Issue 49

G. Meneghetti et alii, Frattura ed Integrità Strutturale, 49 (2019) 53-64; DOI: 10.3221/IGF-ESIS.49.06 59 In more detail, the PSM enables to rapidly estimate the SIFs K I , K II and K III (Eqns. (3)-(5)) from the crack tip singular, linear elastic, opening, sliding and tearing FE peak stresses σ yy,peak , τ xy,peak and τ yz,peak , respectively, which are referred to the bisector line according to Fig. 4a, concerning the in-plane stress components, and Fig. 4c, as to the out-of-plane stress component. More precisely, the following expressions are valid [18–20]: * 0.5 , 1.38 I FE yy peak K K d     (11) ** 0.5 , 3.38 II FE xy peak K K d     (12) *** 0.5 , 1.93 III FE yz peak K K d     (13) where d is the so-called ‘global element size’ parameter to input in Ansys ® FE code, i.e. the average FE size adopted by the free mesh generation algorithm available in the FE code. Eqns. (11)-(13) were derived using particular 2D or 3D finite element types and sizes, so that a range of applicability exists, which has been presented in detail in previous contributions [18–20], to which the reader is referred. Here it is worth recalling that for mode I loading (Eqn. (11)) the mesh density ratio a/d that can be adopted in numerical analyses must be a/d  3, a being the minimum between the crack and the ligament lengths; for mode II loading (Eqn. (12)) more refined meshes are required, the mesh density ratio a/d having to satisfy a/d  14; in case of mode III loading (Eqn. (13)) the condition a/d  3 must again be satisfied. As an example, Fig. 4b shows a free mesh where d = 0.15 mm was input in Ansys ® software, while Fig. 4d shows a free mesh where the average FE size d is in constant proportion with the crack length a, i.e. a / d = 3. The mesh patterns shown in Figs. 4b,d are as coarse as possible to estimate the averaged SED with a 10% error using next Eqn. (17). It is important to underline that no additional input parameters other than d and no additional special settings are required to generate an FE mesh according to the PSM. When Eqns. (11)-(13) were calibrated [18–20], the ‘exact’ K I , K II and K III SIFs were evaluated using definitions (3)-(5), respectively, applied to FE results of numerical analyses characterized by very refined meshes, where the element size close to the crack tip was reduced to approximately 10 -5 mm. Therefore, the FE size required to estimate K I , K II and K III from σ yy,peak , τ xy,peak and τ yz,peak , respectively, is likely to be some orders of magnitude larger than that needed to directly calculate the local stress fields in order to apply definitions (3)-(5). Moreover, while Eqns. (3)-(5) require to process a number of stress-distance numerical results, the PSM requires a single stress value to evaluate the SIFs. A FE- BASED TECHNIQUE TO EVALUATE RAPIDLY THE T- STRESS he analytical expressions of the stress components σ xx along the crack free edges are obtained substituting the polar coordinate θ = +π and –π in Eqn. (1) and are given in Eqns. (14a) and (14b), respectively [12]:   1/2 II xx θ=π 2K σ =- +T+O(r ) 2πr (14a)   1/2 II xx θ=-π 2K σ =+ +T+O(r ) 2πr (14b) Therefore, the T-stress contribution can be derived according to Eqn. (15), as previously highlighted by Ayatollahi et al. [6], Lazzarin et al. [11] and Radaj [21]:     xx xx θ=π θ=-π σ + σ T= 2 (15) T