III. 2 Strain energy release
The semi-infinite crack propagates during the rate of release of the stored elastic strain energy is equal to the rate of creation of strain energy of the crack area [39]. The released strain energy is defined as being the released energy during the propagation of the semi-infinite crack at presence of a neighboring dislocation [40]. Based on the energy state of atoms before and after cracking, Griffith has shown that failure is an consuming phenomenon of energy [11]. Using mathematical approaches, Rice [7] used a scalar quantity called J-integral which predicts that the rate of energy release increases proportionally with the propagation of the semi-infinite crack at presence of the disturbance zone. Consequently, the J-integral can be used to determine the rate of energy release related to the translational deformation of the damage zone at crack tip [41]. This energy is applied to materials with brittle behavior and it is considered to be an essential parameter of failure [6]. In linear elasticity, the strain energy is given by the following formula according to constraints and strains of material [42]:
\(W=\frac{1}{2}\left(\sigma_{11}\varepsilon_{11}+\sigma_{22}\varepsilon_{22}+2\sigma_{12}\varepsilon_{12}\right)\)(13)
In a brittle material, the external loads produce very high constraints but the strains are very small, that is to say ;\(\varepsilon_{11}\backsimeq\varepsilon_{12}\backsimeq\varepsilon_{22}\backsimeq\varepsilon\backsimeq 0\). For this reason, the strain energy (Equation (13)) can be written in the following form:
\(W=\frac{\varepsilon}{2}\left(\sigma_{11}+\sigma_{22}+2\sigma_{12}\right)\)(14)
Based on the plane constraints, one can determine the strain energy generated by the interaction between the semi-infinite crack and the neighboring microcrack. The substitution of Equations (10) to (12) into Equation (14), brings us the energy of deformation under the following expression:
\(W=\frac{2.\mu b\varepsilon}{\pi\left(1+k\right)\rho}\left[\cos\left(\alpha+\theta\right)+sin\left(\theta\right)\sin\left(\alpha-2\theta\right)\right]\)(15)