PSI - Issue 13

Ramdane Boukellif et al. / Procedia Structural Integrity 13 (2018) 85–90

87

3

Boukellif et al. / Structural Integrity Procedia 00 (2018) 000–000

( ˜ x (c) ; ˜ y (c) ) = (18; − 20) mm, as shown in Fig. 1 (c) . The ”unknown” parameters from the inverse problem solution based on a genetic algorithm [Holland (1992)] are given in Tables 1, 2, 3. The parameters have successfully been determined by the solution of the inverse problem. The SIFs K I and K II have been calculated subsequently based on the identified parameters.

∞ yy

∞ yy

∞ xy

∞ xy

σ

¯ σ yy

¯ σ xy

σ

σ

σ

¯ σ xx

( ¯ x (b) ; ¯ y (b) )

( ¯ x (c) ; ¯ x (c) )

( + )

( ¯ x (a) ; ¯ y (a) )

2 a

α 2

α

( x a ; y a )

2 a 2

measuring point P m

( x 2 ; y 2 )

y

α

y

x

x

( − )

a

2 a 1

α 1

w

( x 1 ; y 1 )

( ˜ x (a) ; ˜ y (a) )

( ˜ x (c) ; ˜ x (c) )

( ˜ x (b) ; ˜ y (b) )

y

x

(b)

(a)

(c)

Fig. 1: (a) : Finite plate with two inclined cracks and P m ( m = 1 , ..., 12) measuring points; (b) : Semi-infinite plate with inclined interior crack and P m ( m = 1 , ..., 4) measuring points; (c) : Semi-infinite plane with inclined edge crack and P m ( m = 1 , ..., 4) measuring points.

Table 2: Results of the crack detection and parameter identification, see Fig. 1 (b) ; K I , K I I [MPa √ mm].

Table 3: Results of the param eter identification, see Fig. 1 (c) ; K I , K I I [MPa √ mm].

Table 1: Results of the crack detection and pa rameter identification, see Fig. 1 (a) .

parameters

given

identified

parameters

given

identified

parameters given identified

¯ σ xx [MPa] ¯ σ yy [MPa] ¯ σ xy [MPa]

25 50 10

24 , 99 49 , 99

σ ∞ σ ∞

σ ∞ σ ∞

yy [MPa] xy [MPa]

50 30 12

50 , 00 29 , 99 11 , 99

yy [MPa]

50

50 , 00 29 , 99

xy [MPa] 30

10 , 0

a 1 [mm] a 2 [mm]

3 2

3 , 0 2 , 0

x a [mm] y a [mm] a [mm]

a [mm]

6

5 , 99

α [ ◦ ]

4 6

3 , 99 6 , 00

40

39 , 99

◦ ] ◦ ]

α 1 [ α 2 [

30

30 , 0

− 21 , 95

− 21 , 86

K I

α [ ◦ ] K I ( + ) K I ( − ) K II ( + ) K II ( − )

150

150 , 0

40

39 , 99

156 , 71 156 , 60

K II

x 1 [mm] x 2 [mm] y 1 [mm] y 2 [mm]

8

7 , 99 13 , 0

176 , 54 176 , 62 168 , 55 168 , 62 133 , 85 133 , 86 114 , 40 114 , 40

13

6

6 , 0

13

12 , 99

3.2. Sensitivity analysis for a finite plate

3.2.1. Crack detection and parameter identification depending on number of strain gauges To study the influence of the number of strain gauges on the accuracy of parameter identification for an inclined center crack in a finite plate (20 mm x 20 mm), the configurations depicted in Fig. 2 have been investigated with ( ¯ x ; ¯ y ) = (1; 19) mm, ( ˜ x ; ˜ y ) = (19; 1) mm. Four, eight and twelve virtual strain gauges are used to solve the inverse problem, see Fig. 2(a), (b), (c). Tables 4, 5, 6 show the error between the given problem and obtained results by using four gauges, eight gauges and twelve gauges for di ff erent crack lengths. Tab. 7 shows the error between the given problem and obtained results for di ff erent inclination angles of a center crack a = 6 mm in a plate with twelve strain gauges, see Fig. 2 (c) . From these results, it appears that four strain gauges may be su ffi cient for the identification of crack parameters only for the smallest crack. Eight strain gauges are enough to identify the small and medium crack lengths. With twelve strain gauges, all parameters could be identified very well. Also for di ff erent inclinations, the parameters were successfully determined by using twelve strain gauges, see, Tab. 7.