PSI - Issue 54

Francisco Q. de Melo et al. / Procedia Structural Integrity 54 (2024) 585–592 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

591

7

1 × 1

∫ 1 ℎ ∫ 1 × 1 0 ℎ 0

11 = 1 ∫

(13) Total 11 expression is (note that the relative displacement refers to mutual values between both crack face, hence previous 11 must be doubled): 11 =2× { 3 [ 3 2 2(ℎ −− 2 ℎ ) 2 + (ℎ −ℎ )] ⏟ . + ⏟ (ℎ −ℎ ) . − ℎ ⏟ ′ ′ } (14) We note that value of 11 is double of the expression between keys because it represents the mutual or relative displacement between the two mating surfaces of the part-through crack of the plate.

a)

b)

c)

Figure 4: a) Bending moment diagram due to crack opening tensile forces: modified edge cracked plate as a variable section beam where SDZ where removed, b) Diagram for uniform tensile force and c) Pure bending diagrams of the modified edge crack plate with SDZ removed. In order to evaluate of coupled displacement/rotation, these terms are the off-diagonal factors of the flexibility matrix: 12 is the rotation at the crack plane due to a tensile load =1 or conversely, is the displacement at the crack plane due to application of a pure bending moment =1 . 12 = 1 ∫ ( ) × ( 1 =1) ( ) = 0 1 ∫ (ℎ − ( 6 − )) 3 ( − ) 0 (15) Doubling it and making variable substitution for integral evaluation (by parts), it gives: 12 = 2 [ 3 (ℎ− ) 2 +∫ 3 (ℎ− ) 2 0 ] = 2 [ 3 (ℎ− ) 2 + ℎ(ℎ− ) ] ; C 21 =C 12 (16) On other hand, for evaluation of mutual rotation of crack faces due to pure bending moment, the following expression may be derived (without removing the “no - crack” term): 22 = 2 ∫ 12 [ℎ − ( − )] 3 0 = 12 [ 2ℎ− ℎ 2 (ℎ− ) 2 ], (17)