Issue 48

M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 600 the least-squares method for determining the NSIFs using the stresses obtained from finite element method (FEM). As these least-squares methods [14, 15] use the stress field, they require either very fine meshes at the notch tip or higher order Williams coefficients to attain the accurate NSIFs. Triefi et al. [16] developed a strain energy method to determine the mixed mode NSIFs of sharp V-notches. In another research, Lazzarin et al. [17] proposed a strain energy density technique for the rapid evaluation of NSIFs. Many researchers used fractral-like finite element method [18-20] and extended finite element method [21, 22] to calculate the NSIFs. Other researchers employed various path independent integrals [23-26] to estimate the NSIFs of sharp V-notches. Apart from these methods, Ayatollahi and Nejati [27] determined the NSIFs along with the higher order terms of Williams coefficients using an overdeterministic method utilizing FE displacements. Recently, Hussain and Murthy [28, 29] developed a collocation technique and a point substitute displacement technique (PSDT) to obtain the mixed mode (I/II) NSIFs using the FE displacements along the notch flank. In their method [29], certain optimal point(s) of the notch tip element along the notch flanks have been identified by minimizing the error in displacements, and interestingly, the displacements and its slope are found to be more accurate at those points. The PSDT utilizes the displacements on the notch flanks at these optimal point(s) to calculate NSIFs accurately. This method is efficient, simple and straightforward to be implemented in the available FE code and can be employed to find the NSIFs using manual calculations. It is worth noting here that, unlike the availability of well-known quarter point elements [30, 31] in the crack problems, no such popular singular elements are currently available for use in the sharp V-notched configurations. Consequently, any post-processing techniques for the determination of the NSIFs should be capable enough to determine the accurate NSIFs yet using the conventional elements at the notch tip and concurrently it should be simple enough to implement in the existing code. The main thrust of the present work is to demonstrate further the efficacy of the above technique, PSDT proposed by Hussain and Murthy [29] in terms accurate estimation of the NSIFs of the specimens having straight and curved boundaries under mode I and mixed mode (I/II) loading conditions. T HEORETICAL B ACKGROUND short description of the formulations for PSDT is presented in this section. However, for more details, one may refer to Ref. [29]. For a homogeneous elastic 2D problems containing sharp V-notch (in Fig. (1)), the displacement field at any nearby point    , P r under any arbitrary in-plane loading can be given as [14]                                                                   1 1 Re cos 2 cos 2 cos cos 2 2 Re cos 2 cos 2 sin sin 2 2 2 for I n II n I I I I I I n n n n n n R n II II II II II II n n n n n n R n n A u r u G B r u G n B (1)                                                                 1 1 Re cos 2 cos 2 sin sin 2 2 Re cos 2 cos 2 cos cos 2 2 2 for I n II n I I I I I n n n n n n n II II II II II II n n n n n n R n n A v r G B r v G n B (2) where n A and n B are Williams’ coefficients corresponding to n -th term for mode I and mode II, respectively,    , r denotes the polar coordinate components,   Re denotes real part of the variables, Kolosov constant  is equal to         3 1 for plane stress and   3 4 for plane strain conditions,     ( 180 2 ) is the notch angle (Fig. (1)),      2 1 G E is the shear modulus,  and E are the Poisson’s ratio and Young’s modulus, respectively.  I n and  II n are the mode I and mode II eigenvalues, respectively, and can be obtained by solving the following characteristic Eqn. (3). The constants displacements I R u , II R u and II R v can be shown in Eqn. (4) [27]. A