Issue 39

J. Sobek et alii, Frattura ed Integrità Strutturale, 39 (2017) 129-142; DOI: 10.3221/IGF-ESIS.39.14 130 Tensile failure of the quasi-brittle materials is accompanied with a crack propagation together with a nonlinear zone [2] (fracture process zone – FPZ) development, where the decrease of material integrity takes place. The size of the FPZ is not negligible in comparison with the rest of the body. Processes of failure therefore occur in a wider area around the crack tip. The description of failure mechanism must be in agreement with the stress and displacement field around the crack tip in an extended area. Recent works of many authors show the relevance of the topic of the crack tip fields accurate description [3–6], moreover, extending it into 3D and taking into account the effects of various loading modes [7–12]. Special focus on brittle and quasi-brittle fracture is summarized in [13]. Works [3,14,15] reported the fact, that for the description of the stress/displacement field in a cracked body in a more distant surroundings from the crack tip the necessity of usage of the several terms of Williams expansion (WE), not only the first or the first two terms, is crucial. Procedures enabling the multi-parameter description of the near crack-tip fields (using e.g. hybrid crack elements [16] formulation, over-deterministic method [17] based on standard FE computation, or other techniques based e.g. on extrapolation of displacements of selected nodes of FE mesh) usually process results from FE nodes selected from the close vicinity of the crack tip. This is adequate for determination of the classical/two-parameter LEFM characteristics (SIF, T -stress). However, several questions arise if the area of accurate enough description of the stress and displacement field extends to a greater distance from the crack tip, where K + T dominance vanishes? How many terms of the series should be taken into account? How to select the FE nodes considered for the regression technique? And how to optimize the mutual relationship between the area from where the nodes are considered for the regression (and how are they located/distributed in that area) and the extent where the approximation of the fields is of relevant accuracy? Answering these questions presents the actual motivation of this work. This paper investigates the above-described issue via a parametric study evaluating the influence of the nodal selection on the quality of the obtained approximation of the field. The work presented here further builds on previous studies. A classical (common) way of nodal selection, where the nodes are selected from a ring in the vicinity of the crack tip was used in recent papers. Influences of several parameters on the description of stress/displacement fields in cracked bodies by Williams series were investigated. Reconstruction of the stress field with the help of a software tool developed by the author’s team was shown in [14,15] where the accuracy of the approximation by WE was verified by a visual comparison (which was regarded as sufficient for its intended purpose) with the FE solution (which was regarded as the exact solution). However, the nodal selection that was used for obtaining the WE terms was performed from the ring around the crack tip (with the distance given by recommendation from [17,18]). Published study [19] on WPS approximation with nodal selection from more than one ring around the crack tip resulted into next studies. Another type of nodal selection was considered in subsequent works by the authors [20,21,22]; the nodes were selected from specific parts of the test specimen body with specific distribution functions of their distance and angular position from the crack tip. This way was employed with expectations that the fields will be better (more accurately/efficiently) approximated. An automatic utility to determine the values of coefficients of the higher order terms of WPS using the over-deterministic method was developed [20] to enable multi-parameter description of stress field, where the coefficients of WPS terms are calculated from several layers and angular sections. And the distance distribution of the nodal selection is governed by various functions. Comparison between those used variants was given by visual technique again [21]. Hence, a new detailed way for evaluation of the accuracy of the reconstruction was used in [22]. Method based on the plot of the relative deviation (percent difference) of the stress field between the correct solution (the FE solution) and the solution given by the approximation using WPS with a certain variant of nodal selection was introduced. Main motivation of this study is to find the easiest way for the multi-parameter stress field description to obtain the sufficiently accurate solution valid for the as far as possible area from the crack tip. M ETHODS Multi-parameter linear elastic fracture mechanics (MP-LEFM) he stress and displacement fields in a planar homogeneous isotropic cracked body can be formulated as an infinite power expansion – Williams series [1] by Eq. (1) and (2), respectively – for more details see [14,21,22,23].              1 2 , 1 , , , , 2 n ij n ij n nA r f n i j x y (1) T