Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

Abstract

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.