III. Interaction between microcrack and a semi-infinite crack
The interaction between microcrack and a semi-infinite crack, generates a high concentration of stress, straints and displacements at the crack tip [30]. The semi-infinite crack is determined by their length (l ). On the other hand, the microcrack is defined by Burger’s vector (b = bx +iby ). During the propagation of a semi-infinite crack towards the neighboring microcrack, a damage zone surrounding the initial crack occurs with a high concentration of stresses. It is considered to be a highly disturbed area and also called the fracture process zone [31]. The extension of semi-infinite crack is envisaged in a small zone near to the initial crack in which, a strain energy is released [32]. The elastic behavior of cracking during the presence or absence of micro-crack has been studied by several researchers from various disciplines such as metallurgy, mechanics, physics [33-35]. In our case, the interaction is ensured by a semi-infinite craeck with a neighboring microcrack varies around the crack tip with angle and around itself with angle α (see Figure 1). The loading is applied according to the first mode of fracture which opens the crack perpendicular to their lips (Mode I). The mathematical problem of the interaction (crack-dislocation) is formulated in terms of complex potentials of plane stress [36]. The total plane constraints generated by this interaction, are given as a function of global complex potentials;
\(\sigma_{11}+\sigma_{22}=2\left[{\phi^{{}^{\prime}}}_{O}\left(z\right)+\overset{}{{\phi^{{}^{\prime}}}_{O}\left(z\right)}\right]\)(6)
\(\sigma_{22}-\sigma_{11}+2i\sigma_{12}=2\left[\overset{}{z}{\phi^{{}^{\prime\prime}}}_{O}\left(z\right)+{\psi^{{}^{\prime}}}_{O}\left(z\right)\right]\)(7)
with
\(\phi_{O}\left(z\right)\mathbf{=}\phi_{\text{SIC}}\left(z\right)+\phi_{d}\left(z\right)\)(8)
\(\psi_{O}\left(z\right)=\psi_{\text{SIC}}\left(z\right)+\psi_{d}\left(z\right)\)(9)
where\(\ \phi_{\text{SIC}}\left(z\right)\),\(\psi_{\text{SIC}}\left(z\right)\) ; complex potential functions of semi-infinite crack. \(\phi_{d}\left(z\right)\),\(\psi_{d}\left(z\right)\ \);
complex potential functions of microcrack,\(\phi_{\text{Int}}\left(z\right)\),\(\psi_{\text{Int}}\left(z\right)\) ; presentents the complex
potential functions of the interaction between semi-infinite crack and microcrack.