PSI - Issue 39

Mahsa Sakha et al. / Procedia Structural Integrity 39 (2022) 792–800

795

4

M. Sakha et al. / Structural Integrity Procedia 00 (2019) 000–000

n

y

y

τ rθ

σ

t

θ

σ r

Kink Crack

τ rθ

L c

u t

σ

σ θ

θ

θ 0

τ rθ

Main Crack

Main Crack

θ 0

Kink Crack

σ

σ

x

x

r

θ

τ rθ

u n

n

n

t

t

( k )

( k )

σ θ

G θ ( ) II

G θ ( ) I

τ rθ

y

y

u t

+

2

1

σ θ

2

1

τ rθ

β

β

x

u n

x

Fig. 2. Schematic representation of the two main crack growth criteria when the kinked crack is subjected to the mixed tensile and shear stresses: a) Maximum tangential stress (MTS); (b) Maximum energy release rate (MERR).

3.1. Maximum tangential stress

The MTS criterion postulates that a crack extends along a radial direction at which the tangential stress σ θ reaches the tensile strength of the material, σ c . This criterion was originally proposed by Erdogan and Sih (1963) for isotropic solids, and then extended to anisotropic solids by adopting a direction-dependent tensile strength (Saouma et al., 1987). The T-stress inclusion in the MTS formulation has already been shown to improve its predictions in isotropic solids (Williams and Ewing, 1972; Smith et al., 2001) as well as anisotropic rocks (Nejati et al., 2020a; Sakha et al., 2021). The tensile strength σ c is often determined using σ c = K Ic / √ 2 π L c , where K Ic is the mode I fracture toughness and L c is the length of the fracture process zone (FPZ). In transversely isotropic solids, however, the parameters σ c ψ and K Ic ( ψ ) are both direction-dependent, where the subscript ψ is the angle between the crack growth direction and the principal direction 1 (see Figure 2a). Under the mixed-mode I / II loading, the tangential stress σ θ at the distance L c from the crack tip can be expressed as

( µ 1 − µ 2 )   −  

K eff √ 2 π L c  

ℜ   1

2 II   ( µ 1 sin θ + cos θ )

K II

K I

3 / 2

σ θ =

+ µ 2

K 2

K 2 3 / 2     + I + K

2 II

I + K

(1)

2 II   ( µ 2 sin θ + cos θ )

+  

sin 2 θ   .

T √ 2 π L c K 2 I + K 2 II

K II

K I

+ µ 1

K 2

K 2

2 II

I + K

I + K

Here, K I and K II respectively denote the mode I and II stress intensity factors, and K eff is the effective stress intensity factor. The complex parameters µ 1 and µ 2 are given by Sakha et al. (2021). The MTS criterion is the most widely-used fracture growth criterion in anisotropic materials. This is due to its simplicity and accurate predictions. Nevertheless, the MTS criterion has inherently one potential drawback, that is related to the assumption that tensile fracturing always prevails. As long as a tensile-based failure precedes a shear based one, the crack growth path can accurately be captured by MTS, even under the pure mode II loading. In cases where the shear-induced fracturing precedes tensile-induced one, the MTS criterion, however, fails to accurately predict fracture growth (Bahrami et al., 2020).

3.2. Maximum energy release rate

The energy release rate (ERR) is defined as the decrease of the total potential energy, Π , with respect to the increase in the fracture surface A : G ( θ ) = − d Π / dA . According to the maximum energy release rate (MERR) criterion, a crack