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RESEARCH PAPERS

Uncertainty, Model Error, and Order Selection for Series-Expanded, Residual-Stress Inverse Solutions

[+] Author and Article Information
Michael B. Prime

Engineering Sciences and Applications Division, Los Alamos National Laboratory, Los Alamos, NM 87545prime@lanl.gov

Michael R. Hill

Mechanical and Aeronautical Engineering Department, University of California, Davis, CA, 95616

J. Eng. Mater. Technol 128(2), 175-185 (Nov 07, 2005) (11 pages) doi:10.1115/1.2172278 History: Received April 11, 2005; Revised November 07, 2005

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.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Some of the measurement methods that can use a series-expansion solution for the stresses

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Figure 2

Through-thickness residual stress profiles used in beam simulations

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Figure 3

Simulated strain data for slitting experiment simulation and residual stress profiles from Fig. 2

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Figure 4

Estimated root-mean-square (rms) average uncertainties compared to rms actual error for n=4 polynomial stress profile with no added noise

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Figure 5

The n=2 solution for the n=4 polynomial stress profile plotted with one standard deviation estimates of total uncertainty. Even for the underfit solution, the uncertainty estimate is appropriate

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Figure 6

Estimated rms average uncertainties compared to rms actual error for Gaussian stress profile with no added noise

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Figure 7

Estimated rms average uncertainties compared to rms actual error for polynomial stress profile and noise added to strain data. These values are averaged over 500 trials with different values of random noise.

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Figure 8

Estimated rms average uncertainties compared to rms actual error for polynomial stress profile with noise added to strain data. This is a typical result for a single set of random noise added to the measured strains.

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Figure 9

Estimated rms average uncertainties compared to rms actual error for Gaussian stress profile with noise with standard deviation of 0.006 added to strain data. These values are averaged over 500 trials with different values of random noise.

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Figure 10

Estimated rms average uncertainties compared to rms actual error for Gaussian stress profile with noise with standard deviation of 0.012 added to strain data. These values are averaged over 500 trials with different values of random noise.

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Figure 11

Estimated rms average uncertainties compared to rms actual error for polynomial stress profile with one bad data point and no added noise

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Figure 12

Solution from noisy data compared with actual, Gaussian stress distribution. Uncertainties calculated (a) including covariances correctly reflect the actual errors and (b) without covariances, the uncertainties are much less accurate.

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Figure 13

The rms average strain misfit as a function of expansion order for the five test cases. The points corresponding to minimum total uncertainty are indicated. (a) Full view; (b) zoomed in on results for more realistic test cases with noise added to the strain.

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