How to Enhance Efficiency and Accuracy of the Over-Deterministic Method Used for Determination of the Coefficients of the Higher-Order Terms in Williams Expansion

Abstract

Using higher-order terms of the Williams expansion is necessary for assessment of fracture behavior of quasi-brittle materials. Multi-parameter fracture mechanics enables more accurate determination of the stress/displacement field even in a larger distance from the crack tip, thus the extended zone with non-elastic behavior typical for this kind of material can be well described. The so-called over-deterministic method (ODM) seems to be a suitable tool for the higher-order terms coefficients calculation, but its utilization exhibits some limitations. Therefore, extensive analyses have been performed in order to summarize recommendations regarding the mesh sensitivity, boundary conditions influence, etc. List of pieces of advice and author’s experiences presented in the end of this work should contribute to more accurate and effective utilization of the ODM.

Williams Expansion Terms and their Importance for Accurate Stress Field Description in Specimens with a Crack

“…2 Note that numerous authors' experience with the ODM application have been applied during the analysis (corresponding for instance to the number of nodes used for the ODM evaluation, or FE mesh parameters etc., see e.g. [3][4][5][6][7][8]). However, although a relatively high number of nodes has been used for investigations (k = 50) and their distance from the crack tip was large enough, it has not led to better convergence.…”

“…More details about the ODM can be found for instance in [2] or in some of the authors' works devoted to parametric studies on ODM accuracy, convergence, or mesh sensitivity etc., see e.g. [3][4][5][6][7][8].…”

Multi-parameter fracture criteria for the estimation of crack propagation direction applied to a mixed-mode geometry

“…More details about the method, its accuracy and reliability or recommendations on its most effective application can be found for instance in [33] or in some of the author's works, e.g. [34,35]. Generally, the following holds: the higher the order of the WE coefficient, the worse its convergence behaviour.…”

“…The values of the first five coefficients A n (or/and B m ) were calculated via ODM and compared to data published by various researchers and obtained by means of other numerical methods (such as hybrid crack element method (HCE), boundary collocation method (BCM) or others). The selected results can be found for instance in [13] or [34] and are included also in this work in order that the readers feel confident in the accuracy of ODM. The results presented in Tables 1 and 2 correspond very well to each other and therefore, the accuracy of the ODM can be considered to be proven.…”

Williams expansion-based approximation of the stress field in an Al 2024 body with a crack from optical measurements

“…The procedure for determination of the coefficients of the higher-order terms was programmed in a commercial mathematical package Wolfram Mathematica [25]. The ODM is thoroughly described and tested previously [3], [26], [27]. In this study, the displacements of nodes closest to the crack tip (obtained experimentally) were considered for application of the ODM; the total number of nodes in the radius of 1 mm around the crack tip was approximately 60 for each configuration.…”