On the Implementation of the Integral Method for Residual Stress Measurement by Integrated Digital Image Correlation

Abstract

The Integrated Digital Image Correlation method (iDIC) is a simple and effective approach for residual stress measurement. iDIC differs from Digital Image Correlation because it replaces the “generic” displacement functions used to describe the displacement field around the measurement point with problem-specific ones. By this simple modification, stress components become the unknowns of the problem, thus allowing a single-pass analysis. Advantages are significant in terms of accuracy, robustness and ease of implementation. However, the implementation of the Integral Method for estimation of depth-dependent residual stress components is difficult. This work suggests two alternative approaches to solve this problem; in the former, the direct solution of the triangular linear system is employed to incrementally identify the stress distribution. In the latter, a global spatio-temporal minimization involving all the acquired images is suggested.

Appendix: Notes on Numerical Implementation

The described algorithms require iterating the solution procedure, thus, an efficient implementation should cache the calibration coefficients. However, this has to be done with care: P^u, Q^u, …, T^v depend on 𝜃 and on the A, B, …, G coefficients. These in turn depend on material properties and on the (normalized) radius. Thus, it is possible either to cache the A, …, G coefficients (the intermediate values) or the final set (P^u, …, T^v). The former solution corresponds to a minimal level of caching because the A, …, G coefficients depend on material and normalized radius only, thus, they can be easily stored either as a one-dimensional function of the radius (an interpolating function) or as a (small) list of values sampled at increasing radii. Using this solution, the $P^{u}_{i,j}$, …, $T^{v}_{i,j}$ coefficients have to be (re)computed every time they are required, with a large computational overhead because each pixel k of the image has a different polar coordinate with respect to the center of the hole.¹⁵

Instead, if the computed $P^{u}_{i,j}$, …, $T^{v}_{i,j}$, are stored in matrices, the above described procedure has to be performed only once, with obvious improvements in terms of CPU usage. However, this approach (full caching) requires a significant memory storage: indeed, assuming to perform m drill increments, the number of matrices is

$$ 6 (1 + 2 + 3 + {\cdots} + m) = 6 \frac{m(m + 1)}{2} = 3 m(m + 1) $$

where the six appearing in front of the summation accounts for the six terms (P^u, Q^u, …, T^v) involved in equations (1), (1).

To give some numbers, let assume that a 1 Mpixel camera is used to image 15 drilling increments; using these parameters, each of the P^u, …, T^v matrices requires 4 MiB (we are assuming that 32bit floating point numbers are used to store real values). Thus, to store all calibration coefficients in memory, you need 3 × 15 × (15 + 1) × 4 = 2880 MiB of storage.¹⁶ Our empirical tests show that full caching is not required because solving the problems shown in the article requires a few seconds using an i7@3.4GHz processor and a minimal level of caching.