13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Improvement of crack-tip stress series with Padé approximants Gaëtan Hello1,*, Mabrouk Ben-Tahar2, Jean-Marc Roelandt2 1 UFR Sciences et Technologies - LMEE, Université d’Evry Val d’Essonne, 91020, France 2 Laboratoire Roberval, Université de Technologie de Compiègne, 60200, France * Corresponding author: gaetan.hello@ufrst.univ-evry.fr Abstract The most favored description of bi-dimensional crack-tip stress fields relies on Williams expansion. In this framework, each stress component is defined as a series which has a certain convergence behavior. Generally, the series is truncated after its first term since it is the most influential one at the vicinity of the crack-tip because of its well-known singularity. However, for some applications, the need for higher order terms arises and the study of truncation influence becomes important. The investigations performed by the authors for a specific fracture configuration have shown the existence of a convergence disk and of rather low convergence rates far from the crack-tip. In this communication, we propose to transform truncated stress series into Padé Approximants (PA) in order to improve both convergence domains and convergence rates. These approximants are rational functions whose coefficients are defined so as to fit the prescribed truncated series. The PA may be obtained following two different procedures. In practical tests, PA stemming from crack-tip stress series exhibit wider convergence domains and higher convergence rates. Keywords Crack-tip, Williams series, Padé approximant 1. Introduction The stress field at the vicinity of a crack-tip in a plane medium may be described using the so-called Williams series [1, 2]. Each term of the series is the product of three factors. Two of them concern respectively the angular and radial dependencies of the field. Their general expressions are known analytically and are the same for all fracture configurations [3, 4]. All the specific information related to the actual problem (geometry and loading conditions) is held by the third factor. Hence, to each fracture problem corresponds a specific infinite set of multi-order stress intensity factors. Concerning the determination of these sets, research has been mainly focused on their first element (the Stress Intensity Factor “Ki” associated to the stress singularity) and on their second one (the T-stress “T” associated to a constant stress state). In [5], closed form asymptotic expansions for the problem of a finite straight crack in an infinite plane medium submitted to uniform remote loads have been proposed. Expressions are provided for the multi-order stress intensity factors in either mode I or mode II problems using both power series (Williams series) and Laurent series. Thanks to these expressions, the convergence behavior of crack-tip stress expansions may be studied. The existence of the expected radii of convergence is observed. Rates of convergence are quantified. With the description of the crack-tip stress field by series arises the problem of truncation influence. The accuracy of the series representation improves as the number of terms increases. However, the convergence is rather slow and the summation procedure is numerically limited to a few hundreds terms. The aim of this work is then to improve the accuracy of the stress description based on the a-priori knowledge of a given number of terms in the series. A method based on Padé approximants (PA) is proposed [16-20]. These approximants are rational functions whose coefficients are defined so as to fit the prescribed truncated series. The communication starts with the description of the procedure leading to closed-form crack-tip stress solutions, the presentation of techniques related to Padé approximation follows and the efficiency of the method is assessed for a practical fracture configuration.
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