Papers by Keyword: Williams Expansion

Abstract: A simplified model of a crack approaching a bi-material interface is modelled by means of the finite element method in order to investigate the significance of the higher-order terms of the Williams expansion for the proper approximation of the opening crack-tip stress near the bi-material interface. The discussion on results is presented and the importance of the higher-order terms proved.

Abstract: The paper presents a study on accuracy of the multi-parameter approximation of the stress field in a cracked body using Williams power series calibrated according to results of FEM computation. Main attention is paid to a detailed analysis of suitable selection of FE nodes, whose results serve as inputs to the over-deterministic method (ODM) employed for determination of the coefficients of terms of the Williams expansion describing the crack-tip field. Two different ways of FE nodal selection are compared – nodes selected from the crack tip vicinity versus nodes selected from a specific part of the test specimen body in a specific way. Comparison is made with the usage of the authors’ own developed procedures enabling both the determination of the coefficients of the analytical approximation of stress field based on the FE results and the backward reconstruction of the field from those determined terms’ coefficients/functions. The wedge-splitting test (WST) specimen with a crack is taken as example for the study.

Abstract: Three cracked geometries loaded in mode I are investigated and the plastic zone size calculated. For estimation of the plastic zone size, two fracture criteria are used (Rankine and von Mises). Whereas the classical criteria give the same results (the stress intensity factor being identical for each geometry), the data from numerical simulations exhibit differences for various geometries. It is shown that the multi-parameter form of the criteria enables to obtain results that agree better to the numerical ones. Particularly, the Williams expansion is utilized for approximation of the stress components that serve as inputs for the fracture criteria. It is concluded that taking into account several more initial terms of the series can help to predict the plastic zone size more accurately.

Abstract: A mixed-mode cracked configuration is investigated using the multi-parameter fracture mechanics concept based on the analytical description of the stress/displacement field near a crack tip by means of the Williams series expansion. It is shown that using only one (singular) parameter as it is usual for brittle materials is not sufficient if the accurate stress distribution also further from the crack tip shall be known. Tangential stress distribution in various distances from the crack tip is presented and importance of the higher-order terms of the Williams expansion emphasized. Moreover, initial crack propagation direction is investigated by means of the MTS criterion and utilization of its generalized form is discussed.

Abstract: Multi-parameter description of crack behavior in quasi-brittle materials offers still enough space for investigations. Several studies have been carried out by the authors in this field [1-3]. One part of the publications by the authors (this work included) contain analyses of the accuracy, convergence and/or tuning of the over-deterministic method that enables determination of the coefficients of the higher-order terms in Williams expansion approximating the stress and displacement fields in a cracked body without any complicated FE formulations. These intermediate studies should bring together a list of recommendations how to use the ODM as effectively as possible and obtain reliable enough values of coefficients of the higher-order terms. Thus, the stress/displacement field can be determined precisely even in a larger distance from the crack tip, which is crucial for assessment of the fracture occurring in quasi-brittle materials.

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.

Abstract: Traditionally, applications of fracture mechanics have been mainly focused on cracks growing under the opening or mode I mechanism (three point bend specimen, middle tension specimen, crack tension specimen, etc.). When investigating instability under mixed mode loading conditions, three different alternatives can be envisaged consisting of: a) machining a hole in standard specimens, b) creating a crack oriented under a given angle, and c) using non-standard Arcan-Richard specimens. In this work, finite element calculations are performed to analyze the initial values of the fracture parameters in Arcan-Richard specimens. First, the influence of the normal stress mode to the shear stress mode ratio is analyzed, then the effect of the constraint level is discussed, and finally, the initial propagation angle of the daughter crack is derived.