13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) ∫ ∫ ∫ ∫ − − ′ ′ ′ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ − ⎟ ′ ′ ′ ⎠ ⎞ ⎜ ⎝ ⎛ ′ − = R I R rdrd r K R r rdrd R r r 0 0 1 2 sin 2 1 cos 2 1 1 1 ~ , π π π π θ θ θ π θ θ σ (4) where I K is the stress intensity factor, r and θ are the polar coordinates with the origin at the crack tip. Under mode I loading, the maximum non-local principal stress is located at a point on the crack axis near the crack tip , 0 0 = =θ r r due to the symmetry. We assume that 0r is small such that the stress at its vicinity is still governed by the crack-tip asymptotic field. On the one hand, according to the damage criterion (3), the element at ( ) , 0 0 = =θ r r is broken when c σ σ ≥ 1 ~ . On the other hand, from the Griffith-Irwin criterion of fracture [20-21], the crack grows when Ic IK K≥ , where Ic K is the critical stress intensity factor. This condition permits us to determine the non-local action radius R by resolving numerically the following equation: with ( ) ( ) θ θ θ θ θ ′ + ′ ′ ′ = ′ ′ + ′ = + ′ cos sin tan sin cos 0 2 2 0 r r r r r r r Thus, the non-local damage criterion (3) is exactly equivalent to Griffith-Irwin criterion when the non-local action radius R is determined by (5). Consequently, we can confirm that in the cases of uniform tensile loads and mode-I cracks, the fracture can exactly be predicted by using the criterion (3). From this point of view, the proposed non-local criterion can be used to predict the crack initiation as well as the crack propagation. In practice, we just need to find the point where the non-local principal stress is maximal: this point is broken when the non-local stress attaints the material strength. The proposed non-local fracture criterion was implemented into a finite element code. As the criterion (3) is an instantaneous damage model, an element is linearly elastic before its complete failure. Therefore, the crack propagation prediction is very similar to that adopted in the linear elastic fracture mechanics: A linear elastic calculation is first carried out for the cracked structure; then the crack propagation is determined according to a suitable criterion. This procedure is then repeated after each small crack progression in the structure. In the present work, the crack propagation is represented by successive eliminations of groups of damaged elements. 3. Thermal shock problem The proposed non-local criterion was used to evaluate the cracking process in ceramic specimens under thermal shock. 0 2 sin 2 1 cos 2 1 3 max ( ) 0 2 0 ⎟ ′ ′ ′ = ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ − = − ∫ ∫ − R Ic r c rdrd r K R r R f R π π θ θ θ π π σ (5)
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