13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Integrating Eq 4, one has 0 0 pl a pl pl pl a P J d J da Bb b η γ Δ Δ Δ = Δ − ∫ ∫ (6) In which a0 is the initial crack size. Figure 1 illustrates a typical pl P−Δ curve for a growing crack. This figure includes the deformation paths for an original crack length a0 and also for two arbitrarily fixed crack lengths ai and ai+1. Since the pl J in Eq 6 is valid for any loading path leading to the current values of ai and i pl Δ , its value at point A(or B) can be determined by following path OA (or OB) for the fixed crack length ai to the corresponding i pl Δ (or 1 i pl + Δ ) in the actual P−Δ curve. Because da=0 on this loading path, from Eq 6, one has 0 i pl i A pl pl i J Pd Bb η Δ Δ = Δ ∫ (7) and 1, pl i B A i i pl pl i J J A Bb ηΔ + Δ = + (8) where 1, pl i i A+ Δ represents the area under the pl P−Δ curve between i pl Δ and 1 i pl + Δ with an error of the area of triangle ABC Δ . Integration of Eq 6 along BC obtains an approximate result ( ) 1 1 i C B pl pl i i i J J a a b γ Δ + ⎛ ⎞ = − − ⎜ ⎟ ⎝ ⎠ (9) From Eqs 7 to 9, one obtains ( ) ( 1) 1, ( ) 1 1 pl i i i i i pl i pl i i i i J J A a a Bb b η γ + Δ + Δ Δ + ⎛ ⎞⎛ ⎞ = + − − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (10) This incremental expression is the LLD-based (Load Line Displacement) J estimation equation that was adopted in ASTM E1820-11and all its predecessors, where the specimen thickness B is replaced by the net thickness BN for specimens with side grooves.
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