13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- weighted average over a representative volume V [11]: ( ) ( ) ( ) ∫ ∫ − − = V V d d x y y y x y y x ( ) 1 ~ 1 1 σ α α σ (4) In Equation (4), α(x- y) is a space weighting function which describes the mutual non-local interactions and depends only on the distance between the source point x and the receiver point y. By simplicity, we write: ( ) ⎪⎩ ⎪ ⎨ ⎧ − ≤ > = r R R r r R r 1 0 α (5) where x y = − r ; R is the radius of non-local action which defines the size of interaction zone for failure processes. By means of the non-local principal stress, the damage model (3) can be rewritten as follows: ⎩ ⎨ ⎧ ≥ < = c c D σ σ σ σ 1 1 1 ~ 0 ~ (6) 3: Connection to the Griffith-Irwin criterion In this section, we will prove that the non-local damage model (6) can be connected to the Griffith-Irwin criterion and therefore, can be used to predict crack growth. Consider a mode-I loaded macro-crack. According to the Williams asymptotic solution [16], the near-tip first principal stress is modulated by the stress intensity factor KI as follows: ( ) 2 sin 2 1 cos 2 , 1 θ θ π θ σ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = r K r I (7) Due to the symmetry, the maximal non-local principal stress is located at a point on the crack axis near the crack tip: , 0 0 = =θ r r . Therefore, from (4), (5), and (7): 0 2 sin 2 1 cos 2 1 3 ~ ( ) max 0 2 1 0 0 ′ ′ ′ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ − = ∫ ∫ − R I r r drd r K R r R r π π θ θ θ π π σ (8) On the one hand, if we assume that ( ) 1 0 ~ r σ is the maximal value of ( )θ σ ~ , 1 r near the crack tip, the element at the crack-tip is damaged when ( ) c r σ σ = 1 0 ~ according to the damage criterion (6). On the other hand, from the Griffith criterion of fracture, the crack grows when the energy release rate Gattains its critical value cG . For mode-I cracks, this criterion is equivalent to the Irwin criterion Ic IK K≥ , where Ic K is the critical stress intensity factor. Therefore, the parameters Rand 0r
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