Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel
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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-07-01
|
| Series: | Applied Mechanics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-3161/5/3/27 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850259122938707968 |
|---|---|
| author | Marco Beghini Tommaso Grossi |
| author_facet | Marco Beghini Tommaso Grossi |
| author_sort | Marco Beghini |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-dbd1565d572c487aa338fdb13b82e113 |
| institution | OA Journals |
| issn | 2673-3161 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Mechanics |
| spelling | doaj-art-dbd1565d572c487aa338fdb13b82e1132025-08-20T01:55:57ZengMDPI AGApplied Mechanics2673-31612024-07-015347548910.3390/applmech5030027Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form KernelMarco Beghini0Tommaso Grossi1DICI—Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, 56122 Pisa, ItalyDICI—Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, 56122 Pisa, ItalyBy 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.https://www.mdpi.com/2673-3161/5/3/27residual stressrelaxation methodsill-posednessill-conditioninginverse problems |
| spellingShingle | Marco Beghini Tommaso Grossi Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel Applied Mechanics residual stress relaxation methods ill-posedness ill-conditioning inverse problems |
| title | Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel |
| title_full | Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel |
| title_fullStr | Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel |
| title_full_unstemmed | Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel |
| title_short | Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel |
| title_sort | measuring residual stresses with crack compliance methods an ill posed inverse problem with a closed form kernel |
| topic | residual stress relaxation methods ill-posedness ill-conditioning inverse problems |
| url | https://www.mdpi.com/2673-3161/5/3/27 |
| work_keys_str_mv | AT marcobeghini measuringresidualstresseswithcrackcompliancemethodsanillposedinverseproblemwithaclosedformkernel AT tommasogrossi measuringresidualstresseswithcrackcompliancemethodsanillposedinverseproblemwithaclosedformkernel |