqueness of eigenstrain tomography for reconstructing 3D residual stress from diffraction data in isotropic solids. It shows that using a single measured component yields non-unique solutions, but that three independent components (diagonal or shear) can enforce a unique full elastic strain/stress field under equilibrium and zero-traction conditions. It proves that any residual stress field can be represented by a diagonal eigenstrain and that some fields cannot be generated by isotropic eigenstrains, implying that restricting to hydrostatic eigenstrains is not robust. Together, these results define rigorous minimal experimental and computational requirements for well-posed inverse eigenstrain problems and inform practical strategies for eigenstrain tomography.

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.

Figures (4)

• Figure 1: At high energies, x-ray diffraction angles become small and the radius of Debye Scherrer rings can be used to measure elastic strain in the (approximately) transverse direction relative to the incident beam. In the vertical direction $\kappa$, the only component that contributes to this measurement is invariant to the rotation angle and can be reconstructed through standard scalar filtered back projection korsunsky_strain_2011uzun_critical_2025.
• Figure 2: Stress and the corresponding elastic strain within the cube $[0,2\pi]^3$ over a cross section at $x_3=\pi/2$ (with $E=1$ and $\nu=0.28$). This corresponds to the single null vector of $A$ in the case of $N=2$. It represents a simple example of a residual stress field on the cube with $\epsilon_{22}=\epsilon_{33}=0$ while simultaneously satisfying equilibrium and a zero-traction boundary condition.
• Figure 3: Similar to Figure \ref{['fig:firstbasisvector']}, a residual stress/strain field shown on the planar slice at $x_3=\pi/2$, but now corresponding to a weighted sum of the 343 null basis fields for $N=8$. Coefficients in the sum have been chosen through a least-squares process fitting $\sigma_{11}$ to the binary ($\pm1$) image shown. Note that all boundary conditions are satisfied and $\epsilon_{22}=\epsilon_{33}=0$ at every point in this cube.
• Figure 4: Boundary conditions on the subset $\partial\Omega_1\subset\partial\Omega$ with normal vector orthogonal to the $x_1$-axis (i.e. $n_1=0$).

Theorems & Definitions (8)

• Remark 1
• Conjecture 1
• Theorem 1: The span of diagonal eigenstrains
• proof
• Corollary 1
• proof
• Theorem 2
• proof