PSI - Issue 39

796 Mahsa Sakha et al. / Procedia Structural Integrity 39 (2022) 792–800 M. Sakha et al. / Structural Integrity Procedia 00 (2019) 000–000 5 extends along the direction θ 0 once the ERR function becomes maximum and exceeds the critical value G c ψ . The energy release rate due to the kink formation at the angle θ is determined from Yang and Yuan (2000)

I

I

1 2

L

K ( k ) II K ( k )

K ( k ) II K ( k )

G ( θ ) =

(2)

.

Here, K ( k ) ( k ) II respectively represent mode I and II stress intensity factors at the kink and can be calculated using the relations given in detail by Sakha et al. (2021). L is a rank two tensor related to the elasticity constants, and is defined in the kink coordinate system (tn) as L = L 11 L 12 L 21 L 22 I and K

(3)

E

cos θ sin θ − sin θ cos θ

ℑ ( µ 1 + µ 2 ) ℑ ( µ 1 µ 2 ) ℑ ( µ 1 µ 2 ) ℑ ( µ 1 µ 2 ( ¯ µ 1 + ¯ µ 2 ))

cos θ sin θ − sin θ cos θ .

˜ S 11

=

Again the complex parameters µ j and the compliance component ˜ S 11 are defined in Sakha et al. (2021). The energy release rate function G ( θ ) can be decomposed into two components as

1 2

1 2

K ( k ) ( k ) I ) G ( k ) I + I ( L 21 K ( k ) II + L 22 K

K ( k ) ( k ) I ) G ( k ) II . II ( L 11 K ( k ) II + L 12 K

G ( k ) =

(4)

Here, G ( ( k ) II respectively indicate the ERR contributions of tensile opening and shear sliding of the kink. denotes the release of the tensile stress, σ θ , upon formation of the kink, while G ( k ) II is the ERR contribution due to the release of the shear stress, τ r θ . Note that due to kinking, the stresses σ θ and τ r θ along the kink are released to form the displacement discontinuities ∆ u n (normal to the kink) and ∆ u t (along the kink), meaning that the stresses shown in Figure 2b depicts the stress state before kinking. The critical value G c ψ has so far been assumed as the critical tensile ERR function G Ic . This classical approach, however, appears to have a significant drawback upon its application to quasi-brittle materials which have signifi cantly different tensile and shear strength values. Indeed, while G ( θ ) includes the energy released due to both tensile opening, and shear sliding, the critical fracture energy G c ψ is only related to tensile failure. To avoid this incompatibil ity, Shen and Stephansson (1994) introduced the indicator F = G ( k ) I ( θ ) / G Ic + G ( k ) II ( θ ) / G IIc as the modified version of the normalized ERR function for isotropic materials. Shen and Shi (2016) then adopted this formulation to predict the direction of hydraulic fracture propagation in transversely isotropic rock. The robustness of the modified MERR cri terion has been shown through the numerical studies (Shen and Stephansson, 1994; Shen and Shi, 2016; Zhang et al., 2017). However, due to the absence of T-stress in their formulation, none of these studies constrained the contribution of G ( k ) I ( θ ) / G Ic to an interval in which the opening is physically possible. Here, we consider the interval of validity as θ 1 ≤ θ ≤ θ 2 . Sakha et al. (2021) highlighted the importance of deriving the interval of validity, especially when T-stress is highly compressive, and introduced a modified piecewise ERR function as As shown in Figure 2b, G ( k ) I k ) I and G