PSI - Issue 39

6

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

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

797

    

   

ˆ G I ( θ ) G ( k ) I ( θ ) G Ic ( ψ ) k ) II ( θ ) G IIc ( ψ ) G (

ˆ G II ( θ ) G ( k ) II ( θ ) G IIc ( ψ )

d 2 ˆ G ( θ ) d θ 2

d ˆ G ( θ ) d θ

θ 1 ≤ θ ≤ θ 2

+

ˆ G ( θ ) =

= 0 ,

< 0 ,

,

(5)

otherwise

ˆ G ( θ ) = 1 .

4. Results and discussion Figure 3 illustrates the variations of the experimental data on the effective fracture toughness ( K c eff ) and the kink angle ( θ 0 ) against β for the four sets of experiments. Recall that β is the angle between the main crack and the foliation plane (principal direction 1). As previously mentioned, the loading constraint and anisotropy are the main influencers of the fracture growth direction when the geometry is kept fixed and the fracture grows under dry conditions. The interplay between the loading and anisotropy is best observed in Figure 3. While the anisotropy tends to drive the crack path toward the foliation plane, the positive mixed-mode loading (i.e. K II / K I > 0) opposes this tendency and thus the crack kinks at negative angles ( θ 0 < 0). The negative loading ratios, on the other hand, facilitate fracture growth toward the plane of isotropy which results in the positive kink angles. Figure 3 compares the experimental data on K c eff and θ 0 with the predictions of MTS along with three different versions of the MERR criterion that incorporate different assumptions into the normalized ERR function. The versions of MERR are defined as follows. 1) The original modified version as formulated in Eq. 5. 2) A version of Eq. 5 where tensile- and shear-based fracture energies are assumed to be equal: G IIc ( ψ ) = G Ic ( ψ ) . 3) A version of Eq. 5 where no interval of validity is considered: − π ≤ θ 0 ≤ π . Figure 3 shows a good agreement between the experimental data and the MTS and the modified MERR predictions when T-stress is included in the formulation. Nevertheless, it is expected that the MTS criterion cannot provide good predictions on the fracture path as soon as the shear-based failure precedes the tensile-based fracturing. In this case, the modified version of the MERR theory is superior to the MTS criterion since the ERR function considers the energy release rate associated with both opening and shearing deformations at the kink. Figure 4 reveals that the classical form of the MERR criterion (i.e. when G IIc ( ψ ) = G Ic ( ψ ) ) cannot yield acceptable predictions in anisotropic rocks if the fracture energy in tensile-based failure is much smaller than the fracture energy in shear-based failure. Therefore, this classical form may only be accurate for materials that have comparable shear and tensile fracture toughness. Figure 4 also shows that considering the maxima outside the interval of validity for θ 0 results in significant errors in the kink angle predictions. Figure 4 illustrates the predictions of the MTS and the modified MERR criteria when the effect of T-stress is neglected. This figure clearly demonstrates the significant influence of T-stress on the robustness of the MTS and MERR predictions. In this regard, a higher inaccuracy in prediction is foreseen when higher levels of T-stress are ignored. This clarifies why the deviation of the predictions from the experimental data is more pronounced in Sets III and IV, where higher compressive T-stress exists. Indeed, a significant negative T-stress (compression) can hinder opening by reducing the tangential stress, σ θ , in the MTS formulation. On the other hand, a more compressive T stress can shrink the interval of validity defined in Section 3.2 for the modified MERR theory, thereby causing the crack opening to occur near the notch bisector. Summing up, the reliability of the MTS and modified MERR criteria strongly rely on inclusion of T-stress in the formulation, where the T-stress is highly compressive.

5. Conclusion

The reliability of the MTS and MERR predictions of the fracture growth path in an anisotropic rock has been addressed in this article. Comparing the predictions with the new experimental data, one can conclude that the MTS