13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Note that the boundary conditions implemented in the finite element model have not integrated the experimental flaws, in terms of rigid body motion and crack tip orientation. Let's recall that these parameters have already been integrated into the optimized displacement fields obtained from the adjustment procedure described above. According to the finite element approach, we therefore place ourselves in an "ideal" test configuration. In mixed-mode configurations, SIF calculations are often performed by implementing Mq method (Figure 4). The Mq-integral is an energy parameter established in order to analyze crack growth in a mixed-mode fracture by isolating various fracture modes, such as opening and shear, through a pseudo-potential that combines real displacements and kinematically admissible auxiliary displacements. ( ) ( ) ( ) ( ) ( ) , , , 1 2 V ij i k ij k i k j real aux aux real M dV u u θ θ σ σ = ⋅ ⋅ − ⋅ ⋅ ⋅ ∫ (5) The relationships between the stress tensor and the displacement vectors require introducing orthotropic elastic properties. By imposing an external force loading however, we can assume that SIF are not really influenced by elastic properties. Given these conditions, we have opted for an arbitrary elastic property rated ‘~’. V 1x 2x 1 q x = r r 0 q = r r Figure 4. Integral domain 2.3. New formalism of energy release rate The coupling between the kinematic and stress approaches allows calculating an energy release rate by the surrounding elastic proprieties. This coupling also serves to identify the actual reduced elastic compliance leading to the Young's modulus determination. Now, by replacing virtual displacements by real displacements, it has been shown that the Mq integral is mistaken for the energy release rate G [15]. More precisely, thanks to the superposition principle, in letting ( ) real Ks a and ( ) aux Ks a be the real and virtual Stress Intensity Factor, respectively, we can adopt the following expression: ( ) × = å r r , 8 real aux K K M u v C s s a a a a q (6) where Ca is the reduced elastic compliance that allows defining local behavior in terms of Stress
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