Uncertainty, model error, and order selection for series-expanded, residual-stress inverse solutions

Measuring the spatial variation of residual stresses often requires the solution of an elastic inverse problem such as a Volterra equation. Using a maximum likelihood estimate (least squares fit), a series expansion for the spatial distribution of stress or underlying eigenstrain can be an effective solution. Measurement techniques that use a series expansion inverse include incremental slitting (crack compliance), incremental hole drilling, a modified Sach's method, and others. This paper presents a comprehensive uncertainty analysis and order selection methodology, with detailed development for the slitting method. For the uncertainties in the calculated stresses caused by errors in the measured data, an analytical formulation is presented which includes the usually ignored but important contribution of covariances between the fit parameters. Using Monte Carlo numerical simulations, it is additionally demonstrated that accurate uncertainty estimates require the estimation of model error; the ability of the chosen series expansion to fit the actual stress variation. An original method for estimating model error for a series expansion inverse solution is presented. Finally, it is demonstrated that an optimal order for the series expansion can usually be chosen by minimizing the estimated uncertainty in the calculated stresses.