Ill-Posedness, Bias, and the Case for Average Stresses in Residual Stress Evaluations

Abstract The evaluation of residual stresses using relaxation methods is a central problem in experimental mechanics, yet the conventional pursuit of pointwise stress values creates persistent physical and mathematical difficulties. Physically, the isotropic continuum models underlying these methods lose validity near the microstructural grain scale, rendering highly localized stress values physically dubious. Mathematically, deriving these pointwise values from measured relaxed strains constitutes an ill-posed inverse problem. To achieve a manageable signal-to-noise ratio, established inversion methods—such as the Integral Method or Tikhonov regularization—must enforce a bias–variance trade-off. This introduces an inherently unquantifiable bias, severely compromising the reliability of standard uncertainty quantification. To resolve this misalignment between physical reality and mathematical formulation, this paper advocates a fundamental shift: Redefining the measurement target from pointwise extremes to the spatial average of the residual stress field over a physically significant characteristic length ( $$l_\sigma $$ l σ ). Because engineering failure criteria (such as fatigue and fracture) are driven by stress integrals over specific volumes rather than single points, this redefinition is physically natural. Most importantly, it makes the inverse problem mathematically well-posed: the bias–variance trade-off vanishes, and standard uncertainty propagation yields exact, highly reliable confidence intervals. Building on recent theoretical frameworks [1], this work develops the physical justification for this scale-dependent reformulation. It provides a critical review of the systematic biases embedded in classical inversion techniques and presents a comprehensive numerical simulation of a hole-drilling residual stress measurement on a shot-peened steel component. This simulation explicitly shows how the spatial averaging procedure guarantees nominal confidence interval coverage, contrasting with the hidden biases of established approaches.