Issue 41

S. Seitl et alii, Frattura ed Integrità Strutturale, 41 (2017) 323-331; DOI: 10.3221/IGF-ESIS.41.43 326 conditions. Coefficients A n are functions of relative crack length  = a / W and need to be calculated numerically in most cases. The origin of the coordinates was positioned at the crack tip [16] and no crack front curvature was considered [17]. It is worth mentioning that no bulk effects were taken into account in this study [18]. The truncated form of the Williams power series considering N terms of the expansion is commonly used for the stress/displacement field approximation. When the crack is subjected to mixed-mode loads [19], Eq. 2 and 3 also include a shear mode component that allows the shear mode SIF to be estimated [20], [21]. Over-deterministic method (ODM) Several methods have been derived and suggested for estimation of the coefficients of the higher-order terms of the Williams expansion. Most of those methods (such as hybrid crack element (HCE) method [22], [23], boundary collocation method (BCM) [24] or others) require advanced mathematical procedures and extensive knowledge of special crack elements or FE code. Therefore, the so-called over-deterministic method (ODM) has been chosen for calculation of the coefficients of the higher-order terms in this paper. This method is based on knowledge of the displacement field data ( u , v ) in a set of k nodes around the crack tip. These values can be obtained either numerically from finite element simulations on the cracked specimen or experimentally via optical measurements (which is the case of this paper). The ( u , v ) data can be then together with the nodes coordinates ( r , θ ) put into Eq. 2. Thus, a system of 2 k equations arises (in 2D) and the only variables are the coefficients A n . 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. Crack tip fields reconstruction When the coefficients of the higher-order terms of the Williams expansion are known, the crack tip stress and/or displacement field can be reconstructed by means of Eq. 2 and 3, respectively. In this paper, the approximation of the principal stress  1 and von Mises stress  HMH is presented because these stresses are often used in fracture mechanics criteria and thus, they are important for description of the crack behaviour. Following relations were used for calculation of the stress components:                  2 2 1 2 2 x y x y xy (4)           2 2 2 HMH 3 x x y y xy (5) In order to verify the experimental data, the comparison of the crack tip stress distribution is introduced. The coefficients of the Williams expansion determined via the hybrid crack elements for a normalized CT specimen were taken from the literature [5], for mixed mode see [28]. R ESULTS AND DISCUSSION n the first step of the verification process, the stress intensity factor (SIF) values were compared. The values obtained directly from the fracture tests are compared to values calculated from the displacements measured via DIC and to values obtained by Bednář and Knésl by means of the hybrid crack elements [5]. Data from measurements on three cracked specimens are introduced. A good agreement can be observed in the results shown in Tab. 2. Thus, further analysis taking into account also the higher- order term (the second one) of the Williams expansion can be performed. In Fig. 2 to 5 the stress distribution around the crack tip is presented. Particularly, the stress levels with the values of 100, 150, 200 and 250 MPa are plotted for two stress components σ 1 and σ HMH . The values are calculated by means of the Williams expansion taking into account its one or two initial terms and the comparison between the results based on the theoretical values of the coefficients calculated by means of the hybrid elements [5] and the results based on the experimentally I