PSI - Issue 13
Katharina Dibblee et al. / Procedia Structural Integrity 13 (2018) 322–327 Katharina Dibblee et al./ Structural Integrity Procedia 00 (2018) 000 – 000
324
3
Fig. 2. Crack propagation in a 3-dimensional fracture mechanical graded structure: (a) application of the new 3D-criterion, (b) determination of the relevant stress function
In order to calculate the required contact of the two functions, the stress at the crack front is first converted to a cyclic stress function and afterwards compared with the local material function. For this purpose, the maximal principle stress σ 1 ' is converted into a cyclic stress function
2 1
2 3
Δ
cos Δ K
sin Δ K
' 1
2
σ
r
2
cos
I
II
2
2
2
2 3
(1).
cos Δ K
sin Δ K
Δ 4
2
2 III
K
I
II
2
The relevant stress function which shows the first contact with the material function results from the consideration of the two stress functions f 1 ( K I th,3D ) and f 2 ( K I th,3D ) from Fig. 2 (b). These two functions are determined as a function of the local threshold K I,th for the material ranges M1 and M2 as well as the kinking angle φ 0 and the grading angle φ M .
B A A B A B
0
2 1
2 1 3
0
0
Δ
Δ
th,3D I
2 0
K
K
cos
cos
sin
I,th 0
2
2
1
2
2
B A A
2 1 3
(2)
0
2 0
cos
sin
4
2
1
B A A
2 1
2 1 3
M
M
Δ
Δ
th,3D I
2 M
K
K
cos
cos
sin
M
I,th M
2
2
1
2
2
B A A
B A B
2 1 3
M
(3)
2 M
cos
sin
4
2
1
A and B represent the mixed-mode ratios:
II Δ Δ Δ Δ
III Δ Δ Δ Δ
K K K K
K K K K
A
B
and
(4).
I
II
III
I
II
III
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