Issue 69
S. Cao et alii, Frattura ed Integrità Strutturale, 69 (2024) 1-17; DOI: 10.3221/IGF-ESIS.69.01
(
)
(
)
(
)
( f r f r ( 1
)
(
)
(
)
θ θ
θ θ
θ θ
θ θ
θ θ
θ θ
f
r
f
r
f
r
f
r
f
r
, ,
, ,
, ,
, ,
, ,
, ,
A 1 1
B 1 1
A 1 1
B 1 1
A 1 1
B 1 1
N = 1 2 D u u u
T
T
0
0
1
N
N
(
)
(
)
(
)
)
(
)
(
)
N N 0 0 1 1 A B A B A B T T
f
r
f
r
f
r
f
r
f
r
B 2 2
A 2 2
B 2 2
A 2 2
B 2 2
A 2 2
T
T
0
0
1
1
N
N
(
)
(
)
(
)
(
)
(
)
(
)
θ θ θ
θ θ θ
θ θ θ
θ θ θ
θ θ θ
θ θ θ
f
r
f
r
f
r
B f r
f
r
f
r
, , ,
, , ,
, , ,
, , ,
, , ,
, , ,
D
N N N D D
D D
D D
D D
D D
D
B
A
B
A
A
(15)
T
T
N
0
N N
0
N N
1
N N
1
N N
N N
N 1 2 D v v v
( g r g r ( 0
)
( g r g r ( 0
)
( g r g r ( 1
)
( g r g r ( 1
)
(
)
(
)
g
r
g
r
B 1 1
A 1 1
B 1 1
A 1 1
B 1 1
A 1 1
T
T
N
N
)
)
)
)
(
)
(
)
g
r
g
r
B 2 2
A 2 2
B 2 2
A 2 2
B 2 2
A 2 2
T
T
0
0
1
1
N
N
(
)
(
)
(
)
(
)
(
)
(
)
θ
θ
θ
θ
θ
θ
A g r
g r
A g r
B g r
g
r
f
r
,
,
,
,
,
,
D D
D D
D D
D D
D D
D D
A
B
T
T
0
N N
B
N N
1
N N
1
N N
N N
N N
0
N
N
where the functions in the coefficient matrix are given by
n
/2
r
2 n
cos( 2 2 2 n n n n n n θ − 2 2 2 n n n 2 2 2 θ θ + + sin( sin(
( ) , r θ
n
κ
2) , θ
=
+ + −
−
f
( 1) cos
A
/2 G
2
n
n
r
2 n
( ) , θ
n
κ − − + −
2) , θ
=
−
B f r
( 1) sin
G
2
n
(16)
n
/2
r
2 n
( ) , θ
n
κ
2) , θ
=
− − −
−
A g r
( 1) sin
G
2
n
n
/2
cos( 2 2 n n
r
2 n
2 n
( ) , θ
n
+
−
θ
2) . θ
κ
=
B g r
( 1) cos
− + −
G
2
n
Eqn. (16) can be written as:
N 2 ,1 D = U C
A
(17)
N N 2 ,2 2 2 +2,1 D T T + N
where vector A contains the unknown coefficients of the Williams expansion. If 2 D N >2 T N +2, then matrix C is rectangular, and the system is overdetermined. Utilizing the generalized inverse of C , vector A can be obtained via
1 − = A C C C U ( ) T T
(18)
In fact, Eqn. (18) yields the solution of Eqn. (17) with the minimal least-squares error.
C YLINDRICAL SHELLS ylinder shell structures are used in this chapter to execute and validate the method for the detection of SIF on curved surface shells. For the experiments, the tensile test of polymethyl methacrylate (PMMA) cylindrical shell specimens (shown in Fig. 3(a)) was carried out; the mechanical properties and dimensions are shown in Tab. 1. Through cracks of specimens were cut by 0.25 mm diameter diamond wire saws, and the length of the crack is denoted by its central angle 2 α (2 α =30 º , 60 º , 90 º , 120 º , 150 º , and 180 º , respectively). Fig. 3 (b) shows that the specimen is under displacement-controlled tension; the loading rate is kept at 0.25 mm/min using a Zwick/Roell Z-150 testing machine to prevent the interference of dynamic actions (i.e., a quasi-static load). Fig. 3 (c) shows the clamped support of the cylinder specimens. The upper and lower parts of the specimen are fixed by specially designed fixtures, which are fastened with two sets of hose clamps. Sand the ends of the specimen with sandpaper for firmer clamps. The length of the clamps are 15mm. The 3D deformation data on the surface of specimens are obtained by a 3D-DIC system produced by Correlated Solutions, Inc., the capture frequency of photos was 1 Hz. To reduce the environmental effect, the DIC-3D system calculated the average of 5 sets of photos at a time. The vertical distance between two cameras and the specimen is 0.6 m, and the angle C
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