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RESEARCH PAPER

Use of Power Series Expansion for Residual Stress Determination by the Incremental Hole-Drilling Technique T.C. Smit1 · R.G. Reid1 Received: 9 March 2020 / Accepted: 20 July 2020 © Society for Experimental Mechanics 2020

Abstract Background The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Objective Investigate the use of series expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Methods Power series expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the series order with the lowest RMS uncertainty in stress to be selected from those series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Results Series expansion is shown to be stable up to 8th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. Conclusions The use of series expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods. Keywords Residual stress · Incremental hole-drilling · Series expansion · Eigenstrain

Introduction Laser Shock Peening (LSP) is a surface treatment technique that generates beneficial residual compressive stress that is typically greater in magnitude and reaches a greater depth [1] with less surface roughness in comparison with conventional shot peening [2]. LSP produces a steep variation in residual stress from the surface with a local minimum usually within 0.4 mm from the surface. This

 R.G. Reid

[email protected] 1

School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

type of residual stress field can be challenging to measure accurately using relaxation methods. The incremental hole-drilling technique (IHD) is one of the most widely used relaxation methods [3] because it is practical to implement, low cost and yields reliable and accurate results [4]. Another advantage of IHD is that it is only semi-destructive, unlike other relax