III.1 Constraints field
The constraints field generated during the interaction between the
semi-infinite crack and neiboring microcrack, represents the
superposition of two constraints fields produced by the semi-infinite
crack and the microcrack [37]. The constraints found at
semi-infinite crack tip, are characterized by a factor called stress
intensity factor (SIF). The orientation of microcrack around
semi-infinite crack, can generate the stress and consequently release
the energies in each zone of the material [38]. Using Equations (6)
to (9), H. Hamli Benzahar [4] determined the constraints field of
interaction between microcrack and a semi-infinite crack in the
following form:
\(\sigma_{11}=\frac{2\mu b}{\pi\left(1+k\right)\rho}\left[4sin\left(\alpha-\theta\right)-sin\left(\alpha+\theta\right)-sin\left(\theta\right)\cos\left(\alpha-2\theta\right)\right]\)(10)
\(\sigma_{22}=\frac{2\mu b}{\pi\left(1+k\right)\rho}\left[\sin\left(\alpha+\theta\right)+sin\left(\theta\right)\cos\left(\alpha-2\theta\right)\right]\)(11)
\(\sigma_{12}=\frac{2\mu b}{\pi\left(1+k\right)\rho}\left[\cos\left(\alpha+\theta\right)-2sin\left(\alpha-\theta\right)+sin\left(\theta\right)\sin\left(\alpha-2\theta\right)\right]\)(12)