II. 1 Brittle failure theory
For a element of elastic material having a semi-infinite crack, the propagation of this one modifies the surface of their lips. Griffith [11] approached the energy problem and thus proposed a theory of failure based on energy consumption during the crack propagation process. The total energy conservation of the cracked system is given as follows [19]:
\(dW_{t}=dW_{e}+dW_{\text{ex}}+dW_{s}+dW_{k}=0\) (2)
where, \(dW_{e}\) ; elastic energy, \(dW_{\text{ex}}\) ; potential energy due to external forces, \(dW_{s}\) ; separation energy,\(dW_{k}\) ; kinetic energy.
In case of kinetic energy greater than zero (\(dW_{k}>0)\), the semi-infinite crack will unstably propagates. The energy rate is defined as being the energy released during the propagation of the semi-infinite crack [20]. By definition, it is given as follows:
\(G=-\frac{\partial}{\partial S}\left({W_{e}+W}_{\text{ex}}\right)\)(3)
According to Griffith [11], the propagation of the semi-infinite crack is ensured when the energy rate is greater than twice the characteristic surface energy of the material (G > 2 ). Irwin [21] used the term surface energy or energy absorbed in the fracture process to apply it to plastic materials. It also showed that at the crack tip, there is a relation between the strain energy rate and the stress intensity factor.