13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- growth. The numerical simulations successfully reproduce the cracking patterns in ceramic specimens after quenching. The periodical and hierarchical characteristics of the crack patterns are predicted with satisfactory accuracy. Moreover, the direct numerical simulations faithfully describe whole the cracking process, including the crack initiation, crack growth and crack arrest during quenching tests. By comparing to the previous experimental results, the accuracy and efficiency of the proposed model are examined and discussed. Finally, we give some concluding remarks and directions to follow in future works. 2: Non-local damage model We first outline briefly the non-local fracture model proposed in Li et al. [16-17]. The basic idea of this model consists in replacing the local damage driving force, an effective stress eσ for example, by its weighted average over a representative volume V [18]: where α is a weighting function. In the present work, a cone-shape function is adopted for simplicity: where x y = − r ; R is the radius of non-local action, representing a material characteristic length which defines the size of interaction zone in failure process. We assume reasonably that the failure in ceramic materials under uniform stress fields obeys the maximum principal stress criterion. However, it can not directly be utilized to predict crack growth due to the stress singularity near the crack tips. This shorthand can be overcome by a non-local formulation such like Eq. (1). Thus, the non-local maximum principal stress criterion can be written as follows: where D is the damage, cσ is the ultimate stress of the material, 1 ~σ is the non-local first principal stress. We enforce the validity of criterion (3) in two special cases: First, it should be valid in the case of a uniform tensile load. It is clear that this condition is automatically satisfied since in this case, we have 1 1 ~ σ σ = . Second, it should be valid for the growth of a mode-I crack. To this end, we assume that the near-tip stress field is governed by the Williams asymptotic expansion [19]. Therefore, for a mode-I loaded crack, the non-local first principal stress near the crack tip writes, according to (1) and (2): ( ) ( ) ( ) ∫ ∫ − − = V e V e d d x y y y x y y x ( ) 1 ~ σ α α σ (1) ( ) ⎪⎩ ⎪ ⎨ ⎧ − ≤ > = r R R r r R r 1 0 α (2) ⎩ ⎨ ⎧ ≥ < = c c D σ σ σ σ 1 1 1 ~ 0 ~ (3)
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