13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The small perturbation parameter ε is defined as ε=ap/WS<<1, where WS is the characteristic size of the specimen (e.g. specimen height). A second scale to the problem can be introduced, represented by the scaled-up coordinates (y1 ε, y 2 ε)=(x 1/ε, x2/ε) which provide a zoomed-in view into the region surrounding the crack, so-called inner domain Ωin (see Figure 3). The energy release rate is defined by: ( ) ( ) 0 1 lim . ε ε δε δε δΠ δΠ δε p b S G a W + → − = (2) Assuming that the loading is constant during crack extension, the change of the potential energy δΠε between the unperturbed state U0 (without the crack extension) and perturbed state εU (with the small finite crack extension) can be obtained from the asymptotic expansion with respect to a small parameter ε as: ( ) ( ) ( ) ( ) 2 1 2 0 2 1 ...., 0 for 0 and 1 Π Π=Π−Π=Π +Π + → → < Π ε ε ε ε ε ε δ δ δ δ ε δ δ , (3) where ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 1 2 2 2 2 1 2 2 2 2 1 1 ( ) 1 2 1 ( ) 2 ( ) 2 2 ( ) . 2 2 2 δ δ δ δ δ δ δ δ ε δ φ ε ε φ φ φ ε + + ′ ′ Π = + ⋅ + + S S S p b p p b p p b p p b p W W W H K HH K K H K (4) Where H1 and H2 are generalized stress intensity factors (GSIF) and δ1, δ2 are the corresponding singularity exponents (δ1<δ2) in the stress asymptotic expansion (see [20,21]). The coefficients K1d(p) and K2d(p) are computed in the inner domain Ωin, which is unbounded for ε→0 but in the model employed in the finite element calculation, Ωin is approximated by a circular region with radius R much larger than the crack extension length ad(p). On the circle boundary, the condition of the type ( ) 1 1 in δ ∂Ω =ρ U u θ is prescribed. ( ) , 1,2 id p K i = are calculated using the path independent integral: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , , , 1,2 i i h h s h kl l ik kl l k ip b i K n u n ds i δ δ Γ =σ ρθ ρ θ−σρ θ ρ θ = ∫ u V V (5) Similarly, the coefficients K´1d(p), K´2d(p) are calculated in the inner domain whose remote boundary ∂Ωin is subjected to the boundary condition ( ) 2 2 in δ ∂Ω =ρ U u θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 , , , 1,2 i i h h s h kl l ik kl l k ip b i K n u n ds i δ δ Γ ′ =σ ρθ ρ θ−σρ θ ρ θ = ∫ u V V (6) where h kl σ , , 1,2 h i i = V denotes FE approximation to the functions kl σ , i V. The second term of the change of the potential energy δΠε (2) depends on crack extension geometry. Two specific crack extension patterns are considered – crack bifurcation and crack deflection. For the case of the crack bifurcation δΠε (2) is given by:
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