PSI - Issue 23

Stepanova Larisa et al. / Procedia Structural Integrity 23 (2019) 322–327 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Conventional fracture mechanics approach

The Williams asymptotic description for the stress field in the vicinity of the crack tip has the form

2      1 m k  m

( ) r,   ij

m k a r f

( ), 

/2 1 ( ) , k m ij 

k



where index m associated to the fracture mode; m

k a amplitude coefficients related to the geometric configuration,

load and mode; ( ) , ( ) k m ij f  angular functions depending on stress component and mode. Analytical expressions for angular eigenfunctions are available (Hello (2012), Hello (2018)). The multi-parameter fracture mechanics concept consists in the idea that the crack-tip stress field is described by means of the Williams expansion. Hello (2012) considered the central crack in an infinite plane medium, where analytical determination of coefficients in crack-tip expansion for a finite crack in an infinite plane medium is given. Since all the coefficients in the Williams asymptotic series expansion for the geometry considered are known it is possible to estimate the crack propagation direction angle by means of the multi-parameter fracture mechanics concept. In this paper two fracture criteria were chosen for the estimation of the initial crack growth direction: maximum tangential stress (MTS) criterion and strain energy density (SED) criterion. The results obtained by MTS and SED criteria are given in (Stepanova and Bronnikov (2018), Stepanova et al. (2017)). The maximum tangential strain criterion postulates that crack propagation initiates from the crack tip in direction of the maximum tangential strain 2 2 / 0, / 0.             The results given by the maximum tangential strain criterion for plane strain conditions are shown in Table 1 and Table 2 where the first column shows the crack propagation direction angle given by the one-term asymptotic expansion ( 1 N  ). The columns 2-9 in Table 1 and Table 2 show the results from multi-parameter fracture criterions where 100 terms in the Williams asymptotic series expansion have been kept. Table 1. Crack propagation direction angles obtained from MTS criterion at various radial distances from the crack tip, 0.3   . 1 N  0.1 c r  0.2 c r  0.3 c r  0.4 c r  0.5 c r  0.6 c r  0.7 c r  0.8 c r  e M -59.99 -52.30 -48.54 -46.38 -45.05 -44.18 -43.60 -43.21 -42.93 0.1 -57.70 -48.70 -45.14 -43.22 -42.07 -41.35 -40.89 -40.57 -40.36 0.2 -55.21 -44.89 -41.58 -39.91 -38.95 -38.37 -38.01 -37.78 -37.63 0.3 -52.35 -40.72 -37.73 -36.32 -35.56 -35.12 -34.86 -34.70 -34.62 0.4 -48.91 -36.05 -33.44 -32.31 -31.74 -31.44 -31.28 -31.19 -31.16 0.5 -44.51 -30.69 -28.53 -27.68 -27.30 -27.13 -27.06 -27.04 -27.06 0.6 -38.50 -24.44 -22.80 -22.23 -22.02 -21.96 -21.96 -22.01 -22.05 0.7 -29.82 -17.17 -16.08 -15.76 -15.68 -15.69 -15.74 -16.98 -17.02 0.8 -17.00 -8.91 -8.37 -8.24 -8.22 -8.26 -8.30 -9.56 -9.62 0.9 Table 2. Crack propagation direction angles obtained from MTS criterion at various radial distances from the crack tip, 0.5   . 1 N  0.1 c r  0.2 c r  0.3 c r  0.4 c r  0.5 c r  0.6 c r  0.7 c r  0.8 c r  e M -54.52 -48.12 -45.39 -43.92 -43.06 -42.54 -42.23 -42.03 -41.92 0.1 -52.70 -44.90 -42.31 -41.01 -40.30 -39.90 -39.67 -39.54 -39.48 0.2 -50.70 -41.47 -39.06 -37.96 -37.40 -37.10 -36.95 -37.78 -37.63 0.3 -48.39 -37.71 -35.53 -34.63 -34.21 -34.03 -33.96 -34.70 -31.16 0.4 -45.56 -33.47 -31.56 -30.87 -30.61 -30.53 -30.54 -31.20 -34.62 0.5 -41.86 -28.58 -26.99 -26.52 -26.40 -26.41 -26.49 -27.04 -27.06 0.6 -36.69 -22.83 -21.63 -21.36 -21.35 -21.44 -21.56 -21.68 -21.80 0.7 -28.92 -16.08 -15.29 -15.18 -15.24 -15.36 -15.49 -15.68 -15.73 0.8 -16.89 -8.36 -7.97 -7.95 -8.01 -8.10 -8.19 -8.27 -8.34 0.9