Issue 35

S S. El Kabir et alii, Frattura ed Integrità Strutturale, 35 (2016) 64-73; DOI: 10.3221/IGF-ESIS.35.08 66 Where Γ 1 is arbitrary curvilinear contour oriented by its normal vector n  , F denotes the Helmholtz strain energy density, u i is the displacement component and σ ij is the stress component. M and M θ integral M-integral is an energetic approach that allows studying stress field around crack tip. It is developed by Chen and Shield [8] in order to separate mixed mode fracture [8] and is a combination between real and virtual strain fields which enables computing fracture parameter. Based on a conservative law, M-integral enables to separate fracture mode under creep mixed load [9]:   ,1 ,1 1 2 v u ij j ij i j M u v n d           (2) Where ij  and ,1 v ij  are real and virtual stresses associated with the real and virtual displacements fields u and v . The curvilinear integral (2) is difficult to realize in the finite element method [1, 2, 9]. In order to make easy numerical integration Destuynder [10] proposed M θ integral. This integral is obtained by Ostrogradski transformation and defined at a surface contour containing the crack tip. Figure 2 : Integral domain of M θ .   , , , 1 2 u v ij i k ij k i k j V M v u dV           (3) where θ is a continuous and derivable scalar field. It forms a crown around the crack tip as shown in Fig. 2. Energy release rate In general case the energy release rate, denoted G , traduces the energy release by the material during an unitary crack growth process. According to this definition, the energy release rate, is the energy consumed by the crack growth Δ a [16] and takes the following expression pot W G a    (4) dW pot is the variation of the potential energy. In simple mode solicitation, the energy release rate can be defined by Eq. (1) or Eq. (4). However, in mixed mode loading, using the superposition principle one can write the following relation [11]:   1 2 , 8 8 u v u v I I II II K K K K M u v C C      (5)