Researchers at the Laboratory of Hierarchically Structured Materials of the Skoltech Engineering Center (part of the VEB.RF group) have developed an approach based on rational experimental-computational correlation (RECC) that makes residual stress estimation more stable against measurement noise. For instance, at a substantial experimental error of up to 10 percent, the error in stress determination dropped from 40 to 6 MPa. The study's findings have been published in the scientific journal Measurement.

Why Residual Stress Matters

Residual stress develops in materials after a part has been manufactured, processed, or loaded, and it can have a considerable impact on the part's durability, fatigue resistance, and crack propagation. Accounting for it is particularly important in stress concentration zones, such as the areas around fastener holes in aviation and transport structures.

One technique for determining how such stresses are distributed through the thickness of a material is the so-called "contour method." A specimen is cut, the displacements on the cut surface caused by the release of stress are measured, and numerical modeling is then used to reconstruct the original residual stress field from these data.

The Inverse Problem at the Heart of the Contour Method

What sets the contour method apart is that it solves an inverse problem: stresses are calculated from the measured surface displacements. As a result, even fairly small errors in the input data can be amplified during reconstruction. The outcome is affected both by statistical measurement noise and by the specifics of data processing, interpolation, and the condition of the surface after cutting. In this study, the researchers quantitatively evaluated how random noise in the measured displacements influences the determination of stress fields.

The study's simulations show the effect clearly. As the level of simulated measurement noise rises from 5 to 30%, the reconstruction produced by the contour method departs further and further from the reference direct solution, and at 20–30% distinct oscillations and regions of non-physical values emerge.

"Traditionally, the accuracy of experimental data in the contour method is accepted as sufficient based on comparison with other approaches (for example, diffraction-based), but the stability and reliability of the solution are rarely assessed quantitatively. We created an experimental-computational scheme that allows us to quantitatively trace how the error in displacement measurement affects the determination of stress fields. For the representative noise level of 10% that we selected, the reconstruction error was about 40 megapascals for aluminum alloy 2024-T4. The proposed approach (Sur-RECC) describes the displacement profile with a smooth analytical function — a Gaussian curve with a quadratic background. This allows suppressing high-frequency deviations and, in the case considered, reduces the effective error to approximately 5–6 MPa," says the article's first author, Igor Zorin, a PhD student in the Engineering Systems program.

When the normalized recovery error (NRMSE) is plotted against measurement noise, applying Sur-RECC considerably lowers the sensitivity of the result to noise compared with the conventional contour method.

How the Approach Was Tested

The approach was tested on specimens of high-strength 2024-T4 aluminum alloy. Double-sided dimpling produced a characteristic residual stress field in the specimens, which served as a model problem. The specimens were then cut on a wire electrical discharge machine using special clamping tooling that limited their movement during cutting. Confocal laser profilometry was used to measure the deformation of the cut surface. The resulting residual stress field was independently verified using X-ray diffraction and focused-ion-beam digital image correlation.

At the same time, the scientists used the finite element method to reproduce the deformation, unloading, and subsequent cutting of the specimen. The field produced by direct modeling served as a reference for a numerical experiment: controlled random noise was added to the input data step by step, and the degree to which it distorted the result of the contour method was evaluated. This made it possible to isolate the influence of statistical noise itself from other sources of uncertainty.

Smoothing Out the Noise

"Sur-RECC effectively regularizes the solution of the inverse problem: it preserves the smooth component of deformation, which is essential for determining the macroscopic stress field, and suppresses high-frequency deviations. At the same time, we don't claim that all local fluctuations are related to measurement error — some of them may be related to material, cutting, or data processing features. The article compares direct modeling and inverse problem approaches for determining residual stresses by specimen sectioning methods, which are extremely important for assessing the distribution of internal stresses through thickness. The proposed data processing makes these approaches more reliable and resistant to noise," said the article's co-author, Professor Alexander Korsunsky, who heads the Laboratory of Hierarchically Structured Materials at the Skoltech Engineering Center.

Collaborators and Funding

The research was conducted by Skoltech scientists together with specialists from the Moscow Aviation Institute and the Central Institute of Aviation Motors. The work received support from the Ministry of Science and Higher Education of the Russian Federation as part of a major scientific project.